Implicit Quiescent Solitons in Optical Metamaterials with Nonlinear Chromatic Dispersion and an Array of Self-Phase Modulation Structures with Generalized Temporal Evolution by Lie Symmetry

Abstract

The current paper retrieves implicit quiescent soliton solutions to optical metamaterials with nonlinear chromatic dispersion with generalized temporal evolution. Seven forms of self-phase modulation structures, as proposed by Kudryashov with time, are taken up. The implemented integration algorithm is Lie symmetry. A few of the solutions are in quadratures, while others are in terms of special functions. We also characterize the parameters that constrain the existence of such solutions.

1. Introduction

The study of soliton propagation in optical metamaterials, alongside conventional optical fibers, photonic–crystal fibers (PCFs), and other waveguide configurations, has attracted considerable attention in recent years [1]. Numerous results on optical metamaterials and related nonlinear photonic structures have been reported over approximately the past decade. A substantial body of work has addressed stationary or quiescent soliton solutions in metamaterial-inspired or complex dispersive models, typically incorporating nonlinear group velocity or chromatic dispersion and, in some cases, complex Ginzburg–Landau-type dynamics [2,3,4,5,6,7,8,9]. A second line of research has focused on non-standard intensity-dependent refractive index responses, such as quadratic–cubic and generalized quadratic–cubic nonlinearities, quintuple power law models, and cubic–quintic–septic laws, often in combination with nonlocal refractive index structures [10,11,12,13,14,15,16,17,18]. Further contributions have developed generalized nonlinear Schrödinger equations derived from higher-order dispersive models or endowed with nonlinear dispersion, arbitrary refractive index profiles, or temporal delay and have analyzed their conservation laws together with various classes of exact or closed-form solutions [19,20,21,22,23,24,25,26,27,28]. Additional studies have explored multisoliton bound states in higher-order members of the nonlinear Schrödinger hierarchy, dispersion compensation strategies, and impairment mitigation in fiber-based transmission links; nonlinear surface and interface waves in graded-index or hyperbolic media; and multi-component pulse dynamics involving degenerate four-photon parametric processes and related mechanisms [29,30,31,32,33,34]. We also note related progress on stationary and quiescent optical structures in complex dispersive media; for instance, generalized nonlinear Schrödinger-type models with engineered dispersion and nonlinearity were analyzed within a different theoretical framework in [35]. Although that work focuses on a distinct physical platform and modeling approach, it further illustrates the current interest in tailoring dispersion and nonlinearity in advanced optical media. By contrast, the present contribution systematically investigates metamaterials with nonlinear chromatic dispersion and an extensive catalog of SPM laws and exploits Lie symmetry methods to obtain implicit quiescent soliton solutions together with explicit parameter constraints. In this way, our analysis both complements and extends these recent developments by emphasizing generalized temporal evolution and by treating a substantially broader class of intensity-dependent refractive index responses.

One of the most interesting areas of study in this context is the formation of quiescent optical solitons in metamaterials. There are several issues that lead to the solitons stalling along the optical waveguides during soliton transmission for inter-continental distances. In addition to the random injection of pulses at the initial end of the fibers, there are numerous other factors that contribute to the problems, such as the twisting, bending, or other forms of rough handling of the fibers. These issues can lead to significant performance degradation in fiber optics, resulting in increased signal loss and reduced overall efficiency. Addressing these challenges is key to guaranteeing reliable data transmission and maintaining the integrity of communication networks.

Several initial findings on the formation of perturbed quiescent solitons in optical metamaterials exhibiting linear temporal evolution have previously been documented [2]. This study is an expansion of the previously published findings. The present research examines the development of perturbed quiescent solitons in optical metamaterials that exhibit generalized temporal evolution. This exploration investigates the complex structure and stability of these solitons, considering additional factors such as nonlinearity and dispersive effects. The aim of this research is to provide a deeper understanding of their behavior and potential applications in advanced optical devices. Consequently, the findings of the publication represent a generalized variation of previously documented research. Therefore, when we reduce the generalized temporal evolution parameter to unity, every result of this publication will converge with those of prior research. The Lie symmetry analysis is the integration method used to retrieve the quiescent optical solitons [36]. We express the derived results in terms of special functions and integrals. Eighteen different self-phase modulation (SPM) setups with nonlinear chromatic dispersion (CD) are detailed in the study.

