Optical Solitons, Optimal Systems and Conserved Quantities of the Schrödinger Equation with Spatio-Temporal and Inter-Modal Dispersions

Abstract

In this study, we present a unified symmetry-conservation solution analysis of a well-posed resonant nonlinear Schrödinger (NLS)-type equation incorporating spatio-temporal dispersion and inter-modal dispersion. Working within the truncated M-fractional derivative framework, we first construct exact traveling-wave solution families via the Kudryashov expansion method, together with the corresponding parameter constraints and limiting cases. We then determine the admitted Lie point symmetries and establish the associated Lie algebra, including the commutator structure, adjoint representation, and an optimal system of one-dimensional subalgebras for classification. Using the conservation theorem, we derive conserved vectors associated with the fundamental invariances of the model; in the NLS setting and under suitable conditions, these quantities can be interpreted as generalized power (mass), momentum, and energy-type invariants. Overall, the results provide explicit wave profiles and structural invariants that enhance the interpretability of the model and offer benchmark expressions useful for further qualitative, numerical, and stability investigations in nonlinear dispersive wave dynamics.

1. Introduction

Optical solitons constitute one of the rapidly expanding research directions in optoelectronics and nano-electronics due to their remarkable ability to preserve localized pulse shapes during propagation. Their importance is closely tied to modern optical technologies, including nonlinear optical fibers, birefringent media, and engineered metamaterials, where stable solitary structures arise through a delicate balance between dispersion and nonlinearity. Representative studies on dark/singular solitons and soliton-like structures in Schrödinger-type settings are reported in [1,2,3]. Foundational monographs that establish the physical background of nonlinear fiber propagation and non-Kerr optical solitons can be found in [4,5]. Additional explicit soliton constructions and traveling-wave formulations in related optical models are presented in [6,7,8]. Soliton solutions under generalized dispersive effects and parabolic- or dual-power law media are discussed in [9,10]. Symmetry-based modeling and analysis perspectives relevant to nonlinear wave equations are illustrated in [11,12]. Computational and fractional-method references that support the analytical framework used in such contexts are given in [13,14]. From a mathematical and physical standpoint, nonlinear Schrödinger (NLS)-type equations have become fundamental models for describing nonlinear dispersive wave phenomena in a wide range of contexts such as nonlinear optics, plasma physics, fluid dynamics, and Bose–Einstein condensates. Representative traveling-wave and first-integral-type approaches for nonlinear evolution models are presented in [15,16,17]. Transform-based and equation-reduction techniques used in nonlinear wave theory are discussed in [18,19]. Closely related soliton dynamics and conservation-law studies in dispersive nonlinear systems can be found in [20,21]. Resonant and non-Kerr optical-soliton models, including settings with additional dispersive effects, are reported in [22,23]. Further exact-solution schemes based on Riccati-/trial-function and projective methods are developed in [24,25,26,27]. Additional traveling-wave solution constructions for nonlinear evolution equations are provided in [28]. Unlike the linear Schrödinger equation, the nonlinear counterpart supports rich dynamical behaviors, including self-focusing, modulational instability, and the emergence of coherent structures. In addition to constructing explicit wave patterns, it is often crucial to reveal the structural properties of the governing equations. Lie symmetry analysis provides a systematic framework for identifying invariances, generating similarity reductions, and organizing solution families in a classification-oriented manner [29,30,31,32]. Complementarily, conservation laws supply invariant integral constraints of the evolution, which are useful in qualitative analysis and in validating analytical or numerical computations. While Noether’s theorem is traditionally associated with variational systems, modern conservation-law approaches allow systematic constructions for broader classes of PDEs, including non-variational models, thereby expanding the applicability of conservation analysis in nonlinear science [33,34,35].

It is known that the nonlinear dynamics of optical solitons (and related hydrodynamic-type formulations such as Madelung fluids) can be described by resonant nonlinear Schrödinger-type equations. In particular, resonant terms have been considered relevant in the modeling of chiral solitons arising in the quantum Hall effect. Moreover, it has been emphasized in the literature that the governing equation for optical-soliton propagation becomes well-posed when additional spatio-temporal dispersion (STD) is incorporated; in the presence of inter-modal dispersion (IMD), this leads to a well-posed resonant NLS-type model as adopted in [22]. Motivated by this established modeling framework, we consider the following nonlinear dynamical equation for the complex wave envelope $q(x,t)$:

$$i{q}_t+\alpha q_{xx}+\beta q_{xt}+{dF\left(\right|q|}^2)q+\gamma \left({\displaystyle \frac{{\left|q\right|}_{xx}}{\left|q\right|}}\right)q-i\sigma q_x=0,i=\sqrt{-1}.$$

(1)

Here, x and t denote the spatial and temporal variables, respectively, and $q(x,t)$ is the complex wave profile. The coefficients $\alpha$ and $\beta$ correspond to group-velocity dispersion and spatio-temporal dispersion (STD), respectively, while $\sigma$ characterizes the inter-modal dispersion (IMD). The nonlinear response is described by the term $F\left(\right|q{|}^2)q$, where F is assumed to be a real-valued sufficiently smooth function, and the parameters $\gamma$ and d control, respectively, the strength of the resonant nonlinear contribution and the non-Kerr-type correction appearing in Equation (1). In this study, Equation (1) is investigated within the framework of the truncated M-fractional derivative. This setting enables an analytical examination of the fractional-order dynamics and the associated structural features of the model (symmetries, conservation laws, and exact solution structures).

This paper aims to provide a unified analysis of the considered resonant NLS-type model in terms of both structural features and explicit solution content. The main outcomes can be summarized as follows: (i) we obtain explicit families of exact traveling-wave solutions by means of the Kudryashov expansion method (KEM), together with parameter conditions and relevant limiting cases; (ii) we determine the Lie point symmetries and compute the associated algebraic structure (commutators and adjoint representation) needed for classification; and (iii) we construct the optimal system of one-dimensional subalgebras and derive conservation laws, thereby complementing the solution results with structural invariants. This combined set of results is intended to improve the interpretability of the model and to provide benchmark expressions useful for further analytical investigation and validation in related nonlinear dispersive systems.

The remainder of the paper is organized as follows. Section 2 introduces the truncated M-fractional derivative and fixes the notation used throughout the manuscript. Section 3 presents the analytical-solution framework: Section 3.1 outlines the Kudryashov expansion method (KEM), and Section 3.2 reports the resulting exact solution families for the truncated M-fractional form of the model. In Section 4, we carry out the Lie symmetry analysis, including the Lie bracket computations, the nonzero commutators (together with the remaining brackets), and the adjoint representation. Section 5 constructs the optimal system of one-dimensional subalgebras and derives the associated conservation laws (Section 5.3). Finally, Section 6 concludes the paper and briefly discusses the significance of the results and possible directions for further work.

