Riccati-Type Pseudo-Potential Approach to Quasi-Integrability of Deformed Soliton Theories

Abstract

This review paper explores the Riccati-type pseudo-potential formulation applied to the quasi-integrable sine-Gordon, KdV, and NLS models. The proposed framework provides a unified methodology for analyzing quasi-integrability properties across various integrable systems, including deformations of the sine-Gordon, Bullough–Dodd, Toda, KdV, pKdV, NLS, and SUSY sine-Gordon models. Key findings include the emergence of infinite towers of anomalous conservation laws within the Riccati-type approach and the identification of exact non-local conservation laws in the linear formulations of deformed models. As modified integrable models play a crucial role in diverse fields of nonlinear physics—such as Bose–Einstein condensation, superconductivity, gravity models, optics, and soliton turbulence—these results may have far-reaching applications.

1. Introduction

Certain nonlinear field theory models that have significant physical applications and describe solitary waves are not inherently integrable. Recently, the concept of quasi-integrability has been introduced for specific deformations of integrable models. Within this framework, various properties of deformed soliton models—such as the sine-Gordon (SG), nonlinear Schrödinger (NLS), Toda, Korteweg-de Vries (KdV), Bullough–Dodd, and SUSY-SG models—have been explored using the anomalous zero-curvature formulation [1,2,3,4,5,6,7,8,9,10,11,12], as well as the deformed Riccati-type pseudo-potential approach [13,14,15,16,17]. For previous findings on certain nonlinear field theories involving solitary waves and their collision dynamics, see, for example, ref. [18] and the references therein.

Since the early development of integrable models, the existence of an infinite number of conservation laws has been one of the defining characteristics of integrability [19,20,21,22,23]. The discovery of anomalous conservation laws even in the N-soliton sector of integrable models represents a novel aspect of these systems [13,24]. The anomalous charges in deformations of the SG, NLS, and KdV models have also been studied using the Riccati-type pseudo-potential approach [13,14,15,17], reproducing the results obtained via the zero-curvature method [1,12]. Notably, both the anomalous zero-curvature formulation and the Riccati-type pseudo-potential approach produce anomalous charges that have the same form as those in standard integrable models. Furthermore, the latter approach also generates additional types of anomalous charges.

Standard integrable models such as the SG, KdV, and NLS equations can be derived as special cases of the AKNS system. Consequently, an appropriate deformation of the conventional pseudo-potential approach within AKNS integrable field theory enables the definition of related quasi-integrable models. In [25,26], both Lax equations and Bäcklund transformations for well-known nonlinear evolution equations were generated using the concept of pseudo-potentials and the properties of the Riccati equation. These methods have been applied across various integrable systems, facilitating the Lax pair formulation, the construction of conservation laws, and the derivation of Bäcklund transformations [25,26,27,28].

In this review, we present a unified framework for analyzing quasi-integrable sine-Gordon (SG), KdV, and NLS models by extending the Riccati pseudo-potential formalism of integrable systems to their quasi-integrable deformations. The Riccati-type pseudo-potential method has been applied to these quasi-integrable models within the deformed AKNS framework [13,14,15,17]. The newly identified properties have been explored in deformations of both the relativistic SG model with topological solitons and the non-relativistic KdV model with non-topological, unidirectional solitons, both conventionally defined for real scalar fields. Additionally, these properties have been examined in NLS-type models, which involve a complex field with envelope solitons.

We present a critical review of the Riccati-type pseudo-potential framework as developed in [13,14,17,29], refining several of the original arguments and providing more precise formulations of key results. We show that the formalism can be systematically understood as specific deformations of the AKNS hierarchy, achieved through an appropriate choice of the coefficients in the Riccati-type system and the introduction of a suitable set of auxiliary fields, as well as a quantity that encodes the deformation of the interaction potential of the model. Furthermore, by revisiting the problem of deformed KdV, we derive a more streamlined and conceptually transparent proof of a general deformation of the non-linear and dispersive terms in the Riccati-type approach. This approach encompasses and extends previous constructions formulated within the anomalous zero-curvature and Hamiltonian frameworks [12,30].

Some of these quasi-integrability properties were also explored in deformed SG, NLS, KdV, and potential KdV (pKdV) models in [12,14,15,16,29,31] by directly constructing novel quasi-conservation laws from the corresponding equations of motion. These studies demonstrated that, despite the loss of full integrability due to deformation, the models retain an infinite hierarchy of anomalous conservation laws. Furthermore, the findings highlighted the role of non-local conserved charges and the persistence of soliton-like structures, reinforcing the broader applicability of quasi-integrability in deformed integrable models.

Within the pseudo-potential framework, a linear system is proposed whose compatibility condition yields the modified AKNS (MAKNS) equations of motion. Quasi-integrable models explored in the literature [1,2,4,5,8,9,10,12,13,14,15,29] exhibit key structures, including infinite sets of non-local conserved charges and certain linear formulations. Furthermore, in the Riccati-type pseudo-potential approach, deformed SG, NLS, and KdV models have been shown to arise as compatibility conditions of specific linear systems and to possess infinite towers of exact non-local conservation laws.

This paper is structured as follows. The next section introduces a deformation of the Riccati system within the AKNS formalism. Section 3 explores the deformation of the SG model, its dual formulation, and the linear system approach leading to non-local charges. Section 4 discusses the $sl\left(2\right)$ deformed AKNS system and its reduction to the modified NLS model. Section 5 examines the modified KdV model and reviews two types of deformations studied in the literature. Finally, Section 6 presents conclusions and discussions.

2. Riccati-Type Pseudo-Potentials and Modified Integrable Models

A general scheme to define a quasi-integrable nonlinear partial differential equation in the pseudo-potential approach follows the deformation of the system of Riccati equations associated with the relevant integrable system. In fact, the AKNS approach to defining an integrable system considers the integrability condition for the system of Riccati equations

$$\begin{array}{ccc}\hfill {\partial }_{\xi }u& =& A_0+A_{1}u+A_2{u}^2,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\partial }_{\eta }u& =& B_0+B_{1}u+B_2{u}^2,\hfill \end{array}$$

(2)

where the complex-valued function $u(\xi ,\eta ,ł)$ is the so-called Riccati pseudo-potential and depends on a spectral parameter $ł$, which is assumed to be independent of the $\xi$ and $\eta$ variables, $\left(\xi ,\eta \right)\in \mathrm{I}{\mathrm{R}}^2$. The partial derivatives with respect to $\xi$ and $\eta$ are denoted by ${\partial }_{\xi }u$ and ${\partial }_{\eta }u$, respectively. The coefficients $A_j(\xi ,\eta ,ł)$ and $B_j(\xi ,\eta ,ł)$, on the other hand, do depend on $ł$ and are functionals of the field of the specific integrable model. The parameter $ł$ is assumed to be an arbitrary complex constant, i.e., $ł\in |\mathrm{C}$. So, we will regard u as a holomorphic function of $ł$ in such a way that locally one can Taylor expand it in powers of $ł$. The Riccati equation exhibits some interesting mathematical properties [32] and has a unique character in the theory of integrable systems (see, e.g., [25,26,33]. Additionally, it has important applications in quantum mechanics (see, e.g., [34,35,36]).

Next, the compatibility condition $({\partial }_{\eta }{\partial }_{\xi }u-{\partial }_{\xi }{\partial }_{\eta }u)=0$ provides the equation of motion of the integrable system [27,28]. In fact, specific finite polynomial expansions of $A_j$ and $B_j$ in powers of $ł$ provide the equations of interest, such as the Korteweg-de Vries (KdV), sine-Gordon (SG), and nonlinear Schrödinger equations (NLS).

To achieve a deformed model, one proceeds by incorporating the relevant deformations in Equation (2). Specifically, the quantities $B_0(\xi ,\eta ,\lambda )$ and $B_1(\xi ,\eta ,\lambda )$ encode the deformation fields, along with certain auxiliary fields $r_j(\xi ,\eta ,\lambda )$. The auxiliary fields satisfy specific first-order linear partial differential equations in the $\xi$ variable

$$\begin{array}{c}\hfill F_k(r_j,u,{\partial }_{\xi }{r}_j,X)=0, k=1,2,..n_1; j=1,2,..n_2,\end{array}$$

(3)

where the $F_k$s for $k=1,2,\dots n_1$ are some functions of $r_j(\xi ,\eta ,\lambda ),u(\xi ,\eta ,\lambda ),{\partial }_{\xi }{r}_j(\xi ,\eta ,\lambda ),$ and $X(\xi ,\eta )$. The number of equations ($n_1$) and auxiliary fields ($n_2$) in (3) will be explicitly formulated for each deformed model. The $\lambda$ independent quantity X encodes the deformation away from the relevant integrable model. This function includes both the interaction potential of the deformed model and the associated deformation parameters. These parameters are not required to be small, as the framework does not rely on a small perturbation theory. Instead, the development allows for deformation parameters of arbitrary magnitude.

In the undeformed integrable case, setting $X=0$ leads to $r_j=0$ and the trivial equations $F_k=0$. As a result, System (1) and (2) simplifies to the standard Riccati equations associated with the corresponding integrable system.

The compatibility condition of the Riccati-type System (1) and (2) can be written as

$$\begin{array}{c}\hfill ({\partial }_{\eta }{\partial }_{\xi }u-{\partial }_{\xi }{\partial }_{\eta }u)=0.\end{array}$$

(4)

Using Systems (1) and (2) successively, one can construct the next two terms entering (4)

$$\begin{array}{ccc}\hfill {\partial }_{\eta }{\partial }_{\xi }u& =& {\partial }_{\eta }{A}_0+A_1{B}_0+(2A_2{B}_0+A_1{B}_1+{\partial }_{\eta }{A}_1)u+(2A_2{B}_1+A_1{B}_2+{\partial }_{\eta }{A}_2)u^2+2A_2{B}_2{u}^3,\hfill \end{array}$$

(5)

and

$$\begin{array}{ccc}\hfill {\partial }_{\xi }{\partial }_{\eta }u& =& {\partial }_{\xi }{B}_0+A_0{B}_1+(2A_0{B}_2+A_1{B}_1+{\partial }_{\xi }{B}_1)u+(2A_1{B}_2+A_2{B}_1)u^2+{\partial }_{\xi }{B}_2+2A_2{B}_2{u}^3.\hfill \end{array}$$

(6)

So, subtracting the above expressions and substituting into (4), one gets the equation

$$\begin{array}{ccc}\hfill A_1{B}_0-A_0{B}_1+{\partial }_{\eta }{A}_0-{\partial }_{\xi }{B}_0+u[2A_2{B}_0-2A_0{B}_2+{\partial }_{\eta }{A}_1-{\partial }_{\xi }{B}_1]& +& \\ \hfill u^2[A_2{B}_1-A_1{B}_2+{\partial }_{\eta }{A}_2-{\partial }_{\xi }{B}_2]& =& 0.\hfill \end{array}$$

(7)

Furthermore, from (7), the relevant modified equation of motion is obtained, provided that the equations for the auxiliary fields (3) are considered.

The introduction of the deformation fields $r_j$ and X constitutes a foundational aspect of our framework for modifying integrable systems. The auxiliary relations given in Equation (3), satisfied by the fields $r_j$, guarantee the consistency of the deformed equations of motion via the compatibility condition expressed in Equation (4). The field X incorporates explicit deformation parameters and encapsulates the modified potential structure of the system, as demonstrated below for the sine-Gordon (SG) and nonlinear Schrödinger (NLS) models [13,17]. In the context of the deformed Korteweg–de Vries (KdV) model, the field X additionally encodes deformations in both the nonlinear and dispersive sectors of the equation [14]. For undeformed, integrable limits of the theory, the deformation field X identically vanishes, which in turn implies that Equation (3) admits only trivial solutions $r_j=0$.

An alternative deformation scheme is based on the anomalous zero-curvature framework, wherein the conventional Lax pair $\{L_1,L_2\}$ is modified through deformations of the terms containing the nonlinear and dispersive terms [1,5,12]. This deformation gives rise to an anomalous zero-curvature condition characterized by the emergence of an extra term that breaks the standard flatness condition. Through an Abelianization procedure, this anomalous structure yields an infinite hierarchy of quasi-conservation laws. Whereas, the Hamiltonian deformation method is naturally embedded within this anomalous Lax formalism, where the deformation acts directly on the Hamiltonian functional rather than on the model’s potential [30].

In the next sections, we provide the constructions of the deformations of the sine-Gordon, NLS, and KdV models in the Riccati-type pseudopotential approach, respectively.

3. Deformation of the Sine-Gordon Model

The compatibility condition of a deformed system of Riccati-type Equations (4)–(7), for specific coefficients $A_j$ and $B_j$ and relevant auxiliary fields r and s, reproduces the equation of motion of a deformed sine-Gordon model (DSG). A DSG model has recently been considered quasi-integrable in the anomalous zero-curvature approach [1]. We will present a dual Riccati-type formulation and construct explicitly the first two exact conservation laws and the first three quasi-conservation laws. Additionally, we provide a pair of linear systems of equations for the DSG model and the associated infinite tower of non-local conservation laws.

Consider Lorentz-invariant field theories in $(1+1)$-dimensions with equations of motion expressed in light-cone coordinates $(\eta ,\xi )$, given by [1] (in the x and t laboratory coordinates: $\eta ={\displaystyle \frac{t+x}{2}}, \xi ={\displaystyle \frac{t-x}{2}}, {\partial }_{\eta }={\partial }_t+{\partial }_x, {\partial }_{\xi }={\partial }_t-{\partial }_x, {\partial }_{\eta }{\partial }_{\xi }={\partial }_t^2-{\partial }_x^2$.)

$$\begin{array}{ccc}\hfill {\partial }_{\xi }{\partial }_{\eta } w+V^{\left(1\right)}\left(w\right)& =& 0.\hfill \end{array}$$

(8)

Here, w is a real scalar field, $V\left(w\right)$ represents the scalar potential, ${\partial }_{\xi }$ and ${\partial }_{\eta }$ denote partial derivatives, and $V^{\left(1\right)}\left(w\right)\equiv {\displaystyle \frac{d}{dw}}V\left(w\right)$. The family of potentials $V\left(w\right)$ corresponds to specific deformations of the conventional sine-Gordon (SG) model, and Equation (8) defines the deformed sine-Gordon (DSG) model equation of motion. The purpose is to explore the properties of this theory by employing modified techniques from integrable field theory, particularly through deformations of Riccati-type equations [25,26].

3.1. Riccati-Type Pseudo-Potential and Conservation Laws

Next, let us consider the Riccati-type System (1) and (2) with the following quantities [13]

$$\begin{array}{ccc}\hfill A_0& =& w_{\xi }, A_1=-2{\lambda }^{-1}, A_2=w_{\xi },\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill B_0& =& -{\displaystyle \frac{1}{2}}\lambda {\displaystyle \frac{d}{dw}}V\left(w\right)+r, B_1=-2\lambda (V\left(w\right)-2)-s, B_2={\displaystyle \frac{1}{2}}\lambda {\displaystyle \frac{d}{dw}}V\left(w\right).\hfill \end{array}$$

(10)

Notice that $\lambda$ is the spectral parameter and r and s are the auxiliary fields. Then, Systems (1) and (2) become [13,24]

$$\begin{array}{ccc}\hfill {\partial }_{\xi }u& =& -2{\lambda }^{-1} u+ {\partial }_{\xi }w+ {\partial }_{\xi }w u^2,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill {\partial }_{\eta }u& =& -2\lambda (V-2) u-{\displaystyle \frac{1}{2}} \lambda V^{\left(1\right)}+{\displaystyle \frac{1}{2}}\lambda V^{\left(1\right)} u^2+r-us.\hfill \end{array}$$

(12)

With the coefficients (9) and (10), Equation (7) can be written as

$$\begin{array}{c}\hfill \Delta +\left[\lambda X-2{\lambda }^{-1}r+(s+u r){\partial }_{\xi }w-{\partial }_{\xi }r\right]+\left[{\partial }_{\xi }s+(r-u s){\partial }_{\xi }w-\lambda uX\right] u+2\Delta u^2=0,\end{array}$$

(13)

with

$$\begin{array}{ccc}\hfill X& \equiv & {\partial }_{\xi }w\left({\displaystyle \frac{V^{\left(2\right)}}{2}}+2V-4\right), V^{\left(2\right)}\equiv {\displaystyle \frac{d^2}{d{w}^2}}V\left(w\right),\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill \Delta & \equiv & {\partial }_{\xi }{\partial }_{\eta } w+V^{\left(1\right)}\left(w\right).\hfill \end{array}$$

(15)

Next, taking into account the following equations for the auxiliary fields $r(\xi ,\eta )$ and $s(\xi ,\eta )$

$$\begin{array}{ccc}\hfill {\partial }_{\xi }r& =& -2{\lambda }^{-1} r+ {\partial }_{\xi }w(u r+s)+\lambda X,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill {\partial }_{\xi }s& =& {\partial }_{\xi }w(u s-r)+\lambda Xu,\hfill \end{array}$$

(17)

in Equation (13), one notices that the terms inside the brackets $\left[\dots \right]$ identically vanish. So, in (13), we are left with a second-order polynomial equation in u of the form $\Delta +2\Delta u^2$, which must vanish identically, i.e., $\Delta =0$. Then, one obtains the equation of motion of the modified SG model (8).