From a physical perspective, the higher-order self-phase modulation laws considered in this work (power, polynomial, and multi-power nonlinearities) provide effective descriptions of engineered optical metamaterials and complex waveguides whose refractive index exhibits a non-Kerr dependence on the field intensity [15,16,17,18]. In particular, composite structures with metallic or dielectric inclusions, photonic–crystal fibers filled with nonlinear liquids, and metamaterials with resonant inclusions can all be modeled by effective susceptibilities that contain several powers of the intensity. In other situations, such as strongly non-paraxial propagation or media with competing focusing and defocusing contributions, polynomial or dual power laws arise as systematic truncations of more general saturation or nonlocal responses. Thus, the variety of SPM laws analyzed here is not only of formal mathematical interest but also reflects experimentally relevant classes of intensity-dependent refractive index behaviors. In the same spirit, recent highly dispersive and higher-order nonlinear Schrödinger-type settings (including cubic–quintic–septic responses and stochastic terms) motivate the consideration of broader polynomial and multi-power refractive-index laws in modern optical models [28].

Governing Model

The perturbed one-dimensional nonlinear Schrödinger Equation (NLSE) that this paper will discuss, which includes nonlinear CD and various SPM structures for finding quiescent optical solitons, is written as

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+H\left({\left|\mathrm{\Phi }\right|}^2\right){\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_3{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}^{\ast }\end{array}$$

(1)

In Equation (1), the dependent variable $\mathrm{\Phi }(x,t)$ represents the wave amplitude, while x and t denote the spatial and temporal coordinates, respectively. The real number a is the coefficient of nonlinear chromatic dispersion, and the function H describes how the refractive index depends on the local intensity ${\left|\mathrm{\Phi }\right|}^2$, thereby generating the self-phase modulation structure. On the right-hand side, the quantities ${\sigma }_j$, $1\le j\le 3$ are effective material parameters that encode higher-order perturbative contributions from the optical metamaterial. The parameter l characterizes the generalized temporal evolution: when $l=1$ the model reduces to the case of linear temporal evolution previously studied in Ref. [37], whereas $l\ne 1$ accounts for more general scaling laws. In this work we treat $l>0$ as a real parameter so that both integer and non-integer values are admissible, and $m>0$ is an arbitrary intensity exponent.

2. Mathematical Preliminaries

Equation (1) does not support mobile solitons. Instead it only supports quiescent optical solitons as proved earlier [2]. Therefore the solution structure is taken to be of the form

$$\begin{array}{c}\hfill q\left(x,t\right)=\phi \left(x\right){\mathrm{e}}^{i\omega t},\end{array}$$

(2)

where $\omega$ is the wave number of the quiescent optical soliton. By substituting (2) into (1) and separating into real and imaginary components, the part that is imaginary produces the ordinary differential Equation (ODE) for the amplitude component $\phi \left(x\right)$ as

$$\begin{array}{c}\hfill l{\sigma }_3{\phi }^{2m}\left(x\right)\left[\phi \left(x\right){\phi }^{″}\left(x\right)+(l-1){\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\right]=0,\end{array}$$

(3)

while the real part gives

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left[l\omega -F\left\{{\phi }^2\left(x\right)\right\}\right]=0.\hfill \end{array}$$

(4)

Here we have introduced the auxiliary function $F\left(u\right)$ as an antiderivative of the refractive index function H, namely,

$$F\left(u\right)={\int }_0^{u}H\left(s\right)\mathrm{d}s,$$

so that the contribution involving $H\left(\right|\mathrm{\Phi }{|}^2)$ in Equation (1) gives rise to the term $F\left({\phi }^2\left(x\right)\right)$ in Equation (4). To avoid confusion with the standard notation $F\left(\psi \right|\mu )$ for elliptic integrals used later in Section 3.6, we shall explicitly refer to the latter as the incomplete elliptic integral of the first kind.

• Equation (3) concludes that

$$\begin{array}{c}\hfill {\sigma }_3=0,\end{array}$$

(5)

for integrability of (1). By virtue of (5), Equation (1) condenses to

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+H\left({\left|\mathrm{\Phi }\right|}^2\right){\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(6)

This ODE given by (4) will be addressed in detail in the subsequent section with a wide range of SPM structures, corresponding to the functional H, that would yield a variety of solution forms.

3. Application to an Array of SPM Structures

The above analysis will be applied to diverse SPM structures, which will lead to various forms of quiescent optical solitons. While most of the results are in terms of quadratures, a few of them are expressible with special functions. The rigorous analyses are enumerated in subsequent eighteen subsections, where the details are exhibited in regards to the retrieval of the solutions.