2. The Truncated $\mathit{M}$-Fractional Derivative

2.1. Definition and Basic Properties

In this section, we recall the basic definitions and properties of the truncated M-fractional derivative needed in the subsequent analysis [36]. The motivation for employing a fractional formulation is that wave propagation in nonlinear dispersive media may exhibit memory and hereditary effects (i.e., nonlocal response in time and/or space), which may not be adequately captured by purely integer-order models. In the present study, the truncated M-fractional framework introduces additional flexibility through the fractional order $\delta$ (and the parameter $\eta$ ), enabling a mathematically tractable treatment of fractional-order behavior within the considered resonant NLS-type model. In particular, the classical model is recovered in the limit $\delta \to 1$, whereas $0<\delta <1$ represents deviations associated with fractional-order dynamics. Accordingly, Equation (1) is investigated in its truncated M-fractional form (see Equation (10)), and the analytical developments in Section 3 are carried out within this setting.

Definition 1.

If we let $h:[0,\infty )\to \mathbb{R}$, then the new M-fractional derivative of h with order δ is

$${\mathcal{D}}_M^{\delta ,\eta }\left\{\left(h\right)\left(t\right)\right\}=\underset{ϵ\to 0}{lim}{\displaystyle \frac{h\left(t{\mathbb{E}}_{\eta }\left(ϵt^{1-\delta }\right)\right)-h\left(t\right)}{ϵ}},\forall t>0,0<\delta <1,\eta >0,$$

(2)

where ${\mathbb{E}}_{\eta }(.)$ is a truncated Mittag–Leffler function of one parameter [36].

Theorem 1

(see [36]). Let $0<\delta \le 1$, $\eta >0$, $q,r\in \mathbb{R}$, and $g,h$ δ-differentiable at a point $t>0$. Then:

1. 

${\mathcal{D}}_M^{\delta ,\eta }\left\{(qg+rh)\left(t\right)\right\}=q{\mathcal{D}}_M^{\delta ,\eta }\left\{g\left(t\right)\right\}+r{\mathcal{D}}_M^{\delta ,\eta }\left\{h\left(t\right)\right\}$.

2. 

${\mathcal{D}}_M^{\delta ,\eta }\left\{(g.h)\left(t\right)\right\}=g\left(t\right){\mathcal{D}}_M^{\delta ,\eta }\left\{h\left(t\right)\right\}+h\left(t\right){\mathcal{D}}_M^{\delta ,\eta }\left\{g\left(t\right)\right\}$.

3. 

${\mathcal{D}}_M^{\delta ,\eta }\left\{\frac{g}{h}\left(t\right)\right\}={\displaystyle \frac{h\left(t\right){\mathcal{D}}_M^{\delta ,\eta }\left\{g\left(t\right)\right\}-g\left(t\right){\mathcal{D}}_M^{\delta ,\eta }\left\{h\left(t\right)\right\}}{{\left[h\left(t\right)\right]}^2}}$.

4. 

${\mathcal{D}}_M^{\delta ,\eta }\left\{c\right\}=0$, where $g\left(t\right)=c$ is a constant.

5. 

${\mathcal{D}}_M^{\delta ,\eta }\left\{g\left(t\right)\right\}={\displaystyle \frac{t^{1-\delta }}{\mathrm{\Gamma }(\eta +1)}}{\displaystyle \frac{dg\left(t\right)}{dt}}$ if g is differentiable.

2.2. Regularity Assumptions on the Nonlinear Term

Soliton structures arise from a balance between dispersion and nonlinearity. In Equation (1) (and its truncated M-fractional counterpart), the regularity of the nonlinear mapping $F\left(\right|q{|}^2)q:C⟶C$ is required. Identifying C with $R^2$, we assume that $F\left(\right|q{|}^2)q$ is k-times continuously differentiable (locally) so that the analytical manipulations in Section 3 are justified. In particular, we may express this assumption in the form

$${F\left(\right|q|}^2)q\in \sum _{m,n=1}^{\infty }{C}^k((-n,n)\times (-m,m);{\mathrm{I}\mathrm{R}}^2).$$

(3)

3. Analytical Solutions

3.1. Analysis of the KEM

In this section, the Kudryashov expansion method (KEM) is summarized and then applied to the truncated M-fractional model. Consider a general local truncated M-fractional partial differential equation of the form

$$P(q,{\mathcal{D}}_{M,x}^{\delta ,\eta }q,{\mathcal{D}}_{M,x}^{2\delta ,\eta }q,{\mathcal{D}}_{M,t}^{\delta ,\eta }q,{\mathcal{D}}_{M,t}^{\delta ,\eta }{\mathcal{D}}_{M,x}^{\delta ,\eta }q,\dots )=0.$$

(4)

To obtain traveling-wave solutions, we use the transformation

$$q(x,t)=K\left(\xi \right),\xi ={\displaystyle \frac{\mathrm{\Gamma }(\eta +1)}{\delta }}\mu (x^{\delta }-\rho t^{\delta }),$$

(5)

where $K\left(\xi \right)$ is an unknown function, $\mu \ne 0$ is a constant, and $\rho$ denotes the wave speed. Substituting (5) into (4) reduces the governing equation to a nonlinear ordinary differential equation

$$D(K,K^{\prime },K^{″},\dots )=0,$$

(6)

where the prime denotes differentiation with respect to $\xi$.

In KEM, the solution is assumed in the polynomial form

$$\begin{array}{c}\hfill K\left(\xi \right)=\sum _{r=0}^N{A}_r{Q}^r\left(\xi \right),A_N\ne 0,\end{array}$$

(7)

where $A_r$ are constants to be determined and $Q\left(\xi \right)$ satisfies an auxiliary first-order equation. In this work, $Q\left(\xi \right)$ is taken to satisfy the following equation:

$$\begin{array}{c}\hfill Q^{\prime }=\varpi Q(Q-1).\end{array}$$

(8)

Solving (8) by separation of variables yields

$$\begin{array}{c}\hfill Q\left(\xi \right)={\displaystyle \frac{1}{1+\zeta e^{\varpi \xi }}}.\end{array}$$

(9)

The positive integer N in (7) is determined by balancing the highest-order derivative term with the highest-power nonlinear term in (6). Substituting (7) and the derivatives generated via (8) into (6) produces a polynomial in powers of $Q\left(\xi \right)$. Equating the coefficients of like powers of Q to zero gives an algebraic system for $A_r$ (and possible parameter relations), whose solution provides explicit, exact traveling-wave solutions of the original truncated M-fractional PDE.