So, one has a set of two deformed Riccati-type equations for the pseudo-potential u (11) and (12) and Systems (16) and (17) for the auxiliary fields r and s.

Notice that, for the usual SG model, one has the potential

$$\begin{array}{c}\hfill V=-2cos\left(2w\right)+2,\end{array}$$

(18)

such that $X=0$, and so the auxiliary Systems (16) and (17) admit the trivial solution $r=s=0$. Taking into account this solution and considering the potential (18) in Systems (11) and (12), one can obtain a pair of Riccati equations for the standard sine-Gordon (SG) model. These equations play a crucial role in analyzing the integrable properties of the SG model, including the derivation of an infinite set of conserved charges and Bäcklund transformations, which relate the field w to another solution $\overline{w}$ [27,28].

Note that only the $\eta –$component ${\partial }_{\eta }u$ of the Riccati equation associated with the ordinary sine-Gordon equation has been modified, deviating from the SG potential (18). This component encapsulates all information regarding the deformation of the model, which is encoded in the potential $V\left(w\right)$ and the auxiliary fields $r(\xi ,\eta )$ and $s(\xi ,\eta )$. In contrast, the $\xi –$component, ${\partial }_{\xi }u$, retains the same form as in the standard Riccati equation of the SG model.

As a verification, one can compute the compatibility condition $[{\partial }_{\eta }{\partial }_{\xi }u-{\partial }_{\xi }{\partial }_{\eta }u]=0$ for the Riccati-type Equations (11) and (12), and taking into account the auxiliary System (16) and (17) one can rederive the equation of motion of the deformed sine-Gordon model (8).

It is important to highlight that, in the standard sine-Gordon (SG) model, Systems (16) and (17) admit the trivial solution $X=0\to r=s=0$. Additionally, the existence of the Lax pair in the conventional SG model is reflected in its equivalent Riccati-type representation, given by Systems (11) and (12) with the well known potential (18) [25,26,27,28].

Next, we examine the relevant conservation laws within the framework of the Riccati-type Systems (11) and (12) and the auxiliary Equations (16) and (17). By substituting the $u^2$ from (11) into (12) and taking into account (17), we obtain the following relationship

$$\begin{array}{c}\hfill {\partial }_{\eta }\left(u {\partial }_{\xi }w\right)+{\partial }_{\xi }\left[\lambda (V-2)-{\displaystyle \frac{1}{2}}\lambda u V^{\left(1\right)}\right]=-{\partial }_{\xi }s.\end{array}$$

(19)

This identity can be regarded as a quasi-conservation law, and in fact, multiple such expressions can be formulated. Here, we construct one in which the non-homogeneous term on the right-hand side explicitly involves the deformation variable s. This choice ensures that, by setting $s=0$, the left-hand side recovers, order by order in $\lambda$, the polynomial conservation laws of the standard sine-Gordon (SG) model. Naturally, it is also possible to write an alternative equation that incorporates both deformation variables s and r.

Equation (19) will be used to reveal an infinite tower of conservation laws for the modified SG model (8). However, its genuine conservation law nature remains to be confirmed. If the function $s(\xi ,\eta )$ on the right-hand side of (19) contained non-local terms like $\int d\xi (\dots )$, the conservation property would break down, leading to anomalies similar to those in anomalous zero-curvature formulations of deformed Lax pairs and quasi-conservation laws [1]. Below, we demonstrate that the right-hand side of (19) can be expressed as $[-{\partial }_{\xi }s]\equiv {\partial }_{\xi }S+{\partial }_{\eta }R$, where S and R are local functions of w and its $\xi$ and $\eta –$derivatives. Thus, a local expression for ${\partial }_{\xi }s$ exists, ensuring that (19) defines a proper local conservation law.

Next, let us consider the expansions

$$\begin{array}{c}\hfill u=\sum _{n=1}^{\infty }{u}_n {\lambda }^n, s=\sum _{n=0}^{\infty }{s}_n {\lambda }^{n+2}, r=\sum _{n=0}^{\infty }{r}_n {\lambda }^{n+2}.\end{array}$$

(20)

Observe that the lowest order terms in $\lambda$ in the above expansions are chosen differently to ensure the consistency of the systems of Equations (11) and (12) and (16) and (17). Specifically, the powers in the spectral parameter $\lambda$ must be compatible with the corresponding equations at each order in ${\lambda }^n$. Note that in (11) the initial terms corresponds to ${\lambda }^0$, providing $u_1={\displaystyle \frac{1}{2}}{\partial }_{\xi }w$ (see the Appendix A in [13]).

The coefficients $u_n$ of the expansion above can be determined order by order in powers of $\lambda$ from the Riccati Equation (11). We refer to the appendices in [13] for the recursion relation for the ${u_n}^{\prime }s$ and the expressions for the first $u_n$. Likewise, using the results for the ${u_n}^{\prime }s$ one can get the relevant expressions for the ${r_n}^{\prime }s$ and ${s_n}^{\prime }s$ from (16) and (17).

So, replacing those expansions into Equation (19), one has that the coefficient of the $n^{\prime }$th order term becomes

$$\begin{array}{ccc}\hfill {\partial }_{\eta }{a}_{\xi }^{\left(n\right)}& +& {\partial }_{\xi }{a}_{\eta }^{\left(n\right)}=-{\partial }_{\xi }{s}_{n-2}, n=1,2,3,....; s_{-1}\equiv 0, {\partial }_{\xi }{s}_0=0,\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill a_{\xi }^{\left(n\right)}& \equiv & u_n{\partial }_{\xi }w, a_{\eta }^{\left(n\right)}\equiv (V-2){\delta }_{1,n}-{\displaystyle \frac{1}{2}}{u}_{n-1}{V}^{\left(1\right)}, u_0\equiv 0.\hfill \end{array}$$

(22)

Let us emphasize that only the coefficients $s_n$ appear on the right-hand side of Equation (21), as the deformation variable $r_n$ does not enter the quasi-conservation Equation (19).

So, from Equation (21), one has that order by order in powers of $\lambda$, the first two exact conservations laws ($n=1,2$) and the first three quasi-conservation laws ($n=3,4,5$) become

$$\begin{array}{ccc}\hfill n& =& 1, {\partial }_{\eta }\left({\displaystyle \frac{1}{2}}{\left({\partial }_{\xi }w\right)}^2\right)+{\partial }_{\xi }\left(V-2\right)=0,\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill n& =& 2, {\displaystyle \frac{1}{4}}{\partial }_{\xi }\left[{\partial }_{\eta }\left({\displaystyle \frac{1}{2}}{\left({\partial }_{\xi }w\right)}^2\right)+{\partial }_{\xi }\left(V-2\right)\right]=0,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill n& =& 3, {\partial }_{\eta }\left[{\displaystyle \frac{1}{8}}{\partial }_{\xi }w\left({\left({\partial }_{\xi }w\right)}^3+{\partial }_{\xi }^{3}w\right)\right]+{\partial }_{\xi }\left({\displaystyle \frac{1}{8}}{\partial }_{\xi }^{2}w{V}^{\left(1\right)}\right)=-{\partial }_{\xi }{s}_1,\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill n& =& 4, {\partial }_{\eta }\left({\displaystyle \frac{5}{16}}{\partial }_{\xi }^{2}w{\left({\partial }_{\xi }w\right)}^3+{\displaystyle \frac{1}{16}}{\partial }_{\xi }w{\partial }_{\xi }^{4}w\right)+{\partial }_{\xi }\left({\displaystyle \frac{1}{16}}{V}^{\left(1\right)}[{\left({\partial }_{\xi }w\right)}^3+{\partial }_{\xi }^{3}w]\right)=-{\partial }_{\xi }{s}_2,\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill n& =& 5, {\partial }_{\eta }\left({\displaystyle \frac{11}{32}}{\left({\partial }_{\xi }^{2}w\right)}^2{\left({\partial }_{\xi }w\right)}^2+{\displaystyle \frac{7}{32}}{\partial }_{\xi }^{3}w{\left({\partial }_{\xi }w\right)}^3+{\displaystyle \frac{1}{16}}{\left({\partial }_{\xi }w\right)}^6+{\displaystyle \frac{1}{32}}{\partial }_{\xi }w{\partial }_{\xi }^{5}w\right)-{\partial }_{\xi }\left[{\displaystyle \frac{1}{2}}{V}^{\left(1\right)}{u}_4\right]=-{\partial }_{\xi }{s}_3,\hfill \end{array}$$

(27)

with

$$\begin{array}{ccc}\hfill -{\partial }_{\xi }{s}_1& =& {\displaystyle \frac{1}{4}}X{\partial }_{\xi }^{2}w-{\displaystyle \frac{1}{4}}{\partial }_{\xi }w {\partial }_{\xi }X,\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill -{\partial }_{\xi }{s}_2& =& {\displaystyle \frac{1}{8}}X{\partial }_{\xi }^{3}w-{\displaystyle \frac{1}{8}}{\partial }_{\xi }w{\partial }_{\xi }^{2}X,\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill -{\partial }_{\xi }{s}_3& =& -{\displaystyle \frac{3}{16}}{\left({\partial }_{\xi }w\right)}^3{\partial }_{\xi }X+{\displaystyle \frac{1}{16}}\left({\partial }_{\xi }^{4}w+{\partial }_{\xi }\left[{\left({\partial }_{\xi }w\right)}^3\right]\right) X-{\displaystyle \frac{1}{16}}{\partial }_{\xi }w{\partial }_{\xi }^{3}X.\hfill \end{array}$$

(30)

Notice that, the right-hand sides in (28)–(30), which also appear in the right-hand sides of (26)–(27), can be rewritten as

$$\begin{array}{ccc}\hfill -{\partial }_{\xi }{s}_1& =& {\partial }_{\eta }{R}_1+{\partial }_{\xi }{S}_1,\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill R_1& \equiv & -{\displaystyle \frac{1}{8}}{\left[{\partial }_{\xi }^{2}w\right]}^2+{\displaystyle \frac{1}{8}}{\left[{\partial }_{\xi }w\right]}^4, S_1\equiv -{\displaystyle \frac{1}{8}}{\left[{\partial }_{\xi }w\right]}^2{V}^{\left(2\right)},\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill -{\partial }_{\xi }{s}_2& =& {\displaystyle \frac{1}{8}}{\partial }_{\xi }[{\partial }_{\xi }^{2}wX-{\partial }_{\xi }w{\partial }_{\xi }X],\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill -{\partial }_{\xi }{s}_3& =& {\partial }_{\eta }{R}_3+{\partial }_{\xi }{S}_3,\hfill \end{array}$$

(34)

where the explicit expressions of $R_3$ and $S_3$ are provided in Equations (E.1)–(E.3) of the Appendix E of [13]. Notice that the right hand sides in (31)–(34) are written as $({\partial }_{\eta }{R}_k+{\partial }_{\xi }{S}_k), (k=1,2,3)$.

Next, let us discuss the equations above.

First order $n=1$. Notice that the right-hand side of (21) vanishes at this order, i.e., by definition, one has $s_{-1}\equiv 0$. In fact, the conservation law (23) provides the first conserved charge

$$\begin{array}{ccc}\hfill q^{\left(1\right)}& =& \int dx \left[{\displaystyle \frac{1}{2}}{\left({\partial }_{\xi }w\right)}^2+(V-2)\right].\hfill \end{array}$$

(35)

Equations (23) and (35), together with their duals and the relevant charge ${\tilde{q}}^{\left(1\right)}=\int dx [{\displaystyle \frac{1}{2}}{\left({\partial }_{\eta }w\right)}^2+(V-2)]$, which will be presented below, give rise to the energy and momentum charges written in laboratory coordinates $(x,t)$, respectively, as

$$\begin{array}{ccc}\hfill E& =& {\displaystyle \frac{1}{2}}\left[q^{\left(1\right)}+{\tilde{q}}^{\left(1\right)}\right],\hfill \end{array}$$

(36)

$$\begin{array}{ccc}& =& \int dx\left[{\displaystyle \frac{1}{2}}{\left({\partial }_{x}w\right)}^2+{\displaystyle \frac{1}{2}}{\left({\partial }_{t}w\right)}^2+(V-2)\right],\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill P& =& {\displaystyle \frac{1}{2}}\left[{\tilde{q}}^{\left(1\right)}-q^{\left(1\right)}\right],\hfill \end{array}$$

(38)

$$\begin{array}{ccc}& =& \int dx\left[{\partial }_{x}w{\partial }_{t}w\right].\hfill \end{array}$$

(39)

Second order $n=2$.

At this order $\mathcal{O}\left({\lambda }^2\right)$, one can write

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{4}}{\partial }_{\xi }\left[{\partial }_{\eta }\left({\displaystyle \frac{1}{2}}{\left({\partial }_{\xi }w\right)}^2\right)+{\partial }_{\xi }\left(V-2\right)\right]=0.\end{array}$$

(40)

As usual, we can define the charge

$$\begin{array}{ccc}\hfill q^{\left(2\right)}& =& \int dx {\displaystyle \frac{1}{4}}\left({\partial }_{\xi }^{2}w{\partial }_{\xi }w+{\partial }_{\xi }w{V}^{\left(1\right)}\right),\hfill \end{array}$$

(41)

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{4}}{\displaystyle \frac{d}{dt}}\left(E-P\right).\hfill \end{array}$$

(42)

So, the Equation (40) does not provide an independent new charge in laboratory coordinates $(x,t)$. So, there is no independent new charge at this order. Notice that the usual SG model also does not possess an independent charge at this order [37]. From this point forward and for the higher order charges, the term encoding the deformation away from the usual SG model, i.e., the right-hand side of (21), will play an important role in the construction of the conservation laws.

Third order $n=3$.

Taking into account (26) and (31) the conservation law turns out to be

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{8}}{\partial }_{\xi }^2\left[{\partial }_{\eta }\left({\displaystyle \frac{1}{2}}{\left({\partial }_{\xi }w\right)}^2\right)+{\partial }_{\xi }\left(V-2\right)\right]=0.\end{array}$$

(43)

Notice that this form of the third-order conservation law holds strictly for deformed SG models, i.e., for models such that $X\ne 0$. In the usual SG model, the Equation (26) with vanishing right hand side—in that case, $X\equiv 0$ implies ${\partial }_{\xi }{s}_1=0$ in (28)—provides the relevant conservation law at this order.

Next, the charge which follows from the above conservation law (43) becomes

$$\begin{array}{c}\hfill q^{\left(3\right)}={\displaystyle \frac{1}{8}}{\displaystyle \frac{d^2}{d{t}^2}}\left(E-P\right).\end{array}$$

(44)

Thus, at this order and within this formulation, unlike the conventional SG model, the deformed SG model (8) does not possess an independent conserved charge.

Nevertheless, it can be shown that the third-order charge and anomaly defined in [1,2,7] can be reformulated in our notation as follows

$$\begin{array}{ccc}\hfill q_a^{\left(3\right)}& =& \int dx[{\displaystyle \frac{1}{8}}{\left({\partial }_{\xi }w\right)}^4+{\displaystyle \frac{1}{8}}{\partial }_{\xi }w{\partial }_{\xi }^{3}w+{\displaystyle \frac{1}{8}}{\partial }_{\xi }^{2}w{V}^{\left(1\right)}+{\displaystyle \frac{1}{4}}X{\partial }_{\xi }w],\hfill \end{array}$$

(45)

and

$$\begin{array}{c}\hfill {\beta }^{\left(3\right)}\equiv {\displaystyle \frac{1}{2}}{\partial }_{\xi }^{2}wX={\partial }_{\xi }{S}_1+{\partial }_{\eta }{R}_1+{\displaystyle \frac{1}{4}}{\partial }_{\xi }\left({\partial }_{\xi }w X\right),\end{array}$$

(46)

with $R_1,S_1$ given in (32). So, one has

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{q}_a^{\left(3\right)}={\int }_{-\infty }^{+\infty }dx{\beta }^{\left(3\right)}.\end{array}$$

(47)

The charge $q_a^{\left(3\right)}$ defined in (45) has been previously computed through numerical simulations of two-soliton collisions for a specific deformation of the SG model [13]. Interestingly, the numerical simulations demonstrate that the charge $q_a^{\left(3\right)}$ is exactly conserved in general two-soliton configurations, with no observable deviation within the precision of the numerical method.

Fourth order $n=4$

Taking into account (27) and (33), one has the conservation law

$$\begin{array}{ccc}\hfill {\partial }_{\eta }\left({\displaystyle \frac{5}{16}}{\partial }_{\xi }^{2}w{\left({\partial }_{\xi }w\right)}^3+{\displaystyle \frac{1}{16}}{\partial }_{\xi }w{\partial }_{\xi }^{4}w\right)+{\partial }_{\xi }\left({\displaystyle \frac{1}{16}}[{\left({\partial }_{\xi }w\right)}^3+{\partial }_{\xi }^{3}w]V^{\left(1\right)}-{\displaystyle \frac{1}{8}}[{\partial }_{\xi }^{2}wX-{\partial }_{\xi }w{\partial }_{\xi }X]\right)& =& 0.\hfill \end{array}$$

(48)

Next, from (48), one can define the charge

$$\begin{array}{ccc}\hfill q^{\left(4\right)}& =& \int dx \left\{{\displaystyle \frac{5}{16}}{\partial }_{\xi }^{2}w{\left({\partial }_{\xi }w\right)}^3+{\displaystyle \frac{1}{16}}{\partial }_{\xi }w{\partial }_{\xi }^{4}w+{\displaystyle \frac{1}{16}}[{\left({\partial }_{\xi }w\right)}^3+{\partial }_{\xi }^{3}w]V^{\left(1\right)}-{\displaystyle \frac{1}{8}}[{\partial }_{\xi }^{2}wX-{\partial }_{\xi }w{\partial }_{\xi }X]\right\},\hfill \end{array}$$

(49)

$$\begin{array}{ccc}& \equiv & 0.\hfill \end{array}$$

(50)

Thus, it has been demonstrated that this charge identically vanishes under appropriate boundary conditions.

Fifth order $n=5$

Similarly, by considering (27) and (34) and performing a lengthy calculation, the fifth-order conservation law becomes

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{32}}{\partial }_{\xi }^4\left\{{\displaystyle \frac{1}{2}}{\partial }_{\eta }{\left({\partial }_{\xi }w\right)}^2+{\partial }_{\xi }(V-2)\right\}=0.\end{array}$$

(51)

From the above conservation law follows the fifth-order conserved charge

$$\begin{array}{ccc}\hfill q^{\left(5\right)}& \equiv & {\displaystyle \frac{1}{32}}{\displaystyle \frac{d^4}{d{t}^4}}\int dx \left[{\displaystyle \frac{1}{2}}{\left({\partial }_{\xi }w\right)}^2+(V-2)\right],\hfill \end{array}$$

(52)

$$\begin{array}{ccc}& \equiv & {\displaystyle \frac{1}{32}}{\displaystyle \frac{d^4}{d{t}^4}}\left(E-P\right).\hfill \end{array}$$

(53)

Thus, the fifth-order charge $q^{\left(5\right)}$ in (52) is not an independent charge of the deformed sine-Gordon model (8), despite arising from a genuine conservation law in the Riccati-type formulation, beyond energy and momentum.

Below, we define a related embedded charge $q_a^{\left(5\right)}$ along with its corresponding anomaly term ${\beta }^{\left(5\right)}$, i.e.,

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{q}_a^{\left(5\right)}={\int }_{-\infty }^{+\infty }dx{\beta }^{\left(5\right)}.\end{array}$$

(54)

The ‘anomaly’ term ${\beta }^{\left(5\right)}$ introduced in [1] has been written, in the notation of [13], as a term of the right-hand side of (30) as

$$\begin{array}{ccc}\hfill -2^4{\partial }_{\xi }{s}_3& =& 2{\beta }^{\left(5\right)}-{\partial }_{\xi }[3{\left({\partial }_{\xi }w\right)}^{3}X+{\partial }_{\xi }^{3}wX-{\partial }_{\xi }^{2}w{\partial }_{\xi }X+{\partial }_{\xi }w{\partial }_{\xi }^{2}X],\hfill \end{array}$$

(55)

$$\begin{array}{ccc}\hfill {\beta }^{\left(5\right)}& \equiv & {\displaystyle \frac{1}{16}}[{\partial }_{\xi }^{4}w+6{\left({\partial }_{\xi }w\right)}^2{\partial }_{\xi }^{2}w] X.\hfill \end{array}$$

(56)

Therefore, the additional terms ${\partial }_{\xi }[. . .]$ appearing in the expression of $(-2^4{\partial }_{\xi }{s}_3)$ and provided in (55) can be incorporated into the left-hand side of the conservation law (27). Since the anomaly ${\beta }^{\left(5\right)}$ can be written in the form ${\partial }_{\eta }[. . .]+{\partial }_{\xi }[. . .]$, one can define the asymptotically conserved charge $q_a^{\left(5\right)}$ as

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{dt}}{q}_a^{\left(5\right)}& =& 2\int dx {\beta }^{\left(5\right)},\hfill \\ & =& {\displaystyle \frac{1}{8}}{\displaystyle \frac{d}{dt}}\int dx \{{\displaystyle \frac{1}{2}}{\left({\partial }_{\xi }w\right)}^6+{\displaystyle \frac{1}{4}}{\left({\partial }_{\xi }^{3}w\right)}^2-{\displaystyle \frac{5}{2}}{\left({\partial }_{\xi }^{2}w\right)}^2{\left({\partial }_{\xi }w\right)}^2-6{\left({\partial }_{\xi }w\right)}^4+6{\left({\partial }_{\xi }^{2}w\right)}^2-{\displaystyle \frac{1}{2}}{\partial }_{\xi }^{4}w{V}^{\left(1\right)}+\hfill \\ \hfill & & 2[2{\partial }_{\xi }w{\partial }_{\xi }^{3}w+{\displaystyle \frac{1}{2}}{\left({\partial }_{\xi }^{2}w\right)}^2+{\displaystyle \frac{3}{2}}{\left({\partial }_{\xi }w\right)}^4]V\}+\hfill \end{array}$$

(57)

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{8}}{\displaystyle \frac{d^2}{d{t}^2}}\int dx \left\{{\displaystyle \frac{1}{2}}{\partial }_{\xi }^{3}w{V}^{\left(1\right)}-2{\partial }_{\xi }w{\partial }_{\xi }^{2}wV\right\}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{d^3}{d{t}^3}}\int dx {\displaystyle \frac{1}{2}}{\left({\partial }_{\xi }w\right)}^2.\hfill \end{array}$$

(58)

Finally, the fifth-order quasi-conservation law from [1] can be expressed as an exact conservation law, provided that the form (34) is utilized. This manipulation results in the exact conservation law (51).

The charge $q_a^{\left(5\right)}$ has been previously computed through numerical simulations of two-soliton collisions for a specific deformation of the SG model [13]. These charges can be considered, in general, as asymptotically conserved ones.

In this way, the right-hand sides $[-{\partial }_{\xi }{s}_n (n=1,2,3,4)]$ of the relevant conservation laws have been written as ${\partial }_{\eta }{R}_n+{\partial }_{\xi }{S}_n$. It has been shown that this property holds in general for each term, $[(-{\partial }_{\xi }{s}_n), n\ge 1$], of the tower of conservation laws. So, the conservation laws (21) in general can be written as

$$\begin{array}{c}\hfill {\partial }_{\eta }[a_{\xi }^{\left(n\right)}-R_{n-2}]+{\partial }_{\xi }[a_{\eta }^{\left(n\right)}-S_{n-2}]=0, n=1,2,3,\dots (S_k=R_k\equiv 0, k=-1,0).\end{array}$$

(59)

Therefore, the construction provides an infinite tower of conservation laws (59).

The notion of quasi-integrability and its associated asymptotically conserved charges, originally introduced in [1] and further developed in [7,8,9,10] through the introduction of subsets of exactly conserved charges, depends on the specific field configurations to which it is applied—such as kink-kink, kink-antikink, and breather configurations in the deformed model. This contrasts with the conventional concept of integrability, where conserved charges are defined for all fields of the model.

3.2. Riccati-Type Pseudo-Potential and Dual Conservation Laws

Since the deformed SG model (8) remains invariant under the transformation $\eta \leftrightarrow \xi$, there naturally exists a dual Riccati-type formulation corresponding to Systems (11)–(17) discussed above. Accordingly, the following system of equations is introduced for the new pseudo-potential $\tilde{u}$ [13]

$$\begin{array}{ccc}\hfill {\partial }_{\eta }\tilde{u}& =& -2\lambda \tilde{u}+ {\partial }_{\eta }w+ {\partial }_{\eta }w {\tilde{u}}^2,\hfill \end{array}$$

(60)

$$\begin{array}{ccc}\hfill {\partial }_{\xi }\tilde{u}& =& -{\displaystyle \frac{2}{\lambda }} (V-2) \tilde{u}-{\displaystyle \frac{1}{2\lambda }} V^{\prime }+{\displaystyle \frac{1}{2\lambda }} V^{\prime } {\tilde{u}}^2+\tilde{r}-\tilde{u} \tilde{s}.\hfill \end{array}$$

(61)

Observe that the transformations $\lambda \to {\lambda }^{-1}$ and $\xi \leftrightarrow \eta$ have been performed into Systems (11) and (12), while also relabeling the pseudo-potential and auxiliary fields. Meanwhile, the field w and the deformed sine-Gordon potential $V\left(w\right)$ remain unchanged.

Next, the following equations for the auxiliary fields $\tilde{r}(\xi ,\eta )$ and $\tilde{s}(\xi ,\eta )$ are considered

$$\begin{array}{ccc}\hfill {\partial }_{\eta }\tilde{r}& =& -2\lambda \tilde{r}+ {\partial }_{\eta }w(\tilde{u} \tilde{r}+\tilde{s})+{\lambda }^{-1}\tilde{X},\hfill \end{array}$$

(62)

$$\begin{array}{ccc}\hfill {\partial }_{\eta }\tilde{s}& =& {\partial }_{\eta }w(\tilde{u} \tilde{s}-\tilde{r})+{\lambda }^{-1}\tilde{X}\tilde{u},\hfill \end{array}$$

(63)

$$\begin{array}{ccc}\hfill \tilde{X}& \equiv & {\partial }_{\eta }w\left({\displaystyle \frac{V^{\left(2\right)}}{2}}+2V-4\right), V^{\left(2\right)}\equiv {\displaystyle \frac{d^2}{d{w}^2}}V\left(w\right).\hfill \end{array}$$

(64)

Thus, one has two deformed Riccati-type equations for the pseudo-potential $\tilde{u}$ (60) and (61), along with System (62) and (63) governing the auxiliary fields $\tilde{r}$ and $\tilde{s}$. Moreover, for the specific potential (18), $\tilde{X}$ vanishes identically, and the linear System (60) and (61) reduces to the ordinary sine-Gordon integrable model, provided that $\tilde{s}=\tilde{r}=0$ in (61).

Similarly to the previous subsection, substituting the expression for ${\tilde{u}}^2$ from (60) into (61) yields the following relationship

$$\begin{array}{c}\hfill {\partial }_{\xi }\left(\tilde{u} {\partial }_{\eta }w\right)+{\partial }_{\eta }\left({\lambda }^{-1} (V-2)-{\displaystyle \frac{1}{2}}{\lambda }^{-1} \tilde{u} V^{\left(1\right)}\right)=-{\partial }_{\eta }\tilde{s}.\end{array}$$

(65)

This equation serves as a tool for revealing an infinite number of new conservation laws associated with the modified SG model (8). Thus, the following expansions are considered:

$$\begin{array}{c}\hfill \tilde{u}=\sum _{n=1}^{\infty }{\tilde{u}}_n {\lambda }^{-n}, \tilde{s}=\sum _{n=0}^{\infty }{\tilde{s}}_n {\lambda }^{-(n+2)}, \tilde{r}=\sum _{n=0}^{\infty }{\tilde{r}}_n {\lambda }^{-(n+2)}.\end{array}$$

(66)

The components ${\tilde{u}}_n$ are determined recursively by substituting the above expression into (60), while the components ${\tilde{s}}_n$ and ${\tilde{r}}_n$ are obtained from Systems (62) and (63). Appendices C and D of [13] provide the explicit expressions for the first few ${\tilde{u}}_n, {\tilde{s}}_n,{\tilde{r}}_n$. Using these components, one can systematically derive the conservation laws order by order in powers of ${\lambda }^{-1}$ by substituting the expansions into (65). Consequently, the conservation law at order $(-n)$ takes the form

$$\begin{array}{ccc}\hfill {\partial }_{\xi }{\tilde{a}}_{\eta }^{\left(n\right)}& +& {\partial }_{\eta }{\tilde{a}}_{\xi }^{\left(n\right)}=-{\partial }_{\eta }{\tilde{s}}_{n-2}, n=1,2,3,....; {\tilde{s}}_{-1}\equiv 0,\hfill \end{array}$$

(67)

$$\begin{array}{ccc}\hfill {\tilde{a}}_{\eta }^{\left(n\right)}& \equiv & {\tilde{u}}_n{\partial }_{\eta }w, {\tilde{a}}_{\xi }^{\left(n\right)}\equiv (V-2){\delta }_{1,n}-{\displaystyle \frac{1}{2}}{\tilde{u}}_{n-1}{V}^{\left(1\right)}, {\tilde{u}}_0\equiv 0.\hfill \end{array}$$

(68)

Next, we provide the first-order conservation law

$$\begin{array}{c}\hfill {\partial }_{\xi }\left({\displaystyle \frac{1}{2}}{\left({\partial }_{\eta }w\right)}^2\right)+{\partial }_{\eta }\left(V-2\right)=0.\end{array}$$

(69)

This equation furnishes the conserved charge

$$\begin{array}{c}\hfill {\tilde{q}}^{\left(1\right)}=\int dx [{\displaystyle \frac{1}{2}}{\left({\partial }_{\eta }w\right)}^2+(V-2)].\end{array}$$

(70)

This charge, combined with its dual, has been used to write the energy and momentum charges as in Equations (36)–(39) in the previous subsection.

The next order term becomes

$$\begin{array}{c}\hfill {\partial }_{\xi }\left({\displaystyle \frac{1}{4}}{\partial }_{\eta }^{2}w{\partial }_{\eta }w\right)+{\partial }_{\eta }\left({\displaystyle \frac{1}{4}}{\partial }_{\eta }w{V}^{\left(1\right)}\right)=0.\end{array}$$

(71)

Notice that the right-hand side vanishes due to ${\partial }_{\eta }{\tilde{s}}_0=0$. As usual, the charge is defined as

$$\begin{array}{ccc}\hfill {\tilde{q}}^{\left(2\right)}& =& \int dx {\displaystyle \frac{1}{4}}\left({\partial }_{\eta }^{2}w{\partial }_{\eta }w+{\partial }_{\eta }w{V}^{\left(1\right)}\right),\hfill \end{array}$$

(72)

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{4}}{\displaystyle \frac{d}{dt}}\left(E+P\right).\hfill \end{array}$$

(73)

Thus, Equation (71) does not yield an independent new charge. Similarly to the dual case discussed in the previous subsection, no independent new charge arises at this order.

As in the construction outlined in the previous subsection, from this point onward and for higher-order charges, the terms representing the deformation from the usual SG model—specifically, $(-{\partial }_{\eta }{\tilde{s}}_{n-2})$ in the r.h.s. of (67)—will play a crucial role.

The next order term provides

$$\begin{array}{c}\hfill {\partial }_{\xi }\left[{\displaystyle \frac{1}{8}}{\partial }_{\eta }w\left({\left({\partial }_{\eta }w\right)}^3+{\partial }_{\eta }^{3}w\right)\right]+{\partial }_{\eta }\left({\displaystyle \frac{1}{8}}{\partial }_{\eta }^{2}w{V}^{\left(1\right)}\right)=-{\partial }_{\eta }{\tilde{s}}_1.\end{array}$$

(74)

The r.h.s. of (74) can be written as

$$\begin{array}{ccc}\hfill -{\partial }_{\eta }{\tilde{s}}_1& =& {\displaystyle \frac{1}{4}}\tilde{X}{\partial }_{\eta }^{2}w-{\displaystyle \frac{1}{4}}{\partial }_{\eta }w {\partial }_{\eta }\tilde{X},\hfill \end{array}$$

(75)

$$\begin{array}{ccc}& =& {\partial }_{\xi }{\tilde{R}}_1+{\partial }_{\eta }{\tilde{S}}_1,\hfill \end{array}$$

(76)

$$\begin{array}{ccc}\hfill {\tilde{R}}_1& \equiv & -{\displaystyle \frac{1}{8}}{\left[{\partial }_{\eta }^{2}w\right]}^2+{\displaystyle \frac{1}{8}}{\left[{\partial }_{\eta }w\right]}^4, {\tilde{S}}_1\equiv -{\displaystyle \frac{1}{8}}{\left[{\partial }_{\eta }w\right]}^2{V}^{\left(2\right)}.\hfill \end{array}$$

(77)

Then, the conservation law (74) becomes

$$\begin{array}{c}\hfill {\partial }_{\xi }\left[{\displaystyle \frac{1}{8}}{\partial }_{\eta }w{\partial }_{\eta }^{3}w+{\displaystyle \frac{1}{8}}{\left[{\partial }_{\eta }^{2}w\right]}^2\right]+{\partial }_{\eta }\left[{\displaystyle \frac{1}{8}}{\partial }_{\eta }^{2}w{V}^{\left(1\right)}+{\displaystyle \frac{1}{8}}{\left[{\partial }_{\eta }w\right]}^2{V}^{\left(2\right)}\right]=0.\end{array}$$

(78)

Thus, the charge, which follows from (78), becomes

$$\begin{array}{ccc}\hfill {\tilde{q}}^{\left(3\right)}& =& \int dx \left[{\displaystyle \frac{1}{8}}{\partial }_{\eta }w{\partial }_{\eta }^{3}w+{\displaystyle \frac{1}{8}}{\left[{\partial }_{\eta }^{2}w\right]}^2+{\displaystyle \frac{1}{8}}{\partial }_{\eta }^{2}w{V}^{\left(1\right)}+{\displaystyle \frac{1}{8}}{\left[{\partial }_{\eta }w\right]}^2{V}^{\left(2\right)}\right],\hfill \end{array}$$

(79)

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{8}}{\displaystyle \frac{d^2}{d{t}^2}}\left(E+P\right).\hfill \end{array}$$

(80)

So, in this dual formulation and at this order, unlike the ordinary SG model, the deformed SG model (8) does not possess an independent conserved charge.