3.2. Solutions

Here, we construct analytical traveling-wave solutions of the truncated M-fractional resonant nonlinear Schrödinger equation given by

$$i({\mathcal{D}}_{M,t}^{\delta ,\eta }q-\sigma {\mathcal{D}}_{M,x}^{\delta ,\eta }q)+\alpha {\mathcal{D}}_{M,x}^{2\delta ,\eta }q+\beta {\mathcal{D}}_{M,x}^{\delta ,\eta }{\mathcal{D}}_{M,t}^{\delta ,\eta }q+{dF\left(\right|q|}^2)q+\gamma \left({\displaystyle \frac{{\mathcal{D}}_{M,x}^{2\delta ,\eta }\left|q\right|}{\left|q\right|}}\right)q=0,$$

(10)

Consider the amplitude–phase form

$$\begin{array}{c}\hfill q(x,t)=K\left(\xi \right)e^{\frac{i\mathrm{\Gamma }(\eta +1)\left(-k{x}^{\delta }+\omega t^{\delta }+\Omega \right)}{\delta }},\xi ={\displaystyle \frac{\mathrm{\Gamma }(\eta +1)\left(x^{\delta }-\rho t^{\delta }\right)}{\delta }}.\end{array}$$

(11)

where $K\left(\xi \right)$ is a real-valued amplitude, k is the wave number, $\omega$ is the frequency, $\Omega$ is a phase constant, and $\rho$ is the wave speed. Substituting (11) into (10) and separating the real and imaginary parts, we obtain the reduced ordinary differential equation

$$(\alpha -\beta \rho +\gamma )K^{″}+(\omega (\beta k-1)-\sigma k-\alpha k^2)K+dF\left(K^2\right)K=0$$

(12)

together with the relation

$$\rho ={\displaystyle \frac{\sigma +2\alpha k-\beta \omega }{\beta k-1}}.$$

(13)

For the Kerr-law nonlinearity $F\left(K^2\right)=K^2$, Equation (12) becomes

$$(\alpha -\beta \rho +\gamma )K^{″}+(\omega (\beta k-1)-\sigma k-\alpha k^2)K+d{K}^3=0.$$

(14)

Balancing the terms $K^{″}$ and $K^3$ gives $N=1$ in the expansion (7). Hence, the KEM trial form is

$$\begin{array}{c}\hfill K\left(\xi \right)=A_0+A_{1}Q\left(\xi \right).A_1\ne 0,\end{array}$$

(15)

where $Q\left(\xi \right)$ satisfies (8) and (9). Substituting (15) and its derivatives (computed using (8)) into (14) yields a polynomial in powers of $Q\left(\xi \right)$. Equating coefficients of like powers of Q to zero produces an algebraic system for $A_0,A_1,\varpi$, and the parameter combinations appearing in (14). Solving this system leads to the explicit exact solutions reported below.

(I): When

$$A_0=-{\displaystyle \frac{i\varpi \sqrt{\alpha +\beta \sigma +\gamma +{\beta }^2\gamma k^2-2\beta \gamma k}}{\sqrt{{\beta }^2(-d){\varpi }^2+2{\beta }^2{k}^{2}d-4\beta kd+2d}}},$$

(16)

$$A_1={\displaystyle \frac{2i\varpi \sqrt{\alpha +\beta \sigma +\gamma +{\beta }^2\gamma k^2-2\beta \gamma k}}{\sqrt{d\left(-{\beta }^2{\varpi }^2+2{\beta }^2{k}^2-4\beta k+2\right)}}},$$

(17)

$$\omega ={\displaystyle \frac{-{\varpi }^2(\alpha +\beta \sigma +\gamma )+2\alpha \beta k^3-2k^2(\alpha -\beta \sigma )+\beta k{\varpi }^2(\gamma -\alpha )-2k\sigma }{2{(\beta k-1)}^2-{\beta }^2{\varpi }^2}},$$

(18)

we have

$$\begin{array}{c}\hfill q^I(x,t)=e^{{\displaystyle \frac{i\mathrm{\Gamma }(\eta +1)\left(\frac{t^{\delta }\left(\alpha {\delta }^2+\beta {\delta }^3+\gamma {\delta }^2-2\alpha \beta k^3+2\alpha k^2-2\beta \delta k^2+\alpha \beta {\delta }^{2}k-\beta \gamma {\delta }^{2}k+2\delta k\right)}{{\beta }^2{\delta }^2-2{\beta }^2{k}^2+4\beta k-2}-k{x}^{\delta }+\Omega \right)}{\delta }}}\\ \hfill \times (-(2i\delta \sqrt{\alpha +\beta \delta +\gamma +{\beta }^2\gamma k^2-2\beta \gamma k}\\ \hfill /\sqrt{d\left(-{\beta }^2{\delta }^2+2{\beta }^2{k}^2-4\beta k+2\right)}\\ \hfill \times \left(\eta e^{\mathrm{\Gamma }(\eta +1)\left(x^{\delta }-{\displaystyle \frac{t^{\delta }\left(\delta -\frac{\beta \left(\alpha {\delta }^2+\beta {\delta }^3+\gamma {\delta }^2-2\alpha \beta k^3+2\alpha k^2-2\beta \delta k^2+\alpha \beta {\delta }^{2}k-\beta \gamma {\delta }^{2}k+2\delta k\right)}{{\beta }^2{\delta }^2-2{\beta }^2{k}^2+4\beta k-2}+2\alpha k\right)}{\beta k-1}}\right)}+1\right))\\ \hfill +{\displaystyle \frac{i\delta \sqrt{\alpha +\beta \delta +\gamma +{\beta }^2\gamma k^2-2\beta \gamma k}}{\sqrt{-{\beta }^2{\delta }^{2}d+2{\beta }^2{k}^{2}d-4\beta kd+2d}}})\end{array}$$

(19)