Additionally, as in (47), it is shown that the third-order charge and anomaly in [1,2,7,38], in our notation, can be rewritten, respectively, in the form

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{\tilde{q}}_a^{\left(3\right)}={\int }_{-\infty }^{+\infty }dx{\tilde{\beta }}^{\left(3\right)}.\end{array}$$

(81)

$$\begin{array}{ccc}\hfill {\tilde{q}}_a^{\left(3\right)}& =& \int dx[{\displaystyle \frac{1}{8}}{\left({\partial }_{\eta }w\right)}^4+{\displaystyle \frac{1}{8}}{\partial }_{\eta }w{\partial }_{\eta }^{3}w+{\displaystyle \frac{1}{8}}{\partial }_{\eta }^{2}w{V}^{\left(1\right)}+{\displaystyle \frac{1}{4}}\tilde{X}{\partial }_{\eta }w],\hfill \end{array}$$

(82)

and

$$\begin{array}{c}\hfill {\tilde{\beta }}^{\left(3\right)}\equiv {\displaystyle \frac{1}{2}}{\partial }_{\eta }^{2}w\tilde{X}={\partial }_{\eta }{\tilde{S}}_1+{\partial }_{\xi }{\tilde{R}}_1+{\displaystyle \frac{1}{4}}{\partial }_{\eta }\left({\partial }_{\eta }w \tilde{X}\right),\end{array}$$

(83)

with ${\tilde{R}}_1,{\tilde{S}}_1$ given in (77).

A relationship between the charge ${\tilde{q}}_a^{\left(3\right)}$, its `anomaly’ ${\tilde{\beta }}^{\left(3\right)}$, and the exactly conserved charge ${\tilde{q}}^{\left(3\right)}$, follows from (74) or (78)

$$\begin{array}{ccc}\hfill {\tilde{q}}_a^{\left(3\right)}& =& {\tilde{q}}^{\left(3\right)}+\int dx\left[{\tilde{S}}_1+{\tilde{R}}_1+{\displaystyle \frac{1}{4}}{\partial }_{\eta }w\tilde{X}\right],\hfill \end{array}$$

(84)

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{8}}{\displaystyle \frac{d^2}{d{t}^2}}[E+P]+\int dx\left[{\displaystyle \frac{1}{8}}{\left({\partial }_{\eta }w\right)}^4-{\displaystyle \frac{1}{8}}{\left({\partial }_{\eta }^{2}w\right)}^2+{\displaystyle \frac{1}{2}}{\left({\partial }_{\eta }w\right)}^2[V-2]\right].\hfill \end{array}$$

(85)

The charge ${\tilde{q}}_a^{\left(3\right)}$, combined with its dual $q_a^{\left(3\right)}$ in (45), has been computed in [13] by numerical simulations of two-soliton collisions for a deformed SG model.

The next order terms, $\mathcal{O}\left({\lambda }^{-4}\right)$ and $\mathcal{O}\left({\lambda }^{-5}\right)$, become, respectively

$$\begin{array}{c}\hfill {\partial }_{\xi }\left({\displaystyle \frac{5}{16}}{\partial }_{\eta }^{2}w{\left({\partial }_{\eta }w\right)}^3+{\displaystyle \frac{1}{16}}{\partial }_{\eta }w{\partial }_{\eta }^{4}w\right)+{\partial }_{\eta }\left({\displaystyle \frac{1}{16}}{V}^{\left(1\right)}[{\left({\partial }_{\eta }w\right)}^3+{\partial }_{\eta }^{3}w]\right)={\partial }_{\eta }{\tilde{s}}_2,\end{array}$$

(86)

with

$$\begin{array}{ccc}\hfill {\partial }_{\eta }{\tilde{s}}_2& =& {\displaystyle \frac{1}{8}}\tilde{X}{\partial }_{\eta }^{3}w-{\displaystyle \frac{1}{8}}{\partial }_{\eta }w{\partial }_{\eta }^2\tilde{X},\hfill \end{array}$$

(87)

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{8}}{\partial }_{\eta }[{\partial }_{\eta }^{2}w\tilde{X}-{\partial }_{\eta }w{\partial }_{\eta }\tilde{X}],\hfill \end{array}$$

(88)

and

$$\begin{array}{c}\hfill {\partial }_{\xi }\left({\displaystyle \frac{11}{32}}{\left({\partial }_{\eta }^{2}w\right)}^2{\left({\partial }_{\eta }w\right)}^2+{\displaystyle \frac{7}{32}}{\partial }_{\eta }^{3}w{\left({\partial }_{\eta }w\right)}^3+{\displaystyle \frac{1}{16}}{\left({\partial }_{\eta }w\right)}^6+{\displaystyle \frac{1}{32}}{\partial }_{\eta }w{\partial }_{\eta }^{5}w\right)-{\partial }_{\eta }\left[{\displaystyle \frac{1}{2}}{V}^{\left(1\right)}{\tilde{u}}_4\right]=-{\partial }_{\eta }{\tilde{s}}_3,\end{array}$$

(89)

with

$$\begin{array}{ccc}\hfill -{\partial }_{\eta }{\tilde{s}}_3& =& -{\displaystyle \frac{3}{16}}{\left({\partial }_{\eta }w\right)}^3{\partial }_{\eta }\tilde{X}+{\displaystyle \frac{1}{16}}\left({\partial }_{\eta }^{4}w+{\partial }_{\eta }\left[{\left({\partial }_{\eta }w\right)}^3\right]\right) \tilde{X}-{\displaystyle \frac{1}{16}}{\partial }_{\eta }w{\partial }_{\eta }^3\tilde{X}\hfill \end{array}$$

(90)

$$\begin{array}{ccc}& =& {\partial }_{\xi }{\tilde{R}}_3+{\partial }_{\eta }{\tilde{S}}_3,\hfill \end{array}$$

(91)

where ${\tilde{R}}_3$ and ${\tilde{S}}_3$ are provided in appendix F of [13].

Equation (86), taking into account (88), can be written as

$$\begin{array}{c}\hfill {\partial }_{\xi }\left({\displaystyle \frac{5}{16}}{\partial }_{\eta }^{2}w{\left({\partial }_{\eta }w\right)}^3+{\displaystyle \frac{1}{16}}{\partial }_{\eta }w{\partial }_{\eta }^{4}w\right)+{\partial }_{\eta }\left({\displaystyle \frac{1}{16}}[{\left({\partial }_{\eta }w\right)}^3+{\partial }_{\eta }^{3}w]V^{\left(1\right)}-{\displaystyle \frac{1}{8}}[{\partial }_{\eta }^{2}w\tilde{X}-{\partial }_{\eta }w{\partial }_{\eta }\tilde{X}]\right)=0.\end{array}$$

(92)

From (92), one can define the trivial charge

$$\begin{array}{ccc}\hfill {\tilde{q}}^{\left(4\right)}& =& \int dx \left\{{\displaystyle \frac{5}{16}}{\partial }_{\eta }^{2}w{\left({\partial }_{\eta }w\right)}^3+{\displaystyle \frac{1}{16}}{\partial }_{\eta }w{\partial }_{\eta }^{4}w+{\displaystyle \frac{1}{16}}[{\left({\partial }_{\eta }w\right)}^3+{\partial }_{\eta }^{3}w]V^{\left(1\right)}-{\displaystyle \frac{1}{8}}[{\partial }_{\eta }^{2}w\tilde{X}-{\partial }_{\eta }w{\partial }_{\eta }\tilde{X}]\right\},\hfill \end{array}$$

(93)

$$\begin{array}{ccc}& \equiv & 0.\hfill \end{array}$$

(94)

Thus, it has been demonstrated that the dual charge at this order also vanishes identically under suitable boundary conditions.

Next, a lengthy calculation enables us to express Equation (89) as the fifth-order conservation law,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{32}}{\partial }_{\eta }^4\left\{{\displaystyle \frac{1}{2}}{\partial }_{\xi }{\left({\partial }_{\eta }w\right)}^2+{\partial }_{\eta }(V-2)\right\}=0.\end{array}$$

(95)

From the above conservation law, it follows the fifth-order conserved charge

$$\begin{array}{ccc}\hfill {\tilde{q}}^{\left(5\right)}& \equiv & {\displaystyle \frac{1}{32}}{\displaystyle \frac{d^4}{d{t}^4}}\int dx \left[{\displaystyle \frac{1}{2}}{\left({\partial }_{\eta }w\right)}^2+(V-2)\right],\hfill \end{array}$$

(96)

$$\begin{array}{ccc}& \equiv & {\displaystyle \frac{1}{32}}{\displaystyle \frac{d^4}{d{t}^4}}\left(E+P\right).\hfill \end{array}$$

(97)

Thus, the fifth-order dual charge ${\tilde{q}}^{\left(5\right)}$ in (96) is not an independent charge of the deformed sine-Gordon model (8), despite arising from a genuine conservation law in the Riccati-type dual formulation, beyond energy and momentum.

In this way, the r.h.s.s $[-{\partial }_{\eta }{\tilde{s}}_n (n=1,2,3,4)]$ of the dual conservation laws have been written as ${\partial }_{\xi }{\tilde{R}}_n+{\partial }_{\eta }{\tilde{S}}_n$. So, the conservation laws (67) can be written as

$$\begin{array}{c}\hfill {\partial }_{\xi }[{\tilde{a}}_{\eta }^{\left(n\right)}-{\tilde{R}}_{n-2}]+{\partial }_{\eta }[{\tilde{a}}_{\xi }^{\left(n\right)}-{\tilde{S}}_{n-2}]=0, n=1,2,3,\dots ({\tilde{S}}_k={\tilde{R}}_k\equiv 0, k=-1,0).\end{array}$$

(98)

Thus, beyond the energy-momentum charges, the higher-order asymptotically conserved charges retain the same form as those in the usual sine-Gordon model, despite being governed by the dynamics of the deformed sine-Gordon model with the potential $V\left(w\right)$, which supports solitary waves.

The existence of an infinite number of conservation laws is one of the defining characteristics of integrable models, as they impose strong constraints on the system’s dynamics and enable the presence of soliton-type solutions. As discussed earlier, in the context of deformed SG models, conservation laws can be derived directly from specific structures, such as the system’s deformed Riccati-type equations or the abelianization procedure in the anomalous Lax pair formulation [1]. However, rigorously proving the mutual independence and non-triviality of the charges associated with the conservation laws (59) is often a challenging task. In the Riccati-type pseudo-potential formulation, these charges do not automatically satisfy these criteria, requiring a detailed order-by-order examination of their non-triviality and mutual independence.

So, let us examine the first-order charges. The first-order dual conserved charges for $n=1$ in (35) and (70), the linear combinations of which provide the energy E in (36) and momentum P in (38), are, in fact, non-trivial and mutually independent.

The dual conserved charges for $n=2$ in (42) and (73), for $n=3$ in (44) and (80), and for $n=5$ in (53) and (97), are not independent charges, since they appear to be ${(n-1)}^{\prime }th$ order time derivatives of the form ${\displaystyle \frac{1}{2^n}}{\displaystyle \frac{d^{n-1}}{d{t}^{n-1}}}(E\pm P), n=2,3,5$. Additionally, for $n=4$, both dual charges (50) and (94) become vanishing trivial charges, respectively.

However, associated with $n=3$, it is possible to define the asymptotically conserved charges $q_a^{\left(3\right)}$ and ${\tilde{q}}_a^{\left(3\right)}$, as in (45) and (85), respectively. Similar expressions $q_a^{\left(5\right)}$ and ${\tilde{q}}_a^{\left(5\right)}$ can be defined for $n=5$ (see Equations (4.48) and (4.93) of [13]).

3.3. Pseudo-Potentials and a Linear System Associated to DSG

This subsection addresses the problem of formulating a linear system of equations associated with the DSG model. The next linear system is proposed [13]:

$$\begin{array}{ccc}\hfill {\partial }_{\xi }\Phi & =& A_{\xi }\Phi ,\hfill \end{array}$$

(99)

$$\begin{array}{ccc}\hfill {\partial }_{\eta }\Phi & =& A_{\eta }\Phi ,\hfill \end{array}$$

(100)

$$\begin{array}{ccc}\hfill A_{\xi }& \equiv & a_0+\lambda a_1, A_{\eta }\equiv b_0+\lambda b_1,\hfill \end{array}$$

(101)

$$\begin{array}{ccc}\hfill a_0& \equiv & -2{\displaystyle \frac{{\left({\partial }_{\xi }w\right)}^3}{{\partial }_{\xi }^{2}w}}, b_0\equiv \zeta ={\int }^{\xi }d{\xi }^{\prime }\left[6V^{\left(1\right)}{\displaystyle \frac{{\left({\partial }_{{\xi }^{\prime }}w\right)}^2}{{\partial }_{{\xi }^{\prime }}^{2}w}}-2V^{\left(2\right)}{\displaystyle \frac{{\left({\partial }_{{\xi }^{\prime }}w\right)}^4}{{\left({\partial }_{{\xi }^{\prime }}^{2}w\right)}^2}}\right],\hfill \end{array}$$

(102)

$$\begin{array}{ccc}\hfill a_1& \equiv & {\displaystyle \frac{1}{2}}{\left({\partial }_{\xi }w\right)}^2, b_1\equiv -2-V.\hfill \end{array}$$

(103)

The compatibility condition of the linear problem (99) and (100) provides

$$\begin{array}{c}\hfill \Delta (\xi ,\eta ) \lambda -6{\displaystyle \frac{{\partial }_{\xi }w}{{\partial }_{\xi }^{2}w}}\Delta (\xi ,\eta )+2{\displaystyle \frac{{\left({\partial }_{\xi }w\right)}^2}{{\left({\partial }_{\xi }^{2}w\right)}^2}}{\partial }_{\xi }\Delta (\xi ,\eta )=0,\end{array}$$

(104)

$$\begin{array}{c}\hfill \Delta (\xi ,\eta )\equiv {\partial }_{\xi }{\partial }_{\eta }w+V^{\left(1\right)}\left(w\right).\end{array}$$

(105)

The first term in the equation above is linear in the spectral parameter $\lambda$, requiring the quantity $\Delta (\xi ,\eta )$ to vanish. This condition naturally leads to the deformed SG equation of motion (8). Once $\Delta (\xi ,\eta )=0$ is imposed, the remaining terms in (104) must also vanish. Consequently, the operators ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$ form a pair of linear operators associated with the deformed SG model (8).

As a first application of the linear problem discussed above, we proceed with the construction of the energy and momentum charges. So, considering the identify ${\partial }_{\eta }\left({\displaystyle \frac{{\partial }_{\xi }\Phi }{\Phi }}\right)-{\partial }_{\xi }\left({\displaystyle \frac{{\partial }_{\eta }\Phi }{\Phi }}\right)=0$ and the linear Systems (99) and (100), one has

$$\begin{array}{ccc}\hfill {\partial }_{\eta }{A}_{\xi }-{\partial }_{\xi }{A}_{\eta }& =& 0.\hfill \end{array}$$

(106)

So, using (101), one can write

$$\begin{array}{ccc}\hfill {\partial }_{\eta }{a}_0-{\partial }_{\xi }{b}_0& =& 0,\hfill \end{array}$$

(107)

$$\begin{array}{ccc}\hfill {\partial }_{\eta }{a}_1-{\partial }_{\xi }{b}_1& =& 0.\hfill \end{array}$$

(108)

In fact, these equations define two conservation laws. Equation (107) can be written conveniently as

$$\begin{array}{c}\hfill {\partial }_{\eta }\left[{\displaystyle \frac{1}{3}}{\left({\partial }_{\xi }w\right)}^3\right]+{\partial }_{\xi }\left[V{\partial }_{\xi }w\right]=V{\partial }_{\xi }^{2}w.\end{array}$$

(109)

This is just a quasi-conservation law provided in [13] (see Equation (2.6) for $N=3$ of that reference).

However, the second Equation (108) provides the energy-momentum conservation law ${\partial }_{\eta }\left[{\displaystyle \frac{1}{2}}{\left({\partial }_{\xi }w\right)}^2\right]+{\partial }_{\xi }[V-2]=0$, which has already been discussed in the pseudo-potential approach (23).

Non-Local Conservation Laws

A set of an infinite number of non-local conservation laws for Systems (99) and (100) can be constructed through an iterative procedure introduced by Brézin, et al. [39]. In fact, this system satisfies: (i) $(A_{\xi },A_{\eta })$ is a “pure gauge”; i.e., $A_{\mu }={\partial }_{\mu }\Phi {\Phi }^{-1}, \mu =\xi ,\eta$; (ii) $J_{\mu }=(A_{\xi },A_{\eta })$ is a conserved current satisfying (106). Thus, an infinite set of non-local conserved currents can be constructed using an inductive procedure. We begin by defining the currents as follows:

$$\begin{array}{ccc}\hfill J_{\mu }^{\left(n\right)}& =& {\partial }_{\mu }{\chi }^{\left(n\right)}, \mu \equiv \xi ,\eta ; n=0,1,2,\dots \hfill \end{array}$$

(110)

$$\begin{array}{ccc}\hfill d{\chi }^{\left(1\right)}& =& A_{\xi }d\xi +A_{\eta }d\eta , \hfill \end{array}$$

(111)

$$\begin{array}{ccc}& \equiv & d{\mathcal{I}}_0(\xi ,\eta )+\lambda d{\mathcal{I}}_1(\xi ,\eta ),\hfill \end{array}$$

(112)

$$\begin{array}{ccc}\hfill J_{\mu }^{(n+1)}& =& {\partial }_{\mu }{\chi }^{\left(n\right)}-A_{\mu }{\chi }^{\left(n\right)}, {\chi }^{\left(0\right)}=1,\hfill \end{array}$$

(113)

where

$$\begin{array}{c}\hfill d{\mathcal{I}}_0(\xi ,\eta )\equiv a_0(\xi ,\eta )d\xi +b_0(\xi ,\eta )d\eta , d{\mathcal{I}}_1(\xi ,\eta )\equiv a_1(\xi ,\eta )d\xi +b_1(\xi ,\eta )d\eta .\end{array}$$

(114)

Then, one can use an inductive procedure to show that the (non-local) currents $J_{\mu }^{\left(n\right)}$ are conserved:

$$\begin{array}{c}\hfill {\partial }_{\mu }{J}^{\left(n\right) \mu }=0, n=1,2,3,\dots \end{array}$$

(115)

The first non-trivial conservation law ${\partial }_{\mu }{J}^{\left(1\right) \mu }=0$ reduces to Equation (106) and then provides the first two conservation laws, (107) and (108). The next conservation law ${\partial }_{\mu }{J}^{\left(2\right) \mu }=0$ becomes

$$\begin{array}{c}\hfill {\partial }_{\eta }\left[A_{\xi }-a_0{\mathcal{I}}_0-(a_0{\mathcal{I}}_1+a_1{\mathcal{I}}_0)\lambda -a_1{\mathcal{I}}_1{\lambda }^2\right]-{\partial }_{\xi }\left[A_{\eta }-b_0{\mathcal{I}}_0-(b_0{\mathcal{I}}_1+b_1{\mathcal{I}}_0)\lambda -b_1{\mathcal{I}}_1{\lambda }^2\right]=0,\end{array}$$

(116)

where ${\mathcal{I}}_0$ and ${\mathcal{I}}_1$ are defined in (114). From (116), in addition to the conservation laws (106), or (107) and (108), one can obtain the next non-local conservation laws order by order: $\lambda$

$$\begin{array}{c}\hfill {\partial }_{\eta }\left(a_0{\mathcal{I}}_0\right)-{\partial }_{\xi }\left(b_0{\mathcal{I}}_0\right)=0,\end{array}$$

(117)

$$\begin{array}{c}\hfill {\partial }_{\eta }(a_0{\mathcal{I}}_1+a_1{\mathcal{I}}_0)-{\partial }_{\xi }(b_0{\mathcal{I}}_1+b_1{\mathcal{I}}_0)=0,\end{array}$$

(118)

$$\begin{array}{c}\hfill {\partial }_{\eta }\left(a_1{\mathcal{I}}_1\right)-{\partial }_{\xi }\left(b_1{\mathcal{I}}_1\right)=0.\end{array}$$

(119)

The formulation of similar linear systems could be significant for studying deformations of well-known integrable models associated with the equation of motion (8), such as the Bullough–Dodd model [4].

4. Modified NLS Model as Reduction of the Deformed $\mathit{Sl}\mathbf{\left(}\mathbf{2}\mathbf{\right)}$ AKNS Model

The compatibility condition (4)–(7), for special coefficients $A_j$ and $B_j$ and relevant auxiliary fields r and s, reproduces the equation of motion of a modified AKNS system (MAKNS). A modified AKNS model has recently been studied as quasi-integrable in the anomalous zero-curvature approach [17]. We will present a dual Riccati-type formulation and explicitly construct the first three exact conservation laws and the first two quasi-conservation laws. Furthermore, a dual formulation allows us to uncover infinite towers of novel anomalous conservation laws. It is shown that certain modifications of the nonlinear Schrödinger model (MNLS) can be obtained using a reduction process starting from the MAKNS model. So, the novel infinite sets of quasi-conservation laws and related anomalous charges for the standard NLS and modified MNLS cases can be constructed using a unified and rigorous approach based on the Riccati-type pseudo-potential method.

It is known that the standard NLS model emerges as a specific reduction of the AKNS system. Therefore, we explore a suitable deformation of the conventional pseudo-potential approach within the $sl\left(2\right)$ AKNS integrable field theory. We then examine the reduction process that leads to the modified NLS model.

In the case of the deformed $sl\left(2\right)$ AKNS system, the Riccati-type Equations (1) and (2) are defined with the following quantities [17]:

$$\begin{array}{ccc}\hfill A_0& =& q, A_1=-2i\zeta , A_2=\overline{q},\hfill \end{array}$$

(120)

$$\begin{array}{ccc}\hfill B_0& =& 2\zeta q+i{\partial }_{x}q+r, B_1=-4i{\zeta }^2-i V^{\left(1\right)}-s, B_2=2\zeta \overline{q}-i{\partial }_x\overline{q},\hfill \end{array}$$

(121)

$$\begin{array}{ccc}\hfill V^{\left(1\right)}& \equiv & {\displaystyle \frac{dV\left[\rho \right]}{d\rho }}, \rho \equiv \overline{q}q.\hfill \end{array}$$

(122)

Note that we will redefine $\xi =x, \eta =t$, such that x and t stand for space and time variables, respectively. ${\partial }_{x}q$ and ${\partial }_x\overline{q}$ stand for partial derivatives with respect to x. Then, one has the system of Riccati-type equations [17],

$$\begin{array}{ccc}\hfill {\partial }_{x}u& =& -2i\zeta u+q+ \overline{q} u^2,\hfill \end{array}$$

(123)

$$\begin{array}{ccc}\hfill {\partial }_{t}u& =& 2(-2i{\zeta }^2-{\displaystyle \frac{1}{2}}i V^{\left(1\right)}) u-(-2\zeta \overline{q}+i{\partial }_x\overline{q}) u^2+(2\zeta q+i{\partial }_{x}q)+r-u s,\hfill \end{array}$$

(124)

$$\begin{array}{ccc}\hfill V^{\left(1\right)}& \equiv & {\displaystyle \frac{dV\left[\rho \right]}{d\rho }}, \rho \equiv \overline{q}q,\hfill \end{array}$$

(125)

where $u(x,t,\lambda )$ is the Riccati-type pseudo-potential, $\zeta$ is the spectral parameter, r and s are the auxiliary fields, and q and $\overline{q}$ are the fields of the model. The potential $V\left(\overline{q}q\right)$ defines the modified AKNS equation (MAKNS). In fact, the form of (1) is similar to the first Riccati equation for the standard AKNS model. The function $B_2$ is the same as in the usual second Riccati equation; whereas, the functions $B_0$ and $B_1$ will depend on a generalized potential as compared to the standard AKNS model [27,28].

The auxiliary fields $r(x,t)$ and $s(x,t)$ satisfy

$$\begin{array}{ccc}\hfill {\partial }_{x}r& =& q s+(-2i\zeta +Q u) r,\hfill \end{array}$$

(126)

$$\begin{array}{ccc}\hfill {\partial }_{x}s& =& Q r-2 \overline{q} r+u\overline{q} s+2iX,\hfill \end{array}$$

(127)

$$\begin{array}{ccc}\hfill X& \equiv & -{\partial }_x\left({\displaystyle \frac{1}{2}}{V}^{\left(1\right)}+\overline{q}q\right),\hfill \end{array}$$

(128)

where Q is an arbitrary field. So, one has a set of two deformed Riccati-type equations for the pseudo-potential u (1) and (2) and linear Systems (126) and (127) for the auxiliary fields r and s.

Note that, for the integrable $sl\left(2\right)$ AKNS model, one has the potential [17]

$$\begin{array}{c}\hfill V_{NLS}\left(\overline{q}q\right)=-{\left(\overline{q}q\right)}^2 \to V_{NLS}^{\left(1\right)}\left(\overline{q}q\right)=-2\left(\overline{q}q\right),\end{array}$$

(129)

and, therefore, $X=0$ in (128), so the auxiliary Systems (126) and (127) possess the trivial solution $r=s=0$. Inserting the trivial solution into (1) and (2) with the potential (129), one has a Riccati system of the standard AKNS model [27,28].

The compatibility condition $[{\partial }_t{\partial }_{x}u-{\partial }_x{\partial }_{t}u]=0$ for the Riccati-type Systems (1) and (2) with (120)–(121). Taking into account the auxiliary Equations (126) and (127), it provides the equations of motion for the fields q and $\overline{q}$

$$\begin{array}{ccc}\hfill i{\partial }_{t}q+{\partial }_x^{2}q-V^{\left(1\right)}q& =& 0,\hfill \end{array}$$

(130)

$$\begin{array}{ccc}\hfill -i{\partial }_t\overline{q}+{\partial }_x^2\overline{q}-V^{\left(1\right)}\overline{q}& =& 0.\hfill \end{array}$$

(131)

This is a modified $sl\left(2\right)$ AKNS system (MAKNS) for arbitrary potential of type $V\left(\overline{q}q\right)$. An important observation in the constructions above is that ${\displaystyle \frac{\partial }{\partial t}}\zeta =0$, as it can be checked by direct computation using Systems (1) and (2) and (126) and (127), provided that Systems (130) and (131) are satisfied. So, the modified system MAKNS possesses an isospectral parameter $\zeta$.

The next identification defines the modified NLS model. So, defining [16,31]

$$\begin{array}{c}\hfill q\equiv i{(-\eta )}^{1/2} \psi , \overline{q}\equiv -i{(-\eta )}^{1/2} \overline{\psi }, \eta \in \mathrm{I}\mathrm{R},\end{array}$$

(132)

in Systems (130) and (131), with $\overline{\psi }$ being the complex conjugate of the field $\psi$, one can obtain the modified NLS model,

$$\begin{array}{ccc}\hfill i{\displaystyle \frac{\partial }{\partial t}}\psi (x,t)+{\displaystyle \frac{{\partial }^2}{\partial x^2}}\psi (x,t)-V^{\left(1\right)}{\left(\right|\psi (x,t)|}^2)\psi (x,t)& =& 0,\hfill \end{array}$$

(133)

$$\begin{array}{ccc}\hfill V^{\left(n\right)}\left(I\right)& \equiv & {\displaystyle \frac{d^n}{d{I}^n}}V\left(I\right), I\equiv \overline{\psi }\psi .\hfill \end{array}$$

(134)

To obtain the standard NLS model, the potential and its derivatives are defined as

$$\begin{array}{c}\hfill V\left(I\right)=-\eta I^2, V^{\left(1\right)}\left(I\right)=-2\eta I \mathrm{and} V^{\left(2\right)}\left(I\right)=-2\eta .\end{array}$$

(135)

Let us emphasize that for the standard NLS model we have the trivial solution of Systems (126) and (127), i.e., $X=0\to r=s=0$, and the existence of the Lax pair of the ordinary NLS model reflects its equivalent Riccati-type representation, provided by Systems (1) and (2), with the well known potential (135) [25,26,27,28].

It is important to highlight that, for the standard NLS model, Systems (126) and (127) admit a trivial solution, namely $X=0\to r=s=0$. Additioanlly, the existence of the Lax pair for the ordinary NLS model is reflected in its equivalent Riccati-type formulation, given by Systems (1) and (2) with the well-known potential (135) [25,26,27,28].

Next, a special space–time symmetry related to soliton-type solutions of the model is introduced. So, a reflection around a fixed point $(x_{\Delta },t_{\Delta })$ becomes

$$\begin{array}{c}\hfill \tilde{\mathcal{P}}:(\tilde{x},\tilde{t})\to (-\tilde{x},-\tilde{t}); \tilde{x}=x-x_{\Delta }, \tilde{t}=t-t_{\Delta }.\end{array}$$

(136)

The operator $\tilde{\mathcal{P}}$ defines a shifted parity ${\mathcal{P}}_s$ for the variable x and a delayed time reversal ${\mathcal{T}}_d$ for the variable t.

The standard NLS model, in the sector described by N-bright solitons possessing $\mathcal{C}{\mathcal{P}}_s{\mathcal{T}}_d$ ($\mathcal{C}\left(\psi \right)\equiv \overline{\psi }$) symmetry and broken space–time translation symmetry, belongs to the family of nonlocal generalization of the NLS model considered in the recent literature [40,41].

We define the quasi-integrable MAKNS model for field configurations q and $\overline{q}$ that satisfy (130) and (131), ensuring that the fields and the deformed potential transform under the space–time transformation (136) as follows:

$$\begin{array}{c}\hfill \tilde{\mathcal{P}}\left(q\right)=\overline{q}, \tilde{\mathcal{P}}\left(\overline{q}\right)=q, \mathrm{and} \tilde{\mathcal{P}}\left[V\left(\rho \right)\right]=V\left(\rho \right), \rho \equiv \overline{q}q.\end{array}$$

(137)

Under this transformation, X from (128) becomes an odd function,

$$\begin{array}{c}\hfill \tilde{\mathcal{P}}\left(X\right)=-X.\end{array}$$

(138)

Next, let us discuss the conservation laws in the context of the Riccati-type Systems (1) and (2) and the auxiliary Equations (126) and (127). So, one can obtain the following relationship:

$$\begin{array}{ccc}\hfill {\partial }_t[i\overline{q} u]-{\partial }_x\left[2i\zeta \overline{q} u-\overline{q}q+u {\partial }_x\overline{q}\right]& =& i\overline{q}(r-s u).\hfill \end{array}$$

(139)

Defining the r.h.s. of (139) as

$$\begin{array}{c}\hfill \chi \equiv i\overline{q}(r-s u),\end{array}$$

(140)

one can write a first-order differential equation for the auxiliary field $\chi$,

$$\begin{array}{ccc}\hfill {\partial }_x\chi & =& \left(-2i\zeta +2u\overline{q}+{\displaystyle \frac{{\partial }_x\overline{q}}{\overline{q}}}\right)\chi +2\overline{q} uX.\hfill \end{array}$$

(141)

Equations (139) and (141) will be utilized below to reveal an infinite hierarchy of quasi-conservation laws associated with the modified AKNS models (130) and (131). We will systematically construct the relevant charges, order by order, in terms of the parameter $\zeta$. Thus, we begin by considering the following expansions

$$\begin{array}{c}\hfill u=\sum _{n=1}^{\infty }{u}_n {\zeta }^{-n}, \chi =\sum _{n=1}^{\infty }{\chi }_n{\zeta }^{-n-1}.\end{array}$$

(142)

The coefficients $u_n$ can be determined order by order in powers of $\zeta$ from the Riccati Equation (1). Likewise, using the results for the ${u_n}^{\prime }s$, we obtain the relevant expressions for the ${{\chi }_n}^{\prime }s$ from (141). The first components of ${u_n}^{\prime }s$ and ${{\chi }_n}^{\prime }s$ are provided in appendices A and B, respectively, of ref. [17].

So, by considering the components $u_n$ and ${\chi }_n$ in Equation (139), one gets the coefficient of the $n^{\prime }$th order as

$$\begin{array}{ccc}\hfill {\partial }_t{a}_x^{\left(n\right)}& +& {\partial }_x{a}_t^{\left(n\right)}={\chi }_{n-1}, n=0,1,2,3,....; {\chi }_{-1}={\chi }_0\equiv 0,\hfill \end{array}$$

(143)

$$\begin{array}{ccc}\hfill a_x^{\left(n\right)}& \equiv & i\overline{q} u_n, a_t^{\left(n\right)}\equiv -\left(2i\overline{q} u_{n+1}-\overline{q}q {\delta }_{0,n}+{\partial }_x\overline{q} u_n\right), u_0\equiv 0.\hfill \end{array}$$

(144)

Setting ${\chi }_{n-1}\equiv 0$ on the right-hand side of Equation (143) yields the tower of exact conservation laws that is characteristic of the standard AKNS system. However, when ${\chi }_n$ is nonzero, the genuinely conserved nature of this equation at each order n remains to be fully understood. We will examine this construction order by order for each ${\chi }_{n-1}$ component. Similar quasi-conservation laws have been derived in the context of the anomalous zero-curvature formulation of the modified NLS model and its corresponding anomalous Lax pair in [5].

The r.h.s. of (143) for ${\chi }_1,{\chi }_2$ and ${\chi }_3$ can be written as ${\chi }_j\equiv {\partial }_x{\chi }_j^x+{\partial }_t{\chi }_j^t$, with ${\chi }_j^x$ and ${\chi }_j^t$ being certain local functions of $\{\overline{q},q,V\}$ and their x and $t–$derivatives; i.e., there exist local expressions for some ${\chi }_j (j=1,2,3)$, such that Equation (143) provides a genuine local conservation law.

The zeroth order provides a trivial identity, since $u_1=-{\displaystyle \frac{1}{2}}iq$ and ${\chi }_{-1}=u_0=0$.