(II): When $A_0=-{\displaystyle \frac{i\varpi \sqrt{\beta \gamma k^3-\gamma k^2+k\sigma +\omega }}{\sqrt{2\beta d{k}^3-2d{k}^2-\beta dk{\varpi }^2-d{\varpi }^2}}},A_1={\displaystyle \frac{2i\varpi \sqrt{\beta \gamma k^3-\gamma k^2+k\sigma +\omega }}{\sqrt{-d\left(-2\beta k^3+2k^2+\beta k{\varpi }^2+{\varpi }^2\right)}}},\alpha ={\displaystyle \frac{\beta {\varpi }^2(\sigma -\beta \omega )+2\beta k^2(\beta \omega -\sigma )+\gamma {\varpi }^2(1-\beta k)+2k(\sigma -2\beta \omega )+2\omega }{2k^2(\beta k-1)-{\varpi }^2(\beta k+1)}},$ we have

$$\begin{array}{c}\hfill q^{II}(x,t)=e^{{\displaystyle \frac{i\mathrm{\Gamma }(\eta +1)\left(-k{x}^{\delta }+\omega t^{\delta }+\mathrm{\Omega }\right)}{\delta }}}(-(\left(2i\delta \sqrt{\beta \gamma k^3-\gamma k^2+\delta k+\omega }\right)\\ \hfill /\sqrt{d\left(-{\delta }^2+2\beta k^3-2k^2-\beta {\delta }^{2}k\right)}\\ \hfill \times \left(\eta e^{\mathrm{\Gamma }(\eta +1)\left(x^{\delta }-{\displaystyle \frac{t^{\delta }\left(-\beta \omega +\delta +\frac{2k\left(-{\beta }^2{\delta }^2\omega +\beta {\delta }^3+\gamma {\delta }^2+2{\beta }^2{k}^2\omega -2\beta \delta k^2-\beta \gamma {\delta }^{2}k-4\beta k\omega +2\delta k+2\omega \right)}{-{\delta }^2+2\beta k^3-2k^2-\beta {\delta }^{2}k}\right)}{\beta k-1}}\right)}+1\right))\\ \hfill +{\displaystyle \frac{i\delta \sqrt{\beta \gamma k^3-\gamma k^2+\delta k+\omega }}{\sqrt{-{\delta }^{2}d+2\beta d{k}^3-2d{k}^2-\beta {\delta }^{2}dk}}}).\end{array}$$

(20)

The 3D and density profiles of Equations (19) and (20) are illustrated in Figure 1 and Figure 2.

4. Lie Symmetry Analysis

In this section, we perform a Lie point symmetry analysis of the governing resonant nonlinear Schrödinger-type model. The symmetry generators obtained here are subsequently used in the structural study of the equation (in particular, in the classification step and in the conservation-law analysis presented later).

To determine Lie symmetries, we express the complex envelope in amplitude-phase form as

$$q(x,t)=u(x,t)e^{i\upsilon (x,t)},$$

(21)

where $u(x,t)$ and $\upsilon (x,t)$ are real-valued functions representing the amplitude and phase, respectively. Substituting (21) into Equation (1) and separating the real and imaginary parts yields the coupled real system

$$\begin{array}{c}\hfill u{\upsilon }_t=d{u}^3+\sigma u{\upsilon }_x-\beta u{\upsilon }_t{\upsilon }_x-\alpha u{\left({\upsilon }_x\right)}^2+\beta u_{xt}+\alpha u_{xx}+\gamma u_{xx},\\ \hfill u_t=\sigma u_x-\beta {\upsilon }_t{u}_x-\beta u_t{\upsilon }_x-2\alpha u_x{\upsilon }_x-\beta u{\upsilon }_{xt}-\alpha u{\upsilon }_{xx}.\end{array}$$

(22)

we analyze system (22) for the functions $u(x,t)$ and $\upsilon (x,t)$. Assume $\beta \ne 0$ whenever division by $\beta$ is required. Let

$$X={\xi }_1{\mathsf{\partial }}_x+{\xi }_2{\mathsf{\partial }}_t+{\eta }_1{\mathsf{\partial }}_u+{\eta }_2{\mathsf{\partial }}_v.$$

be the general Lie point symmetry generator of system (22). Solving the associated determining equations gives the infinitesimal coefficients

$${\xi }_1=x{C}_2+C_4,{\xi }_2=C_1+t{C}_2,{\eta }_1=-u{C}_2,{\eta }_2=-{\displaystyle \frac{x{C}_2+t\sigma C_2-\beta C_3}{\beta }}.$$

where $C_1,C_2,C_3,C_4$ are arbitrary constants. Hence, a basis for the Lie algebra of symmetries can be chosen as

• $C_1=1$ (others 0): ${\xi }_1=0,{\xi }_2=1,{\eta }_1=0,{\eta }_2=0$, so

  $$X_1={\mathsf{\partial }}_t$$
• $C_2=1$ (others 0): ${\xi }_1=x,{\xi }_2=t,{\eta }_1=-u,{\eta }_2=-(x+\delta t)/\beta$, so

  $$X_2=x{\mathsf{\partial }}_x+t{\mathsf{\partial }}_t-u{\mathsf{\partial }}_u-{\displaystyle \frac{x+\delta t}{\beta }}{\mathsf{\partial }}_v$$
• $C_3=1$ (others 0): ${\xi }_1=0,{\xi }_2=0,{\eta }_1=0,{\eta }_2=1$, so

  $$X_3={\mathsf{\partial }}_v$$
• $C_4=1$ (others 0): ${\xi }_1=1,{\xi }_2=0,{\eta }_1=0,{\eta }_2=0$, so

  $$X_4={\mathsf{\partial }}_x$$

These generators admit standard physical interpretations for NLS-type models: $X_1$ and $X_4$ correspond to invariance under translations in time and space, respectively (homogeneity in t and x). The generator $X_3$ expresses invariance under a constant phase shift (global gauge/phase invariance). The scaling-type generator $X_2$ represents a self-similar transformation mixing ($x,t,u,\upsilon$) and is often associated with the scaling behavior of wave profiles. Such invariances reflect the intrinsic structural properties of the model and provide a natural foundation for the subsequent classification and conservation-law analysis.

4.1. Lie Bracket Formula

For two vector fields

$$X={\xi }_1^X{\mathsf{\partial }}_x+{\xi }_2^X{\mathsf{\partial }}_t+{\eta }_1^X{\mathsf{\partial }}_u+{\eta }_2^X{\mathsf{\partial }}_v,Y={\xi }_1^Y{\mathsf{\partial }}_x+{\xi }_2^Y{\mathsf{\partial }}_t+{\eta }_1^Y{\mathsf{\partial }}_u+{\eta }_2^Y{\mathsf{\partial }}_v,$$

Their Lie bracket is defined by

$$[X,Y]=\left(X({\xi }_1^Y)-Y({\xi }_1^X)\right){\mathsf{\partial }}_x+\left(X({\xi }_2^Y)-Y({\xi }_2^X)\right){\mathsf{\partial }}_t+\left(X({\eta }_1^Y)-Y({\eta }_1^X)\right){\mathsf{\partial }}_u+\left(X({\eta }_2^Y)-Y({\eta }_2^X)\right){\mathsf{\partial }}_v,$$

where, for any smooth function $F(x,t,u,\upsilon )$,

$$X\left(F\right)={\xi }_1^X{F}_x+{\xi }_2^X{F}_t+{\eta }_1^X{F}_u+{\eta }_2^X{F}_v.$$

(23)

4.2. Compute Nonzero Commutators

In this subsection, we compute the nonzero commutators $[X_1,X_2]$, $[X_2,X_4]$; the remaining commutators vanish. We use the coefficient representation in the order $({\xi }_1,{\xi }_2,{\eta }_1,{\eta }_2)$. Recall the basis generators

$$X_1={\mathsf{\partial }}_t,X_2=x{\mathsf{\partial }}_x+t{\mathsf{\partial }}_t-u{\mathsf{\partial }}_u-{\displaystyle \frac{x+\sigma t}{\beta }}{\mathsf{\partial }}_v$$

(24)

Thus, the corresponding coefficient vectors are

$$X_1:(0,1,0,0),X_2:(x,t,-u,-\frac{x+\sigma t}{\beta }),X_4=(1,0,0,0).$$

Commutator $[X_1,X_2]$.