The order $n=1$ and the field normalization

At this order, the anomaly is also trivial ${\chi }_0=0$ and $u_2={\displaystyle \frac{1}{4}}{\partial }_{x}q$. So, one has

$$\begin{array}{c}\hfill {\partial }_t\left({\displaystyle \frac{1}{2}}\overline{q}q\right)-{\partial }_x\left({\displaystyle \frac{1}{2}}i\overline{q}{\partial }_{x}q-{\displaystyle \frac{1}{2}}iq{\partial }_x\overline{q}\right)=0.\end{array}$$

(145)

It provides the charge

$$\begin{array}{c}\hfill N=\int dx \overline{q}q.\end{array}$$

(146)

The order $n=2$ and momentum conservation

At this order, one has $u_3={\displaystyle \frac{1}{8}}i(\overline{q}qq+{\partial }_x^{2}q)$, and so

$$\begin{array}{ccc}\hfill {\partial }_t\left({\displaystyle \frac{1}{4}}i\overline{q}{\partial }_{x}q\right)+{\partial }_x\left({\displaystyle \frac{1}{4}}[{\left(\overline{q}q\right)}^2+\overline{q}{\partial }_x^{2}q-{\partial }_x\overline{q}{\partial }_{x}q]\right)& =& {\chi }_1.\hfill \end{array}$$

(147)

The function ${\chi }_1$ can be rewritten as

$$\begin{array}{ccc}\hfill {\chi }_1& =& {\displaystyle \frac{1}{2}}{\partial }_x\left[F\left(\rho \right)\right], \rho \equiv \overline{q}q.\hfill \end{array}$$

(148)

$$\begin{array}{ccc}\hfill F\left(I\right)& \equiv & {\displaystyle \frac{1}{2}}\rho {\displaystyle \frac{d}{d\rho }}V\left(\rho \right)-{\displaystyle \frac{1}{2}}V\left(\rho \right)+{\displaystyle \frac{1}{2}}{\rho }^2.\hfill \end{array}$$

(149)

So, from (147), taking into account (148), one can write the conserved charge,

$$\begin{array}{c}\hfill P=i\int dx \left(\overline{q}{\partial }_{x}q-q{\partial }_x\overline{q}\right).\end{array}$$

(150)

The order $n=3$ and energy conservation

One has $u_4=-{\displaystyle \frac{1}{16}}[4\overline{q}q{\partial }_{x}q+qq\partial -x\overline{q}+\partial -x^{3}q]$; therefore, it follows that

$$\begin{array}{ccc}\hfill {\partial }_t[-{\displaystyle \frac{1}{8}}{\left(\overline{q}q\right)}^2-{\displaystyle \frac{1}{8}}\overline{q}{\partial }_x^{2}q]-{\partial }_x\left(2i\overline{q}{u}_4+{\partial }_x\overline{q}{u}_3\right)& =& {\chi }_2.\hfill \end{array}$$

(151)

The function ${\chi }_2$ can be rewritten as

$$\begin{array}{c}\hfill {\chi }_2\equiv -{\displaystyle \frac{1}{8}}{\partial }_{t}V-{\displaystyle \frac{1}{8}}{\partial }_t{\left(\overline{q}q\right)}^2-{\displaystyle \frac{1}{4}}i{\partial }_x\left[X\overline{q}q-X(q{\partial }_x\overline{q}-\overline{q}{\partial }_{x}q)\right].\end{array}$$

(152)

So, (151) provides the conserved charge,

$$\begin{array}{c}\hfill H_{MNLS}=\int dx [ {\partial }_x\overline{q}{\partial }_{x}q+V\left(\overline{q}q\right) ].\end{array}$$

(153)

This expression (153) is valid for the general MAKNS model. In particular, for the ordinary AKNS, the energy follows directly from the l.h.s. of (151) (provided that ${\chi }_2=0$), i.e., $H_{NLS}=\int dx [ {\partial }_x\overline{q}{\partial }_{x}q+V_{NLS}\left(\overline{q}q\right) ]$, where $V_{NLS}=-{\left(\overline{q}q\right)}^2$ as in (129).

The order $n=4$: A first trivial charge and its associated anomalous charge

One has

$$\begin{array}{ccc}\hfill {\partial }_t\left(-{\displaystyle \frac{3}{16}}i\overline{q}q\overline{q}{\partial }_{x}q-{\displaystyle \frac{1}{32}}i{\partial }_x{\left(\overline{q}q\right)}^2-{\displaystyle \frac{1}{16}}i\overline{q}{\partial }_x^{3}q\right)-{\partial }_x[2i\overline{q} u_5+{\partial }_x\overline{q} u_4]& =& {\chi }_3.\hfill \end{array}$$

(154)

Remarkably, the expression for ${\chi }_3$ can be written as

$$\begin{array}{ccc}\hfill {\chi }_3& \equiv & {\partial }_x\left[{\chi }_x^{\left(3\right)}\right]+{\partial }_t\left[{\chi }_t^{\left(3\right)}\right],\hfill \end{array}$$

(155)

$$\begin{array}{ccc}\hfill {\chi }_t^{\left(3\right)}& =& -{\displaystyle \frac{3}{16}}i\overline{q}q\overline{q}{\partial }_{x}q-{\displaystyle \frac{1}{16}}i\overline{q}{\partial }_x^{3}q,\hfill \\ \hfill {\chi }_x^{\left(3\right)}& =& {\displaystyle \frac{3}{8}}X\overline{q}{\partial }_{x}q+{\displaystyle \frac{3}{8}}{H}_1\left(\rho \right)+{\displaystyle \frac{1}{8}}\overline{q}q{\partial }_{x}X-{\displaystyle \frac{1}{8}}X{\partial }_x\left(\overline{q}q\right)-{\displaystyle \frac{1}{16}}{\left[{\partial }_x\left(\overline{q}q\right)\right]}^2+{\displaystyle \frac{3}{8}}\overline{q}q{\partial }_x\overline{q}{\partial }_{x}q-{\displaystyle \frac{3}{8}}{H}_2\left(\rho \right)+\hfill \\ & & {\displaystyle \frac{3}{32}}i\overline{q}{\partial }_t{q}^2-{\displaystyle \frac{1}{16}}{V}^{\left(1\right)}\left(q{\partial }_x^2\overline{q}+\overline{q}{\partial }_x^{2}q\right)+{\displaystyle \frac{1}{16}}{V}^{\left(1\right)}{\partial }_{x}q{\partial }_x\overline{q}+\hfill \end{array}$$

(156)

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{16}}i[{\partial }_{t}q{\partial }_x^2\overline{q}-{\partial }_x{\partial }_{t}q{\partial }_x\overline{q}+\overline{q}{\partial }_t{\partial }_x^{2}q],\hfill \end{array}$$

(157)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{d\rho }}{H}_1\left(\rho \right)& \equiv & -(V^{\left(2\right)}/2+1){\rho }^2, {\displaystyle \frac{d}{d\rho }}{H}_2\left(\rho \right)\equiv \rho V^{\left(1\right)}, \rho \equiv \overline{q}q.\hfill \end{array}$$

(158)

After a careful examination of Equation (154), one can argue that the charge $Q^{\left(4\right)}$ trivially vanishes.

However, at this and higher orders, an asymptotically conserved charge can be defined for the MAKNS model. In this case, one has

$$\begin{array}{c}\hfill Q_a^{\left(4\right)}={\displaystyle \frac{i}{2}}\int dx \left[3\overline{q}q\left(\overline{q}{\partial }_{x}q-q{\partial }_x\overline{q}\right)+\overline{q}{\partial }_x^{3}q-q{\partial }_x^3\overline{q}\right],\end{array}$$

(159)

such that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{dt}}{Q}_a^{\left(4\right)}& =& -8\int dx {\chi }_3,\hfill \end{array}$$

(160)

$$\begin{array}{ccc}\hfill & =& -\int dx [3{\left(\overline{q}q\right)}^{2}X+\overline{q}q{\partial }_x^{2}X-3{\partial }_x\overline{q}{\partial }_{x}qX].\hfill \end{array}$$

(161)

Notice that the anomaly density in (161) possesses an odd parity under (136) and (137), taking into account that X is an odd function according to (138). Therefore, one has $\int dt\int dx {\chi }_3=0$, implying the asymptotic conservation of the charge $Q_a^{\left(4\right)}$, i.e., $Q_a^{\left(4\right)}(+\infty )=Q_a^{\left(4\right)}(-\infty )$.

The charge $Q_a^{\left(4\right)}$ in (159) has the same form as the fourth-order charge in the standard AKNS model. Specifically, when the right-hand side of (154) vanishes, i.e., ${\chi }_3=0$, the resulting charge takes a form similar to that of (159), suitably rewritten by discarding surface terms. Considering the reduction process in (132), a corresponding anomalous charge for the MNLS model can be obtained, as discussed in ref. [5,8,9,10]. In fact, under the reduction (132), the anomalous charge $Q_a^{\left(4\right)}$ in (159) corresponds to the one derived for the MNLS model in Section 3.5 of ref. [16,31].

The order $n=5$ and the quasi-conserved charge

At this order, one can write [17]

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{32}}{\partial }_t\left[2{\left(\overline{q}q\right)}^3+5{\overline{q}}^2{\left({\partial }_{x}q\right)}^2+6\overline{q}q\left({\partial }_{x}q{\partial }_x\overline{q}+\overline{q}{\partial }_x^{2}q\right)+\overline{q}{q}^2{\partial }_x^2\overline{q}+\overline{q}{\partial }_x^{4}q\right]-{\partial }_x[2i\overline{q} u_6+{\partial }_x\overline{q} u_5]={\chi }_4,\end{array}$$

(162)

with

$$\begin{array}{ccc}\hfill {\chi }_4& \equiv & {\partial }_x\left[{\chi }_x^{\left(4\right)}\right]+{\displaystyle \frac{1}{16}}{\partial }_t\left[Z\left(\rho \right)\right]+{\beta }_1,\hfill \end{array}$$

(163)

$$\begin{array}{ccc}\hfill {\beta }_1& =& {\displaystyle \frac{i}{32}}\left({\displaystyle \frac{2\overline{q}q}{V^{\left(1\right)}}}+1\right)\left(\overline{q}{\partial }_x^{4}q-q{\partial }_x^4\overline{q}\right)V^{\left(1\right)}\hfill \end{array}$$

(164)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{d\rho }}Z\left(\rho \right)& \equiv & 6{\int }_{{\rho }_0}^{\rho }\widehat{\rho }[{\displaystyle \frac{1}{2}}{V}^{\left(2\right)}\left(\widehat{\rho }\right)+1] d\widehat{\rho },\hfill \end{array}$$

(165)

where ${\chi }^{\left(4\right)}$ is provided in Equation (2.44)–(2.46) of ref. [17].

One can define the fifth-order quasi-conservation law as

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{dt}}{Q}_a^{\left(5\right)}& =& \int dx {\beta }_1,\hfill \end{array}$$

(166)

$$\begin{array}{ccc}\hfill Q_a^{\left(5\right)}& \equiv & {\displaystyle \frac{1}{32}}\int dx \left[2{\left(\overline{q}q\right)}^3-8\overline{q}q{\partial }_{x}q{\partial }_x\overline{q}-{\overline{q}}^2{\left({\partial }_{x}q\right)}^2-q^2{\left({\partial }_x\overline{q}\right)}^2+{\partial }_x^2\overline{q}{\partial }_x^{2}q-2Z\left(\rho \right)\right].\hfill \end{array}$$

(167)

Note that the anomaly in (164) possesses an odd parity under (136) and (137). Then, one has $\int dt\int dx {\beta }_1=0$, implying the asymptotic conservation of the charge $Q_a^{\left(5\right)}$ ($Q_a^{\left(5\right)}(+\infty )=Q_a^{\left(5\right)}(-\infty )$).

Note that, in the standard AKNS limit, i.e., when $V^{\left(1\right)}=-2\overline{q}q$ and $V^{\left(2\right)}=-2$ for the AKNS potential (129), the factor $\left({\displaystyle \frac{2\overline{q}q}{V^{\left(1\right)}}}+1\right)$ of the anomaly ${\beta }_1$ in (164) vanishes, and the term $Z\left(\rho \right)$ in (167) can be set to zero (see (165)). So, the quasi-conserved charge $Q_a^{\left(5\right)}$ in (166) becomes the fifth order charge $Q^{\left(5\right)}$ of the usual AKNS model. Actually, in this limit one has that ${\chi }_4=0$ for $X=0$, then the r.h.s. of (162) vanishes,; therefore, this equation is as an exact conservation law.

The set of (quasi-)conservation laws of type (143) has been constructed using the Riccati-type approach of the modified AKNS model. Through a suitable reduction process, these charges correspond to those of the MNLS model. In ref. [5], within the anomalous Lax pair framework of modified NLS models and via the abelianization procedure, an infinite set of asymptotically conserved charges was derived, resembling the exact conserved charges of the standard NLS model.

Dual Riccati-Type Formulation and Novel Anomalous Charges

To discuss a dual formulation, let us consider the system formed by the Riccati-type Equations (123) and (124) and the auxiliary Equation (141),

$$\begin{array}{ccc}\hfill {\partial }_{x}u& =& -2i\zeta u+q+ \overline{q} u^2,\hfill \end{array}$$

(168)

$$\begin{array}{ccc}\hfill {\partial }_{t}u& =& 2(-2i{\zeta }^2-{\displaystyle \frac{1}{2}}i V^{\left(1\right)}) u-(-2\zeta \overline{q}+i{\partial }_x\overline{q}) u^2+(2\zeta q+i{\partial }_{x}q)-i{\displaystyle \frac{\chi }{\overline{q}}},\hfill \end{array}$$

(169)

$$\begin{array}{ccc}\hfill {\partial }_x\chi & =& \left(-2i\zeta +2u\overline{q}+{\displaystyle \frac{{\partial }_x\overline{q}}{\overline{q}}}\right)\chi +2\overline{q} uX.\hfill \end{array}$$

(170)

Note that the system of differential equations of the AKNS models (130) and (131) is invariant under the transformations: $q\leftrightarrow \overline{q}$ and $i\leftrightarrow -i$.

A dual formulation of the Riccati-type System (168)–(170) is achieved by performing the changes $q\leftrightarrow \overline{q}$ and $i\leftrightarrow -i$, $u\to \overline{u}$ and $\chi \to \overline{\chi }$ into Systems (168)–(169) and (170).

So, one has a dual Riccati-type system [17],

$$\begin{array}{ccc}\hfill {\partial }_x\overline{u}& =& 2i\zeta \overline{u}+\overline{q}+ q {\overline{u}}^2,\hfill \end{array}$$

(171)

$$\begin{array}{ccc}\hfill {\partial }_t\overline{u}& =& 2(2i{\zeta }^2+{\displaystyle \frac{1}{2}}i V^{\left(1\right)}) \overline{u}+(2\zeta q+i{\partial }_{x}q) {\overline{u}}^2+(2\zeta \overline{q}-i{\partial }_x\overline{q})+i{\displaystyle \frac{\overline{\chi }}{q}},\hfill \end{array}$$

(172)

$$\begin{array}{ccc}\hfill {\partial }_x\overline{\chi }& =& \left(2i\zeta +2\overline{u}q+{\displaystyle \frac{{\partial }_{x}q}{q}}\right)\overline{\chi }+2q \overline{u}X,\hfill \end{array}$$

(173)

where $\overline{u}$ is a new Riccati-type pseudo-potential and $\overline{\chi }$ is a new auxiliary field. Note that X defined in (128) is an invariant. It is a simple calculation to verify that this dual Riccati-type System (171)–(173) reproduces the AKNS Equations (130) and (131).

Consider the expansions,

$$\begin{array}{c}\hfill \overline{u}=\sum _{n=1}^{\infty }{\overline{u}}_n {\zeta }^{-n}, \overline{\chi }=\sum _{n=1}^{\infty }{\overline{\chi }}_n{\zeta }^{-n-1},\end{array}$$

(174)

where the coefficients ${\overline{u}}_n$ and ${{\overline{\chi }}_n}^{\prime }s$ can be obtained from (171) and (173), respectively. The first components are provided in appendix C of ref. [17].

For the fields $q,\overline{q}$ and X, which satisfy (137) and (138), respectively, one can verify the next symmetry transformations from (168)–(170) and (171)–(173)

$$\begin{array}{ccc}\hfill ,\tilde{\mathcal{P}}\left(u\right)& =& -\overline{u}, \tilde{\mathcal{P}}\left(\overline{u}\right)=-u,\hfill \end{array}$$

(175)

$$\begin{array}{ccc}\hfill \tilde{\mathcal{P}}\left(\chi \right)& =& -\overline{\chi }, \tilde{\mathcal{P}}\left(\overline{\chi }\right)=-\chi .\hfill \end{array}$$

(176)

In fact, by inspecting the first few components in appendix C of ref. [17], one can show

$$\begin{array}{ccc}\hfill \tilde{\mathcal{P}}\left(u_n\right)& =& -{\overline{u}}_n, \tilde{\mathcal{P}}\left({\overline{u}}_n\right)=-u_n, n=1,2,\dots ,6,\hfill \end{array}$$

(177)

$$\begin{array}{ccc}\hfill \tilde{\mathcal{P}}({\chi }_n\pm {\overline{\chi }}_n)& =& \mp ({\chi }_n\pm {\overline{\chi }}_n), n=1,2,\dots ,5.\hfill \end{array}$$

(178)

For both dual systems of Riccati-type Equations (168)–(170) and (171)–(173), one can write the next equations, respectively,

$$\begin{array}{c}\hfill {\partial }_t\left(i\overline{q}u\right)-{\partial }_x(2i\zeta \overline{q}u-\overline{q}q+u{\partial }_x\overline{q})=\chi ,\end{array}$$

(179)

and

$$\begin{array}{c}\hfill {\partial }_t\left(iq\overline{u}\right)-{\partial }_x(2i\zeta q\overline{u}+\overline{q}q-\overline{u}{\partial }_{x}q)=-\overline{\chi },\end{array}$$

(180)

where (179) has already been expressed in (139) with $\chi$ defined in (140). Subtracting the right-hand side of (179) and (180), one has

$$\begin{array}{c}\hfill {\partial }_t[i\overline{q}u-iq\overline{u}]-{\partial }_x[2i\zeta (\overline{q}u-q\overline{u})-2\overline{q}q+u{\partial }_x\overline{q}+\overline{u}{\partial }_{x}q]=\chi +\overline{\chi }.\end{array}$$

(181)

Observe that the right-hand side of the last equation is an odd function under the special space–time operator. Consequently, Equation (181) represents a quasi-conservation law. Indeed, the first five lowest-order components are explicitly odd functions, as shown in (178).