Using the definition in Section 4.1, we compute each component:

$$\begin{array}{cc}\hfill \left(1\right){\mathsf{\partial }}_x\text{-}\mathrm{coef}& :X_1\left({\xi }_1^{\left(2\right)}\right)-X_2\left({\xi }_1^{\left(1\right)}\right)={\mathsf{\partial }}_t\left(x\right)-X_2\left(0\right)=0.\hfill \\ \hfill \left(2\right){\mathsf{\partial }}_t\text{-}\mathrm{coef}& :X_1\left({\xi }_2^{\left(2\right)}\right)-X_2\left({\xi }_2^{\left(1\right)}\right)={\mathsf{\partial }}_t\left(t\right)-X_2\left(1\right)=1-0=1.\hfill \\ \hfill \left(3\right){\mathsf{\partial }}_u\text{-}\mathrm{coef}& :X_1\left({\eta }_1^{\left(2\right)}\right)-X_2\left({\eta }_1^{\left(1\right)}\right)=X_1(-u)-X_2\left(0\right)={\mathsf{\partial }}_t(-u)=0.\hfill \\ \hfill \left(4\right){\mathsf{\partial }}_v\text{-}\mathrm{coef}& :X_1\left({\eta }_2^{\left(2\right)}\right)-X_2\left({\eta }_2^{\left(1\right)}\right)={\mathsf{\partial }}_t\left(-{\displaystyle \frac{x+\sigma t}{\beta }}\right)-0=-{\displaystyle \frac{\sigma }{\beta }}.\hfill \end{array}$$

Collecting

$$[X_1,X_2]=1·{\mathsf{\partial }}_t-{\displaystyle \frac{\sigma }{\beta }}{\mathsf{\partial }}_v=X_1-{\displaystyle \frac{\sigma }{\beta }}{X}_3.$$

By antisymmetry,

$$[X_2,X_1]=-X_1+{\displaystyle \frac{\sigma }{\beta }}{X}_3.$$

Commutator $[X_2,X_4]$.

Since $X_4={\mathsf{\partial }}_x$, we obtain

$$\begin{array}{cc}\hfill \left(1\right){\mathsf{\partial }}_x\text{-}\mathrm{coef}& :X_2\left({\xi }_1^{\left(4\right)}\right)-X_4\left({\xi }_1^{\left(2\right)}\right)=X_2\left(1\right)-{\mathsf{\partial }}_x\left(x\right)=0-1=-1.\hfill \\ \hfill \left(2\right){\mathsf{\partial }}_t\text{-}\mathrm{coef}& :X_2\left({\xi }_2^{\left(4\right)}\right)-X_4\left({\xi }_2^{\left(2\right)}\right)=X_2\left(0\right)-{\mathsf{\partial }}_x\left(t\right)=0-0=0.\hfill \\ \hfill \left(3\right){\mathsf{\partial }}_u\text{-}\mathrm{coef}& :X_2\left({\eta }_1^{\left(4\right)}\right)-X_4\left({\eta }_1^{\left(2\right)}\right)=0-{\mathsf{\partial }}_x(-u)=0.\hfill \\ \hfill \left(4\right){\mathsf{\partial }}_v\text{-}\mathrm{coef}& :X_2\left({\eta }_2^{\left(4\right)}\right)-X_4\left({\eta }_2^{\left(2\right)}\right)=0-{\mathsf{\partial }}_x\left(-{\displaystyle \frac{x+\sigma t}{\beta }}\right)={\displaystyle \frac{1}{\beta }}.\hfill \end{array}$$

Thus

$$[X_2,X_4]=-{\mathsf{\partial }}_x+{\displaystyle \frac{1}{\beta }}{\mathsf{\partial }}_v=-X_4+{\displaystyle \frac{1}{\beta }}{X}_3,$$

and hence

$$[X_4,X_2]=X_4-{\displaystyle \frac{1}{\beta }}{X}_3.$$

Other Brackets

One checks directly (by computing derivatives of constant coefficients) that

$$[X_1,X_4]=0,[X_i,X_3]=0\mathrm{for}\mathrm{all}i,$$

so $X_3$ is central.

Thus, the only nonzero commutators are

$$\begin{array}{cc}\hfill [X_1,X_2]& =X_1-{\displaystyle \frac{\sigma }{\beta }}{X}_3,\hfill \\ \hfill [X_2,X_4]& =-X_4+{\displaystyle \frac{1}{\beta }}{X}_3.\hfill \end{array}$$

We present the commutator table (rows = left argument):

$$\begin{array}{ccccc}[·,·]& X_1& X_2& X_3& X_4\\ X_1& 0& X_1-\frac{\sigma }{\beta }{X}_3& 0& 0\\ X_2& -X_1+\frac{\sigma }{\beta }{X}_3& 0& 0& -X_4+\frac{1}{\beta }{X}_3\\ X_3& 0& 0& 0& 0\\ X_4& 0& X_4-\frac{1}{\beta }{X}_3& 0& 0\end{array}$$

4.3. Adjoint Representation

Recall that $Ad\left(e^{\epsilon X}\right)=exp\left(\epsilon {ad}_X\right)$ where ${ad}_X\left(Y\right)=[X,Y]$. In this problem, since the nested commutators quickly approach zero (or the central element X3), the adjoint effect can be obtained in closed form (exact).

4.3.1. Ad by $X_1$

From $[X_1,X_2]=X_1-\frac{\sigma }{\beta }{X}_3$ and $[X_1,\mathrm{others}]=0$,

$$\mathrm{Ad}\left(e^{\epsilon X_1}\right)X_2=X_2+\epsilon \left(X_1-\frac{\sigma }{\beta }{X}_3\right),$$

and $X_1,X_3,X_4$ are fixed:

$$\mathrm{Ad}\left(e^{\epsilon X_1}\right)X_1=X_1,\mathrm{Ad}\left(e^{\epsilon X_1}\right)X_3=X_3,\mathrm{Ad}\left(e^{\epsilon X_1}\right)X_4=X_4.$$

4.3.2. Ad by $X_4$

From $[X_4,X_2]=X_4-\frac{1}{\beta }{X}_3$ and $[X_4,\mathrm{others}]=0$, we get

$$\mathrm{Ad}\left(e^{\epsilon X_4}\right)X_2=X_2+\epsilon \left(X_4-\frac{1}{\beta }{X}_3\right),$$

with $X_1,X_3,X_4$ fixed.

4.3.3. Ad by $X_3$

Since $X_3$ is the central element, $[X_3,X_j]=0$ (for all j) and therefore $\mathrm{Ad}\left(e^{\epsilon X_3}\right)$ is an identity transformation on the Lie algebra.

4.3.4. Ad by $X_2$

To obtain the effect of $\mathrm{Ad}\left(e^{\epsilon X_2}\right)X_j$, we use the effect of $X_2$ on $X_1$ and $X_4$

$$[X_2,X_1]=-X_1+\frac{\sigma }{\beta }{X}_3,[X_2,X_4]=-X_4+\frac{1}{\beta }{X}_3,$$

Therefore, we solve the flow equation in the subspaces $\{X_1,X_3\}$ and $\{X_4,X_3\}$.

Effect on $X_1$: Let $Y\left(\epsilon \right)=a\left(\epsilon \right)X_1+b\left(\epsilon \right)X_3$; $Y\left(0\right)=X_1$, and ${\displaystyle \frac{d}{d\epsilon }}Y=[X_2,Y]$ From this

$${\displaystyle \frac{d}{d\epsilon }}Y=a\left(\epsilon \right)[X_2,X_1]=a\left(\epsilon \right)\left(-X_1+\frac{\sigma }{\beta }{X}_3\right).$$

and the following ODE system is obtained:

$$a^{\prime }\left(\epsilon \right)=-a\left(\epsilon \right),b^{\prime }\left(\epsilon \right)={\displaystyle \frac{\sigma }{\beta }}a\left(\epsilon \right),$$

with $a\left(0\right)=1,b\left(0\right)=0$.

Solution

$$a\left(\epsilon \right)=e^{-\epsilon },b\left(\epsilon \right)={\displaystyle \frac{\sigma }{\beta }}\left(1-e^{-\epsilon }\right).$$

Therefore

$$\mathrm{Ad}\left(e^{\epsilon X_2}\right)X_1=e^{-\epsilon }{X}_1+{\displaystyle \frac{\sigma }{\beta }}\left(1-e^{-\epsilon }\right)X_3.$$

Effect on $X_4$: similarly, taking $Z\left(\epsilon \right)=p\left(\epsilon \right)X_4+q\left(\epsilon \right)X_3$, we have $Z\left(0\right)=X_4$ and ${\displaystyle \frac{d}{d\epsilon }}Z=[X_2,Z]$. From this,

$${\displaystyle \frac{d}{d\epsilon }}Z=p\left(\epsilon \right)[X_2,X_4]=p\left(\epsilon \right)\left(-X_4+\frac{1}{\beta }{X}_3\right).$$

and the ODE system

$$p^{\prime }=-p,q^{\prime }={\displaystyle \frac{1}{\beta }}p,p\left(0\right)=1,q\left(0\right)=0,$$

is obtained. The solution

$$p\left(\epsilon \right)=e^{-\epsilon },q\left(\epsilon \right)={\displaystyle \frac{1}{\beta }}\left(1-e^{-\epsilon }\right),$$

is therefore

$$\mathrm{Ad}\left(e^{\epsilon X_2}\right)X_4=e^{-\epsilon }{X}_4+{\displaystyle \frac{1}{\beta }}\left(1-e^{-\epsilon }\right)X_3.$$

Also $\mathrm{Ad}\left(e^{\epsilon X_2}\right)X_2=X_2$ and $X_3$ is fixed. This completes the adjoint table.

These adjoint transformations provide the main tool for classifying one-dimensional subalgebras of the Lie algebra. In particular, the adjoint action generates equivalence transformations on the symmetry generators, allowing an arbitrary linear combination to be simplified to canonical representatives. Therefore, in the next section, the computed adjoint maps are used to eliminate equivalent subalgebras and to construct an optimal system of one-dimensional subalgebras.

5. Optimal System of 1-Dimensional Subalgebras and Conservation Laws

In what follows, we classify one-dimensional subalgebras up to conjugacy under the adjoint action. By combining adjoint transformations with rescaling, a general generator is reduced to canonical representatives, yielding an optimal system of one-dimensional subalgebras:

$$X=a_1{X}_1+a_2{X}_2+a_3{X}_3+a_4{X}_4,a_i\in \mathbb{R}.$$

We use adjoint actions and rescaling ($X\sim X,d\ne 0$) to find canonical representatives.

5.1. Case A: $a_2\ne 0$

Scale X by $1/a_2$ so $a_2=1$, i.e.,

$$X=a_1{X}_1+X_2+a_3{X}_3+a_4{X}_4.$$

Apply $\mathrm{Ad}\left(e^{\epsilon X_1}\right)$ with $\epsilon =-a_1$:

$$\mathrm{Ad}\left(e^{-a_1{X}_1}\right)X=a_1{X}_1+\left(X_2-a_1{X}_1+\frac{\sigma }{\beta }{a}_1{X}_3\right)+a_3{X}_3+a_4{X}_4.$$

Collecting terms gives $X\sim X_2+a_3^{\prime }{X}_3+a_4{X}_4$ with

$$a_3^{\prime }=a_3+{\displaystyle \frac{\sigma }{\beta }}{a}_1,\mathrm{and}\mathrm{the}{X}_1\mathrm{coefficient}\mathrm{is}\mathrm{zero}.$$

Next apply $\mathrm{Ad}\left(e^{\eta X_4}\right)$ with $\eta =-a_4$ to eliminate the $X_4$ term:

$$\mathrm{Ad}\left(e^{-a_4{X}_4}\right)(X_2+a_3^{\prime }{X}_3+a_4{X}_4)=X_2+a_4{X}_4-{\displaystyle \frac{a_4}{\beta }}{X}_3+a_3^{\prime }{X}_3+a_4{X}_4-a_4{X}_4,$$

so the final form is

$$X\sim X_2+c{X}_3,c=a_3^{\prime }+{\displaystyle \frac{a_4}{\beta }}.$$

Thus any generator with $a_2\ne 0$ is conjugate to $X_2+c{X}_3$.