A standard computation reveals that the expansion of the quasi-conservation law (181) in powers of ${\zeta }^{-n}$ yields the conserved charges associated with normalization $(n=1)$, momentum $(n=2)$, and energy $(n=3)$. Higher-order terms, on the other hand, produce the same anomalous charges discussed in Section 2.

Notably, the charge densities in the (quasi-)conservation Equation (181) are even functions. This follows from the fact that the expression $[i\overline{q}u-iq\overline{u}]$ within the partial time derivative on the left-hand side of (181) has even parity.

However, the summation of the right-hand side of (179) and (180) will not reproduce a quasi-conservation law, since the anomaly $(\chi -\overline{\chi })$ is an even function according to (176). Additionally, in this case, the expression $[i\overline{q}u+iq\overline{u}]$ of the charge density will be an odd parity function, furnishing a trivial charge.

It is possible to construct new towers of quasi-conservation laws with true anomalies, meaning anomaly densities that exhibit odd parity under the symmetry transformation (136). To achieve this, one must seek conservation equations with charge densities of even parity, such as $\overline{u}u$, $i(\overline{u}{\partial }_{x}u-u{\partial }_x\overline{u})$, ${\partial }_x\overline{u}{\partial }_{x}u$, $i(\overline{u}{\partial }_x^{3}u-u{\partial }_x^3\overline{u})$, $i\overline{u}u(\overline{u}{\partial }_{x}u-u{\partial }_x\overline{u}),\dots$, among others.

Following this approach, an infinite hierarchy of new quasi-conservation laws was constructed in [17], each building upon monomials or polynomials of these forms. In [16,31], an infinite set of anomalous charges for the modified NLS equation was derived through a direct construction method, while [17] uncovered novel anomalous charges via the pseudo-potential approach.

These findings undoubtedly merit a more thorough investigation within the framework of (quasi-)integrability phenomena and, more broadly, in the context of soliton collision dynamics—an avenue that warrants further exploration in future work.

Additionally, formulating the modified AKNS Systems (130) and (131) as a linear system enables the construction of an infinite set of non-local conserved charges [17].

AKNS-type models are widespread in nonlinear science, making it worthwhile to explore the significance and physical implications of the infinite towers of anomalous and non-local charges. In recent years, remarkable and profound connections between integrable models and gauge theories have been uncovered. Notably, a form of triality has been proposed, linking gauge theories, integrable models, and gravity theories in certain UV regimes. In particular, the $(1+1)D$ nonlinear Schrödinger equation corresponds to the $2D \mathcal{N}={(2,2)}^{★}U\left(N\right)$ super Yang–Mills theory (see [42] and references therein).

5. Riccati-Type Pseudo-Potentials and Deformations of KdV

A specific deformation of the KdV model has been analyzed in [12] within the framework of the anomalous zero-curvature formulation. In connection with this, an alternative scheme involving a direct deformation of the model’s Hamiltonian structure was proposed in [30]. In the present work, we demonstrate that both of these frameworks can be systematically embedded within the unified Riccati-type pseudo-potential formalism developed in [14,15], wherein an infinite hierarchy of asymptotically conserved charges emerges.

A deformed KdV system can be defined as the Riccati-type Equations (1) and (2) with the following quantities [14,15]:

$$\begin{array}{ccc}\hfill A_0& =& U, A_1=-2\lambda , A_2=2, B_0=-4{\lambda }^{2}U-4U^2+2\lambda U_x-U_{xx}+Y+\chi ,\hfill \end{array}$$

(182)

$$\begin{array}{ccc}\hfill B_1& =& 2(4{\lambda }^3+4\lambda U-2U_x), B_2=-8({\lambda }^2+U),\hfill \end{array}$$

(183)

where the notation $U_x, U_{xx}$ stand for ${\displaystyle \frac{\partial U}{\partial x}},{\displaystyle \frac{{\partial }^{2}U}{\partial x^2}}$ and the space and time coordinates identifications $\xi =x, \eta =t$ have been made. Then, one has the system of Riccati-type equations [14],

$$\begin{array}{ccc}\hfill u_x& =& U+2u^2-2\lambda u,\hfill \end{array}$$

(184)

$$\begin{array}{ccc}\hfill u_t& =& -4{\lambda }^{2}U-4U^2-8({\lambda }^2+U)u^2+2u(4{\lambda }^3+4\lambda U-2U_x)+2\lambda U_x-U_{xx}+Y+\chi ,\hfill \end{array}$$

(185)

where the field $U(x,t,)$ is a KdV type field, $Y(x,t)$ encodes the deformation away from the KdV model, $u(x,t,\lambda )$ is a Riccati-type pseudo-potential, and $\chi (x,t,\lambda )$ is an auxiliary field satisfying

$$\begin{array}{c}\hfill {\partial }_x\chi +2(\lambda -2u)\chi =-2(\lambda -2u)Y.\end{array}$$

(186)

In the equations above, $\lambda$ plays the role of a spectral parameter. Equation (186) is a non-homogeneous ordinary differential equation for $\chi$ in the variable x, which can be integrated by quadratures.

Next, the compatibility condition (4)–(7) for Systems (184) and (185), i.e., $({\partial }_t{\partial }_{x}u-{\partial }_x{\partial }_{t}u)=0$, together with the auxiliary Equation (186), furnishes the equation

$$\begin{array}{c}\hfill U_t+12U{U}_x+U_{xxx}=Y_x.\end{array}$$

(187)

This is a deformed KdV equation. Note that for $Y=0$ and $\chi =0$, one has the Riccati Systems (184) and (185) approach for the KdV equation,

$$\begin{array}{c}\hfill U_t+12U{U}_x+U_{xxx}=0.\end{array}$$

(188)

Since Y can be regarded as an arbitrary functional of U and its derivatives—encompassing both local and non-local terms, as well as potential deformation parameters—Equation (187) represents a generalized deformation of the KdV model within the pseudo-potential framework. This deformation is characterized by a set of parameters $\left\{{ϵ}_i\right\}$, where setting ${ϵ}_i=0$ recovers the standard KdV model. Thus, the problem reduces to determining the existence of the auxiliary field $\chi$.

To define quasi-integrability in deformed KdV models, it is convenient to introduce some space–time symmetries related to soliton-type solutions. So, consider the space–time reflection around a fixed point $(x_{\Delta },t_{\Delta })$,

$$\begin{array}{c}\hfill \mathcal{P}:(\tilde{x},\tilde{t})\to (-\tilde{x},-\tilde{t}); \tilde{x}=x-x_{\Delta }, \tilde{t}=t-t_{\Delta }.\end{array}$$

(189)

The transformation $\mathcal{P}$ defines a shifted parity ${\mathcal{P}}_s$ for the x variable and the delayed time reversal ${\mathcal{T}}_d$ for the t variable. For the type of deformations Y satisfying

$$\begin{array}{ccc}\hfill \mathcal{P}\left(U\right)& =& U,\hfill \end{array}$$

(190)

$$\begin{array}{ccc}\hfill \mathcal{P}\left(Y\right)& =& Y,\hfill \end{array}$$

(191)

the deformed KdV model (187) will be considered quasi-integrable.

A notable feature of the Riccati-type approach outlined above is its capacity to incorporate more general deformations. In this framework, the field Y can depend on arbitrary functions of U and its derivatives, as well as on auxiliary fields. Among non-local deformations, an interesting avenue for exploration is the recently introduced Alice–Bob (AB) physics [43,44], particularly in the context of quasi-integrability.

Next, we analyze the anomalous conservation laws within the pseudo-potential framework. To this end, we consider the relevant (quasi-)conservation law expressed in terms of the pseudo-potential field u and the auxiliary field $\chi$. From (184) and (185), one can write the following equation (various expressions of this form can be constructed; here, we adopt a formulation where the non-homogeneous terms on the right-hand side explicitly involve the deformation variable $\chi$. This ensures that when $\chi =0(Y=0)$, the left-hand side reconstructs, order by order in ${\lambda }^{-1}$, the standard conservation laws of the usual KdV model.)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}u+{\displaystyle \frac{\partial }{\partial x}}(4{\lambda }^{2}u+4Uu-2\lambda U+U_x)=Y+\chi .\end{array}$$

(192)

From this point onward, we derive the relevant (quasi-)conservation laws in terms of the fields of the deformed KdV model. To this end, we consider expansions in powers of the $\lambda$ parameter,

$$\begin{array}{ccc}\hfill u& =& \sum _{n=0}{c}_n{\lambda }^{-n-1},\hfill \end{array}$$

(193)

$$\begin{array}{ccc}\hfill \chi & =& \sum _{n=1}{d}_n{\lambda }^{-n+1}.\hfill \end{array}$$

(194)

Substituting these expansions into (184) and (186), the first $c_n$’s and $d_n$’s take the form [14,15]

$$\begin{array}{ccc}\hfill c_0& =& {\displaystyle \frac{1}{2}}U,\hfill \\ \hfill c_1& =& -{\displaystyle \frac{1}{4}}{U}_x,\hfill \\ \hfill c_2& =& {\displaystyle \frac{1}{4}}(U^2+{\displaystyle \frac{1}{2}}{U}_{xx}),\hfill \\ \hfill c_3& =& -{\displaystyle \frac{1}{2}}(U{U}_x+{\displaystyle \frac{1}{8}}{U}_{xxx}),\hfill \\ \hfill c_4& =& {\displaystyle \frac{1}{32}}(8U^3+10U_x^2+12U{U}_{xx}+U_{xxxx}),\hfill \\ \hfill c_5& =& -{\displaystyle \frac{1}{64}}(64U^2{U}_x+36U_x{U}_{xx}+16U{U}_{xxx}+U_{xxxxx}).\hfill \end{array}$$

(195)

and

$$\begin{array}{ccc}\hfill d_1& =& -Y,\hfill \\ \hfill d_2& =& {\displaystyle \frac{1}{2}}{Y}_x,\hfill \\ \hfill d_3& =& -{\displaystyle \frac{1}{4}}{Y}_{xx},\hfill \\ \hfill d_4& =& {\displaystyle \frac{1}{8}}(4U{Y}_x+Y_{xxx}),\hfill \\ \hfill d_5& =& -{\displaystyle \frac{1}{16}}(8U_x{Y}_x+8U{Y}_{xx}+Y_{xxxx}),\hfill \\ \hfill d_6& =& {\displaystyle \frac{1}{32}}(24U^2{Y}_x+12U_{xx}{Y}_x+20U_x{Y}_{xx}+12U{Y}_{xxx}+Y_{xxxxx}).\hfill \end{array}$$

(196)

Note that by setting $Y=0$ and $\chi =0$ on the right-hand side of Equation (192), it transforms into a strictly exact conservation law. This allows for the construction of an infinite hierarchy of exact conservation laws for the standard KdV equation. Next, by utilizing the power series expansions of u (193) and $\chi$ (194) in terms of ${\lambda }^{-1}$, and substituting them into Equation (192), one obtains a polynomial in powers of ${\lambda }^{-1}$ for $c_n$’s and $d_n$’s. Considering the expressions for $c_n$’s and $d_n$’s, it follows that the first two terms in this series ($n=-1,0$) yield trivial equations. Similarly, for $n\ge 1$, one can write

$$\begin{array}{c}\hfill {\partial }_t\left(c_{n-1}\right)+{\partial }_x\left(4c_{n+1}+4U{c}_{n-1}\right)=d_{n+1}, n=1,2,3....\end{array}$$

(197)

This is an infinite set of quasi-conservation laws for the deformed KdV model (187) in the Riccati-type pseudo-potential approach.

A careful examination of the conservation laws corresponding to even-order terms in ${\lambda }^{-n}, n=2,4,6$ reveals that these equations are merely the $x–$derivatives of the conservation laws associated with $n=1,3,5$, respectively. Consequently, they do not introduce any new conservation laws. Then, we consider below the first three non-trivial cases for $n=1,3,5$.

First order ($n=1$)

$$\begin{array}{c}\hfill {\partial }_t\left({\displaystyle \frac{1}{2}}U\right)-{\partial }_x\left(4c_2+4U{c}_0-{\displaystyle \frac{2}{2}}Y\right)=0.\end{array}$$

(198)

Then, taking into account (208), one can define the charge

$$\begin{array}{c}\hfill q^{\left(1\right)}={\displaystyle \frac{\alpha }{12}}{\int }_{-\infty }^{+\infty }dx U.\end{array}$$

(199)

This charge is typically linked to the “mass”. Because it arises from an exact conservation law, it remains conserved even in the deformed KdV model (212).

Third order ($n=3$)

$$\begin{array}{c}\hfill {\partial }_t[{\displaystyle \frac{1}{4}}{U}^2+{\displaystyle \frac{1}{3}}{U}_{xx}]+{\partial }_x(4c_3+U{c}_2)={\displaystyle \frac{1}{2}}U{Y}_x+{\displaystyle \frac{1}{8}}{\partial }_x\left(Y_{xx}\right).\end{array}$$

(200)

A remarkable fact is that the r.h.s. of (200), using the equation of motion (187), can be written as

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}U{Y}_x+{\displaystyle \frac{1}{8}}{\partial }_x\left(Y_{xx}\right)={\partial }_t\left({\displaystyle \frac{1}{4}}{U}^2\right)+{\partial }_x[2U^3+{\displaystyle \frac{1}{2}}U{U}_{xx}-{\displaystyle \frac{1}{4}}{\left(U_x\right)}^2].\end{array}$$

(201)

Substituting the last identity into Equation (151) yields a trivial identity, resulting in a vanishing charge density. Consequently, the charge at this order is trivial,

$$\begin{array}{c}\hfill q^{\left(3\right)}=0.\end{array}$$

(202)

However, following the developments in [12,14,15], at this order, one can define the quasi-conservation law

$$\begin{array}{c}\hfill {\displaystyle \frac{d{q}_a^{\left(3\right)}}{dt}}={\alpha }^{\left(3\right)},\end{array}$$

(203)

where

$$\begin{array}{c}\hfill q_a^{\left(3\right)}\equiv {\displaystyle \frac{1}{4}}{\int }_{-\infty }^{+\infty }dx U^2, {\alpha }^{\left(3\right)}\equiv {\int }_{-\infty }^{+\infty }dx {\displaystyle \frac{1}{2}}U{Y}_x\end{array}$$

(204)

is the asymptotically conserved charge $q_a^{\left(3\right)}$, with ${\alpha }^{\left(3\right)}$ being its relevant anomaly. The space–time integral ${\int }_{-\infty }^{+\infty }dt {\alpha }^{\left(3\right)}$ vanishes for field configurations satisfying the symmetry (189)–(191), rendering the charge $q_a^{\left(3\right)}$ an asymptotically conserved charge, i.e., $q_a^{\left(3\right)}(-\infty )=q_a^{\left(3\right)}(+\infty ).$

Fifth order ($n=5$)

At this order, one can write

$$\begin{array}{ccc}\hfill {\partial }_t[{\displaystyle \frac{1}{4}}{U}^3-{\displaystyle \frac{1}{16}}{U}_x^2+{\partial }_x({\displaystyle \frac{3}{8}}U{U}_x+{\displaystyle \frac{1}{32}}{U}_{xxx})]+{\partial }_x(4c_6+4U{c}_4)& =& {\partial }_t({\displaystyle \frac{1}{4}}{U}^3-{\displaystyle \frac{1}{16}}{U}_x^2)+{\partial }_x[{\displaystyle \frac{1}{8}}{U}_x{U}_t+\hfill \\ & & {\displaystyle \frac{3}{4}}{U}^2{U}_{xx}+{\displaystyle \frac{1}{16}}{U}_{xx}^2].\hfill \end{array}$$

(205)

Therefore, one also has a trivially vanishing charge at this order. Following [12,14,15], one can define the quasi-conservation law

$$\begin{array}{c}\hfill {\displaystyle \frac{d{q}_a^{\left(5\right)}}{dt}}={\alpha }^{\left(5\right)},\end{array}$$

(206)

where

$$\begin{array}{c}\hfill q_a^{\left(5\right)}\equiv {\int }_{-\infty }^{+\infty }dx[{\displaystyle \frac{1}{4}}{U}^3-{\displaystyle \frac{1}{16}}{U}_x^2], {\alpha }^{\left(5\right)}\equiv {\int }_{-\infty }^{+\infty }dx ({\displaystyle \frac{3}{4}}{U}^2+{\displaystyle \frac{1}{8}}{U}_{xx})Y_x,\end{array}$$

(207)

are the asymptotically conserved charge $q_a^{\left(5\right)}$ and its relevant anomaly ${\alpha }^{\left(5\right)}$. This charge maintains the same form as in the usual KdV at this order. Similarly, as above, the space–time integral ${\int }_{-\infty }^{+\infty }dt {\alpha }^{\left(5\right)}$ vanishes for solutions with the symmetry (189)–(191), such that the charge $q_a^{\left(5\right)}$ becomes an asymptotically conserved charge, i.e., $q_a^{\left(5\right)}(-\infty )=q_a^{\left(5\right)}(+\infty ).$

A similar construction can be performed for the higher-order asymptotically conserved charges. Note that in the ordinary KdV, i.e., when the anomaly $Y=0$, the charges of (199), (204) and (207) are usually associated with “mass”, “momentum”, and “energy” conservation, respectively.