5.2. Case B: $a_2=0$

Then $X=a_1{X}_1+a_3{X}_3+a_4{X}_4$.

• If $a_1=a_4=0$ we get $X=a_3{X}_3$⇒$\langle X_3\rangle$.
• If $a_1\ne 0$, $a_4=0$: scale to $a_1=1$ and obtain $\langle X_1+c{X}_3\rangle$.
• If $a_4\ne 0$, $a_1=0$: scale to $a_4=1$ and obtain $\langle X_4+c{X}_3\rangle$.
• If $a_1\ne 0$ and $a_4\ne 0$: scale to $a_1=1$ and write

  $$X=X_1+\mu X_4+c{X}_3,\mu ={\displaystyle \frac{a_4}{a_1}}.$$
  Under $\mathrm{Ad}\left(e^{\epsilon X_2}\right)$ both $X_1$ and $X_4$ scale by $e^{-\epsilon }$ (with central shifts), so the ratio $\mu$ is invariant under that scaling. Therefore the canonical family is $\langle X_1+\mu X_4+c{X}_3\rangle$, with $\mu \ge 0$.

Thus an optimal system of 1D subalgebras is

(i)

$\langle X_3\rangle$,

(ii)

$\langle X_2+c{X}_3\rangle$, $c\in \mathbb{R}$,

(iii)

$\langle X_1+c{X}_3\rangle$, $c\in \mathbb{R}$,

(iv)

$\langle X_4+c{X}_3\rangle$, $c\in \mathbb{R}$,

(v)

$\langle X_1+\mu X_4+c{X}_3\rangle$, $\mu \ge 0$, $c\in \mathbb{R}$.

5.3. Conservation Laws

In this subsection, we present conservation laws associated with the Lie point symmetries $X_1\dots X_4$ derived via the conservation theorem in [30]. Since the theorem is formulated in a nonlocal setting, the conserved vectors may involve auxiliary (adjoint) variables; nonetheless, each conserved vector is divergence free and yields an invariant integral under suitable boundary conditions. In NLS-type models, such invariants are commonly interpreted as generalized power (mass), momentum, and energy-type quantities, providing structural insight into wave propagation and soliton dynamics.

• The symmetry $X_1={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}$ yields the nonlocal conserved vector
  $\begin{array}{c}{\mathbf{C}}_{\mathbf{1}}^{\mathbf{x}}={\displaystyle \frac{1}{2}}\left(\beta p{u}_{tt}-\beta ru{\upsilon }_{tt}-2u_t\left({\displaystyle \frac{\beta p_t}{2}}+(\alpha +\gamma )p_x+r\left(-\sigma +\beta {\upsilon }_t+2\alpha {\upsilon }_x\right)\right)-\right.\hfill \\ {\upsilon }_t\left(-u\left(\beta r_t+2\alpha r_x\right)+r\left(\beta u_t+2\alpha u_x\right)+2pu\left(-\sigma +\beta {\upsilon }_t+2\alpha {\upsilon }_x\right)\right)+\hfill \\ \left.2(\alpha +\gamma )p{u}_{xt}-2\alpha ru{\upsilon }_{xt}\right),\hfill \\ \\ {\mathbf{C}}_{\mathbf{1}}^{\mathbf{t}}={\displaystyle \frac{1}{2}}\left(-\beta u_t{p}_x-2\sigma r{u}_x+r{u}_x\left(\beta {\upsilon }_t+4\alpha {\upsilon }_x\right)+p\left(-2d{u}^3+2u{\upsilon }_x\left(-\sigma +\alpha {\upsilon }_x\right)-\right.\right.\hfill \\ \left.\left.\beta u_{xt}-2(\alpha +\gamma )u_{xx}\right)+u\left(\beta {\upsilon }_t{r}_x+r\left(\beta {\upsilon }_{xt}+2\alpha {\upsilon }_{xx}\right)\right)\right).\hfill \end{array}$
  Associated with time-translation invariance, this conservation law can be viewed as an energy-type (Hamiltonian-like) invariant in the generalized/nonlocal formulation.
• The symmetry $X_2=t{\displaystyle \frac{\mathsf{\partial }}{{\mathsf{\partial }}_t}}-u{\displaystyle \frac{\mathsf{\partial }}{{\mathsf{\partial }}_u}}+x{\displaystyle \frac{\mathsf{\partial }}{{\mathsf{\partial }}_x}}-({\displaystyle \frac{x}{\beta }}+{\displaystyle \frac{t\delta }{\beta }}){\displaystyle \frac{\mathsf{\partial }}{{\mathsf{\partial }}_{\upsilon }}}$ yields the nonlocal conserved vector
  $\begin{array}{l}{\mathbf{C}}_{\mathbf{2}}^{\mathbf{x}}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\left(x+t\delta +t\beta {\upsilon }_t+x\beta {\upsilon }_x\right)\left(u\left(\beta r_t+2\alpha r_x\right)-r\left(\beta u_t+2\alpha u_x\right)+2pu\left(\delta -\beta {\upsilon }_t-2\alpha {\upsilon }_x\right)\right)}{\beta }}-\right.\\ \left(u+t{u}_t+x{u}_x\right)\left(\beta p_t+2(\alpha +\gamma )p_x+r\left(-2\delta +2\beta {\upsilon }_t+4\alpha {\upsilon }_x\right)\right)+\\ \beta p\left(2u_t+t{u}_{tt}+x{u}_{xt}\right)-ru\left(\delta +\beta \left({\upsilon }_t+t{\upsilon }_{tt}+x{\upsilon }_{xt}\right)\right)+2(\alpha +\gamma )\\ p\left(2u_x+t{u}_{xt}+x{u}_{xx}\right)-{\displaystyle \frac{2\alpha ru\left(1+\beta \left({\upsilon }_x+t{\upsilon }_{xt}+x{\upsilon }_{xx}\right)\right)}{\beta }}+\\ 2x\left(-p\left(d{u}^3-u\left({\upsilon }_t+\left(-\delta +\beta {\upsilon }_t\right){\upsilon }_x+\alpha {{\upsilon }_x}^2\right)+\beta u_{xt}+(\alpha +\gamma )u_{xx}\right)+\right.\\ \left.\left.r\left(u_x\left(-\delta +\beta {\upsilon }_t+2\alpha {\upsilon }_x\right)+u_t\left(1+\beta {\upsilon }_x\right)+u\left(\beta {\upsilon }_{xt}+\alpha {\upsilon }_{xx}\right)\right)\right)\right),\\ \\ {\mathbf{C}}_{\mathbf{2}}^{\mathbf{t}}={\displaystyle \frac{1}{2\beta }}\left(p\left(-2dt\beta u^3-2u\left(x+t\delta +\beta {\upsilon }_x\left(2(x+t\delta )+(-t\alpha +x\beta ){\upsilon }_x\right)\right)+\right.\right.\\ \left.{\beta }^2\left(2u_x-t{u}_{xt}\right)+\beta (x\beta -2t(\alpha +\gamma ))u_x\right)+\beta \left(-\beta p_x\left(t{u}_t+x{u}_x\right)+\right.\\ u\left(-\beta p_x+r_x\left(x+t\delta +t\beta {\upsilon }_t+x\beta {\upsilon }_x\right)\right)r\left(u_x(-3(x+t\delta )+\right.\\ \left.\left.\left.\left.t\beta {\upsilon }_t+(4t\alpha -3x\beta ){\upsilon }_x\right)+u\left(-3-3\beta {\upsilon }_x+t\beta {\upsilon }_{xt}+(2t\alpha -x\beta ){\upsilon }_{xx}\right)\right)\right)\right).\end{array}$
  Associated with the scaling-type symmetry, it represents a generalized scaling invariant relevant to the structural classification of solution families.
• The symmetry $X_3={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\upsilon }}$ determines the conserved vector
  $\begin{array}{c}{\mathbf{C}}_{\mathbf{3}}^{\mathbf{x}}={\displaystyle \frac{1}{2}}\left(-u\left(\beta r_t+2\alpha r_x\right)+r\left(\beta u_t+2\alpha u_x\right)\right)+pu\left(-\delta +\beta {\upsilon }_t+2\alpha {\upsilon }_x\right),\hfill \\ \\ {\mathbf{C}}_{\mathbf{3}}^{\mathbf{t}}={\displaystyle \frac{1}{2}}\left(-\beta u{r}_x+\beta r{u}_x\right)+pu\left(1+\beta {\upsilon }_x\right).\hfill \end{array}$
  Associated with global phase invariance, it is typically interpreted as a power (mass)-type conserved quantity in NLS-type dynamics.
• The symmetry $X_4={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}$ determines the nonlocal conserved vector
  $\begin{array}{c}{\mathbf{C}}_{\mathbf{4}}^{\mathbf{x}}={\displaystyle \frac{1}{2}}\left(-\left(\beta p_t+2(\alpha +\gamma )p_x\right)u_x+u\left(\beta r_t+2\alpha r_x\right){\upsilon }_x-\right.\hfill \\ p\left(2u\left(d{u}^2-{\upsilon }_t+\alpha {\left({\upsilon }_x\right)}^2\right)+\beta u_{xt}\right)+r\left(-2\alpha u_x{\upsilon }_x+\right.\hfill \\ \left.\left.u_t\left(2+\beta {\upsilon }_x\right)+\beta u{\upsilon }_{xt}\right)\right),\hfill \\ \\ {\mathbf{C}}_{\mathbf{4}}^{\mathbf{t}}={\displaystyle \frac{1}{2}}\left(-\beta p_x{u}_x+u{\upsilon }_x\left(\beta r_x-2p\left(1+\beta {\upsilon }_x\right)\right)+\beta p{u}_{xx}+\right.\hfill \\ \left.r\left(-u_x\left(2+3\beta {\upsilon }_x\right)-\beta u{\upsilon }_{xx}\right)\right).\hfill \end{array}$

Associated with spatial translation invariance, it can be interpreted as a momentum-type invariant (or momentum flux) in the generalized/nonlocal setting.

Overall, the obtained conservation laws complement the symmetry structure reported in Section 4 by providing explicit divergence-free quantities associated with the fundamental invariances of the considered NLS-type model. Although the expressions are lengthy due to the spatio-temporal and inter-modal dispersion terms and the nonlocal formulation, they encode conserved integral constraints that can be useful for qualitative analysis and for validating symbolic computations.

6. Conclusions

In this study, the Lie symmetry structure, conservation laws, and analytical soliton-type solutions of a nonlinear Schrödinger equation incorporating space-time dispersion and inter-mode dispersion effects were investigated. First, Lie point symmetries are identified and the corresponding Lie algebra is constructed. To clarify the algebraic structure, commutator relations, adjoint representation, and an optimal system for one-dimensional subalgebras were presented. Subsequently, using conservation theorems, various conserved vectors (nonlocal if necessary) related to the fundamental invariants of the considered NLS-type model were obtained. These conservation laws can be interpreted as invariants of generalized power (mass), momentum, and energy types, yielding zero divergence under appropriate conditions; they also provide structural verification (control) capabilities for analytical calculations. Finally, new exact moving wave solutions were obtained by applying the Kudryashov expansion method. In general, this approach, which jointly addresses symmetry-conservation exact solution analysis, provides a consistent structural framework for the equation under investigation and can form a basis for qualitative investigations of related nonlinear wave models. In particular, the obtained symmetries, conservation quantities, and exact wave profiles are meaningful in qualitative analyses of dispersive wave propagation; they can also provide reference results for comparison (benchmarking) purposes in numerical schemes and stability investigations in the nonlinear optical context.

Compared with recent solution-oriented studies on complex nonlinear Schrödinger-type models such as [6], where the improved Bernoulli sub-equation function method is employed to construct traveling-wave structures (e.g., periodic and breather-type waves) for a modified unstable complex NLS equation, our work addresses a different resonant NLS-type model incorporating spatio-temporal dispersion and inter-modal dispersion. It complements explicit wave profiles with a full structural analysis. In particular, while the above-mentioned line of research primarily emphasizes the generation and visualization of traveling-wave families, the present study additionally provides the admitted Lie symmetry algebra, the optimal system for one-dimensional subalgebras, and conservation laws, thereby offering a systematic symmetry-invariant-solution framework for the considered equation.

Accordingly, the main novelty of this paper is not limited to producing exact wave solutions; rather, it lies in presenting a unified symmetry, conservation, and solution treatment for the considered resonant NLS-type model with spatio-temporal and inter-modal dispersion. Beyond constructing explicit traveling-wave families via the Kudryashov expansion method (together with the associated parameter constraints and limiting cases), we provide a complete Lie symmetry analysis that clarifies the admitted algebraic structure and yields an optimal system for classification. We further complement these results by deriving conservation laws that lead to physically interpretable invariant quantities of generalized power (mass), momentum, and energy types under suitable conditions.

Although the study does not rely on externally generated datasets, its main contribution is analytical: we deliver reproducible closed-form expressions for new exact traveling-wave solution families, together with structural invariants obtained from symmetry and conservation analysis. These outcomes provide benchmark references that can be used to verify analytical manipulations and to support future numerical simulations and stability investigations in nonlinear dispersive wave dynamics and related nonlinear optical contexts.