Particular Deformations of the KdV

A particular deformation of the KdV model has been considered [12,14,15] by making the identification

$$\begin{array}{ccc}\hfill U& \equiv & {\displaystyle \frac{\alpha }{12}}V+{\displaystyle \frac{\alpha }{144}},\hfill \end{array}$$

(208)

$$\begin{array}{ccc}\hfill Y& =& -{\displaystyle \frac{\alpha }{12}}[{\displaystyle \frac{\alpha }{4}}{ϵ}_2{w}_x{v}_t-{\displaystyle \frac{12{ϵ}_1}{\alpha }}(V_{xt}+V_{xx})],\hfill \end{array}$$

(209)

where Y is defined such that the auxiliary fields satisfy

$$\begin{array}{ccc}\hfill V& =& w_t,\hfill \end{array}$$

(210)

$$\begin{array}{ccc}\hfill V& =& v_x.\hfill \end{array}$$

(211)

So, substituting into (187), one gets the equation [12,14,15]

$$\begin{array}{c}\hfill V_t+V_x+{\left[{\displaystyle \frac{\alpha }{2}}{V}^2+{ϵ}_2{\displaystyle \frac{\alpha }{4}}{w}_x{v}_t+V_{xx}-{ϵ}_1(V_{xt}+V_{xx})\right]}_x=0.\end{array}$$

(212)

This is a deformation of the KdV model. Note that for ${ϵ}_1={ϵ}_2=0$, one recovers the usual KdV model. The real parameters ${ϵ}_1$ and ${ϵ}_2$ serve as deformation parameters that modify the standard KdV equation, while $\alpha$ is an arbitrary real parameter. The formalism imposes no constraint on the magnitudes of the deformation parameters $\{{ϵ}_1,{ϵ}_2\}$; they may take arbitrary values without compromising validity. The soliton-like behavior in these deformed models arises from the underlying quasi-integrable structure rather than perturbative expansions around small deformation parameters. The soliton-like behavior should be contrasted with that of genuine solitons, which preserve both their velocity and spatial profile following interactions, with the only measurable consequence of the collision being a phase shift.

The model (212) encompasses various sub-models. Specifically, when ${ϵ}_1={ϵ}_2=0$, the equation reduces to the integrable KdV model. Setting ${ϵ}_1=1,{ϵ}_2=0$ corresponds to the regularized long wave (RLW) equation, which is non-integrable but admits a one-soliton solution. Although two- and three-soliton solutions for the RLW model have been constructed numerically, their analytical forms remain unknown.

In contrast, when ${ϵ}_1={ϵ}_2=1$, the equation describes the modified regularized long wave (mRLW) equation, which exhibits the notable property of possessing exact analytical two-soliton solutions. Furthermore, for ${ϵ}_2=0, ϵ\ne \{0,1\}$, the model corresponds to the KdV-RLW or Korteweg–de Vries–Benjamin–Bona–Mahony (KdV-BBM) type equations.

So, the Riccati-type pseudo-potential approach includes the model (212) considered in [12,14,15], and it is suitable to explore a deformation of the KdV equation that includes non-local terms such as Y in (209).

To illustrate the vanishing anomaly property and the asymptotically conserved charge, let us take the third-order quasi-conservation law (200) and its associated asymptotically conserved charge $q_a^{\left(3\right)}$ in (203), adapted to the deformed model (212). So, one must take into account the identifications (208) and (209) and (210) and (211). So, let us take the time integral of the quasi-conservation law (203) as

$$\begin{array}{ccc}\hfill {\int }_{-\infty }^{+\infty }dt{\displaystyle \frac{d{q}_a^{\left(3\right)}}{dt}}& =& {\int }_{-\infty }^{+\infty }dt{\alpha }^{\left(3\right)},\hfill \end{array}$$

(213)

$$\begin{array}{ccc}\hfill & =& {\int }_{-\infty }^{+\infty }dt{\int }_{-\infty }^{+\infty }dx {\displaystyle \frac{1}{2}}U{Y}_x\hfill \end{array}$$

(214)

$$\begin{array}{ccc}& =& -{\int }_{-\infty }^{+\infty }dt{\int }_{-\infty }^{+\infty }dx {\displaystyle \frac{1}{2}}{U}_{x}Y,\hfill \end{array}$$

(215)

where the line (215) is achieved after an integration by parts and assuming the boundary condition $U\left(\right|x|=+\infty )=0$. Next, taking into account (208) and (209) the last identity (215) can be written as

$$\begin{array}{ccc}\hfill q_a^{\left(3\right)}(t=+\infty )-q_a^{\left(3\right)}(t=-\infty )& =& {\int }_{-\infty }^{+\infty }dt{\int }_{-\infty }^{+\infty }dx\left[{\displaystyle \frac{{\alpha }^3}{3^2{2}^7}}{ϵ}_2{V}_x{w}_x{v}_t-{\displaystyle \frac{\alpha }{32^4}}{ϵ}_1({\partial }_t{V}_x^2+{\partial }_x{V}_x^2)\right]\hfill \end{array}$$

(216)

$$\begin{array}{ccc}& =& {\displaystyle \frac{{\alpha }^3{ϵ}_2}{3^2{2}^7}}{\int }_{-\infty }^{+\infty }dt{\int }_{-\infty }^{+\infty }dx V_x{w}_x{v}_t,\hfill \end{array}$$

(217)

where the last two terms in the r.h.s. of (216), being total derivatives with respect to space and time, respectively, vanish upon the integral evaluation under the assumptions $V_x^2\left(\right|x|=\infty ,t)=0, V_x^2(x,|t|=\infty )=0$.

Next, let us analyze the symmetries of the fields. The field configurations satisfying the space–time symmetry (189) with even parity in (191), for the model (212) implies

$$\begin{array}{c}\hfill \mathcal{P}\left(V\right)=V, \mathcal{P}\left(w\right)=-w, \mathcal{P}\left(v\right)=-v.\end{array}$$

(218)

Since the integrand in Equation (217) is odd under the transformation (218), the space–time integral on the right-hand side of (217) vanishes when evaluated over the entire space–time domain:

$$\begin{array}{c}\hfill {\int }_{-\infty }^{+\infty }dt{\int }_{-\infty }^{+\infty }dx V_x{w}_x{v}_t=0.\end{array}$$

(219)

So, taking into account (219), from (217), one can write

$$\begin{array}{c}\hfill q_a^{\left(3\right)}(t=+\infty )=q_a^{\left(3\right)}(t=-\infty ).\end{array}$$

(220)

This demonstrates that the charge $q_a^{\left(3\right)}$ is asymptotically conserved, attaining the same constant value in the remote past and far future, despite potential variations during the intermediate interaction. Below, we will numerically verify that $q_a^{\left(3\right)}$ remains effectively constant throughout the entire two-soliton scattering process, up to numerical precision.

Let us consider some numerical results for the model (212). Here, we follow the finite difference method presented in [14]. In Figure 1, we plot the numerical result for 2-soliton collision with parameter values ${ϵ}_1=1.2,{ϵ}_2=0.9,\alpha =4$ of the deformed KdV model (212). In Figure 2, we show the results of the numerical simulations of the quantities related to the anomaly density in (219): (1) The top figure shows the anomaly density $\left(w_x{v}_t{V}_x\right) \mathrm{vs} x$ for three successive times. (2) The middle plot shows ${\int }_0^{L}dx\left(w_x{v}_t{V}_x\right) \mathrm{vs} t$. (3) The bottom plot shows ${\int }_{t_o}^{t}dt{\int }_0^{L}dx\left(w_x{v}_t{V}_x\right) \mathrm{vs} t$. Note that the bottom plot shows the vanishing of the space–time integrated anomaly, with numerical accuracy in the order of ${10}^{-9}$. Therefore, it proves that (220) holds effectively with an accuracy in the order of ${10}^{-9}$.

Another approach to modifying the KdV equation in the context of quasi-integrability has been proposed by a direct deformation of its Hamiltonian in the Hamiltonian formulation of the model [30]. The deformed KdV (187), under $U\to {\displaystyle \frac{1}{2}}U, Y\to {\displaystyle \frac{1}{2}}Y$, becomes $U_t+6U{U}_x+U_{xxx}=Y_x$. So, the identification

$$\begin{array}{c}\hfill Y\equiv 2U^2+{\displaystyle \frac{2}{3}}[{\displaystyle \frac{\delta H_1}{\delta U}}+U_{xx}],\end{array}$$

(221)

with $H_1$ being the deformed Hamiltonian, defines the deformed KdV in this approach. The undeformed Hamiltonian $H_1^{undef}$ is defined as

$$\begin{array}{c}\hfill H_1^{undef}\left[U\right]={\int }_{-\infty }^{+\infty }({\displaystyle \frac{1}{2}}{U}_x^2-U^3).\end{array}$$

(222)

Note that for the undeformed $H_1^{undef}$ (222), one has $Y=0$ in (221). An explicit power modification of the nonlinear term in (222) as $U^{(3+3ϵ)}$, with parameter $ϵ$, has been discussed in [30]. So, the Riccati-type approach, which is suitable for deforming the both nonlinear and dispersive terms, may also include the Hamiltonian deformation method to tackle quasi-integrable KdV models.

Furthermore, using the Riccati-type pseudo-potential approach, a linear system of equations corresponding to the deformed KdV model (187) has been constructed. Within this linear formulation, an infinite set of non-local charges has also been identified [14,15].

Furthermore, given the widespread presence of KdV-type models across various areas of nonlinear science, it would be valuable to explore the significance and physical implications of the tower of asymptotically conserved charges identified above. For instance, a well-established connection exists between gravitation in three-dimensional spacetime and two-dimensional integrable systems. Notably, the KdV-type and KdV-Gardner models have recently been recognized as governing the dynamics of the boundary degrees of freedom in General Relativity on AdS₃ (see, e.g., [45] and references therein).

With regard to the asymptotically conserved charges, it is important to highlight that [14] presents an analytical—rather than a purely numerical—proof of the quasi-integrability of the well-known non-integrable modified regularized long-wave (mRLW) theory, corresponding to the model defined in Equation (212) with ${ϵ}_1={ϵ}_2=1$. Specifically, it has been demonstrated that the two-soliton solution of the mRLW theory, when expressed in a $\mathcal{P}$-invariant form, exhibits quasi-integrable behavior in an analytical sense. This quasi-integrability applies to all anomalous charges associated with the infinite towers of quasi-conservation laws discussed earlier.

The dynamics of soliton-like configurations in quasi-integrable models remain largely unexplored. Key findings are as follows: (1) One-soliton sectors possess an infinite set of anomalous conserved charges, due to vanishing space–time integrals of anomaly densities. (2) For two-soliton type solutions possessing the symmetry $\mathcal{P}$, the charges are asymptotically conserved. This means that these quantities vary in time during the collision process (and sometimes can vary quite a lot) of two one-solitons but return, in the distant future (after the collision), to the values they had in the remote past (before the collision). (3) For multi-soliton configurations with widely separated solitons, the anomalies also integrate to zero, with non-zero contributions localized to interaction regions. (4) A sufficient condition for vanishing integrated anomalies is the presence of definite parity (even/odd) under the shifted parity and delayed time-reversal symmetry $\mathcal{P}$. Anomaly densities odd under this symmetry yield asymptotically conserved charges. (5) Multiple infinite towers of anomalous charges have been identified, expanding on earlier anomalous Lax results [1,5,7,38] with new contributions [13,14,16,17] for deformations of SG, KdV, and NLS models based on Riccati-type formulations. (6) Certain deformed models exhibit subsets of infinite anomalous charges for soliton eigenstates invariant under shifted space-reflection symmetry ${\mathcal{P}}_s$ (${\mathcal{P}}_s\subset \mathcal{P}$), as seen in deformed focusing/defocusing NLS models and deformed sine-Gordon models for two-soliton and breather solutions [7,8,10,38]. These results derive from combined analytical and numerical approaches.

In quasi-integrable models, the soliton boundary conditions studied in the literature fall into two main categories: (1) Vanishing boundary conditions at $x=\pm \infty$, associated with bright solitons in focusing modified NLS models [5,10], breather solutions in deformed sine-Gordon (SG) models [1,2,7,38], and solitons in modified KdV models [12,14]. (2) Non-vanishing boundary conditions at $x=\pm \infty$, relevant to dark solitons in defocusing modified NLS models [8,31], kink solutions in deformed SG models [1,7,38], and kink solitons in deformed potential KdV (pKdV) models [29]. Alternative boundary conditions, such as periodic or open, remain underexplored and warrant further investigation in the context of quasi-integrability.

6. Conclusions and Discussions

We have reviewed the quasi-integrability framework developed in [12,14,15,16,17,29,31] for deformations of the sine-Gordon (SG), nonlinear Schrödinger (NLS), and Korteweg–de Vries (KdV) integrable models, refining and unifying key arguments from these works. Our unified approach is based on the introduction of specific deformation fields, denoted as $r_j$ and X in Section 2, into the Riccati-type pseudo-potential system associated with these integrable models. These deformation fields modify the underlying structure of the equations while preserving essential quasi-integrability properties, such as the existence of infinite towers of anomalous and non-local conservation laws. This formulation provides a systematic way to extend integrability concepts to deformed models, offering new insights into their soliton dynamics and conserved quantities.

We demonstrate that deforming the Riccati-type pseudo-potential equations away from the SG model generates infinite towers of quasi-conservation laws for deformed sine-Gordon models of type (8). The first two conserved charges correspond to energy and momentum, while a related linear system enables the construction of an infinite set of non-local conservation laws. Additionally, through direct construction, further towers of quasi-conservation laws have been identified in [13].

In Section 4, we applied the Riccati-type pseudo-potential approach to deformations of the AKNS model, obtaining the modified NLS (MNLS) through a specific reduction. This framework enabled the construction of infinite towers of quasi-conservation laws related to the MNLS model. Our approach reproduced the NLS-type quasi-conserved charges derived via the anomalous zero-curvature method in [5]. We introduce a dual Riccati-type pseudo-potential approach, revealing a new infinite set of quasi-conservation laws that encompass those obtained using a direct method from the equations of motion in [16,31].

In Section 5, we applied the pseudo-potential approach to KdV model deformations, demonstrating that modifying the Riccati-type pseudo-potential equations produces infinite towers of quasi-conservation laws for general KdV deformations (187). This framework allowed us to construct an infinite set of quasi-conservation laws, recovering the KdV-type quasi-conservation laws derived via the anomalous zero-curvature method in [13]. Additionally, we showed that the recently proposed Hamiltonian deformation approach to quasi-integrability [30] aligns with our general pseudo-potential framework.

Partial differential equations have been interpreted as infinite-dimensional manifolds, leading to the introduction of differential coverings. These coverings have been employed to construct various structures, such as Lax pairs and Bäcklund transformations (see, e.g., refs. [46,47,48,49] ). Notably, an auto-Bäcklund transformation corresponds to an automorphism of the covering. Therefore, it would be valuable to explore the properties of the Riccati-type Systems (1) and (2) as a form of differential covering for the corresponding modified integrable system.

An interesting direction for future research is the dynamics of kinks and bound-state solutions in soliton–fermion quasi-integrable models, such as the deformed SUSY-SG [11]. Notably, the integrable affine Toda model coupled to fermions (ATM) was recently studied in [50], revealing strong-weak duality in soliton–fermion sectors. This investigation could also have significant implications for one-dimensional topological superconductors [51], where understanding the impact of interactions on the shape and lifetime of bound states is crucial. In particular, recent proposals to identify Majorana modes through local integrals of motion in interacting systems [52] highlight the relevance of this study. Therefore, further analysis of quasi-integrable versions of the ATM model is warranted.

Furthermore, the relationship between quasi-integrability and non-Hermiticity deserves further investigation. The non-Hermitian affine Toda model coupled to fermions was studied in [53] using soliton theory techniques to explore pseudo-chiral and pseudo-Hermitian symmetries. The interplay between non-Hermiticity, quasi-integrability, nonlinearity, and topology could play a crucial role in the formation and dynamics of exotic quantum states, with potential applications in topological quantum computation. Additionally, studying quasi-integrability and non-Hermitian solitons may offer new insights into the theory of topological phases of matter.

Recent studies have explored non-Hermitian extensions of certain integrable models. This direction presents a compelling avenue for future investigation within the framework of quasi-integrability. In principle, the Riccati-type pseudo-potential method is applicable to a broad class of integrable systems, provided that their standard Riccati formulation is established. However, the analysis of novel quasi-integrable features emerging from such non-Hermitian deformations lies beyond the scope of the present review.

Finally, a deeper investigation is needed into the physical nature and dynamics of deformed solitons, particularly their behavior in collision regions and the underlying mechanisms governing anomaly cancellation to restore exact conservation laws, as suggested in [16,29,31]. Understanding how these solitons interact and evolve under deformations could provide further insights into quasi-integrability and its broader implications. Additionally, exploring potential applications of these findings in nonlinear physics, such as in Bose–Einstein condensation [54,55], superconductivity [56,57], soliton turbulence [58,59,60,61,62,63], and soliton gas [64], remains an open avenue for research.