Although many of the solutions are written in implicit form and involve special functions, their qualitative behavior can still be understood in simple physical terms. For realistic parameter values the profiles that we obtain are bell-shaped, exponentially localized waveforms whose amplitude is determined by the balance conditions discussed above. In the Kerr and power law cases, for instance, the hypergeometric expressions reduce in appropriate limits to deformations of the standard bright soliton profile known from the cubic nonlinear Schrödinger equation. Moreover, when the generalized temporal evolution parameter is set to $l=1$ and the additional perturbation terms are switched off, our implicit solutions collapse to those reported previously for quiescent solitons with linear temporal evolution [10,38], thereby providing a useful consistency check.

In what follows we consider a broad family of self-phase modulation laws $H\left(s\right)$ and, for each choice, derive the corresponding implicit quiescent soliton solutions. To enhance the physical interpretation, we briefly comment for each model on the type of optical medium or metamaterial that it represents, on typical situations in which such a refractive index response is relevant, and on how the recovered quiescent structures can be exploited or should be avoided in practical telecommunication scenarios. Where available, we also point to representative experimental or modeling studies that motivate the use of a given SPM law.

3.1. Kerr Law

In the present setting, Kerr law nonlinearity (also known as the Kerr effect or optical Kerr effect) is a basic phenomenon in nonlinear optics describing how a material’s refractive index varies with the intensity of the light traversing it. This framework models metamaterials and waveguides, in which the dominant contribution to the refractive index is proportional to the local intensity, as in standard silica fibers, highly nonlinear glass fibers, and many Kerr-type composites. Our quiescent soliton solutions therefore describe stalled wave packets in media where the linear chromatic dispersion is dressed by a purely cubic response, in line with earlier analyses of stationary solitons in models with nonlinear chromatic dispersion and Kerr-type or more general intensity-dependent refractive index laws [4,20]. From a practical viewpoint, the parameter constraints obtained for this case can be used to identify operating regimes (e.g., in dispersion-managed fiber links) in which the formation of quiescent structures is suppressed so that robust mobile soliton transmission is maintained.

For the Kerr law of SPM, one must have

$$\begin{array}{c}\hfill H\left(s\right)=bs,\end{array}$$

(7)

for non-zero real-valued constant b. This would transform the governing Equation (6) to

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+b{\left|\mathrm{\Phi }\right|}^2{\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(8)

Consequently the ODE for $\phi \left(x\right)$ given by (4) simplifies to

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -b{\phi }^2\left(x\right)\right\}=0.\hfill \end{array}$$

(9)

For (9) to be integrable, one must choose

$$\begin{array}{c}\hfill n=2m.\end{array}$$

(10)

This choice modifies the governing Equation (8) to

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+b{\left|\mathrm{\Phi }\right|}^2{\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(11)

Moreover, the ODE (9) simplifies to

$$\begin{array}{cc}\hfill & \left\{\left(a-{\sigma }_1\right)(l+2m)-{\sigma }_{2}l\right\}{\phi }^{2m+1}\left(x\right){\phi }^{″}\left(x\right)\hfill \\ +& \left\{\left(a-{\sigma }_1\right)(l+2m)(l+2m-1)-(l-1){\sigma }_2\right\}{\phi }^{2m}\left(x\right){\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -b{\phi }^2\left(x\right)\right\}=0.\hfill \end{array}$$

(12)

Equation (12) has a unique Lie point symmetry, specifically ∂/∂x. By using this symmetry, the implicit solution to the ODE (12) is obtained as

$$\begin{array}{c}\hfill x=L {}_{2}F_1\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{m}{2}};{\displaystyle \frac{2+m}{2}};{\displaystyle \frac{b{\phi }^2\left\{(l+m)(l+2m)\left(a-{\sigma }_1\right)-l(l-m){\sigma }_2\right\}}{l\omega \left\{(l+m+1)(l+2m)\left(a-{\sigma }_1\right)-l(l-m+1){\sigma }_2\right\}}}\right),\end{array}$$

(13)

where

$$\begin{array}{c}\hfill L=\pm {\displaystyle \frac{{\phi }^m}{m}}\sqrt{{\displaystyle \frac{(l+m)(l+2m)\left(a-{\sigma }_1\right)-l(l-m){\sigma }_2}{l\omega }}}.\end{array}$$

(14)

The Gauss hypergeometric function is defined as follows:

$$\begin{array}{c}\hfill {}_{2}F_1\left(\alpha ,\beta ;\gamma ;z\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(\alpha \right)}_n{\left(\beta \right)}_n}{{\left(\gamma \right)}_n}}{\displaystyle \frac{z^n}{n!}},\end{array}$$

(15)

whereas the Pochhammer symbol can be defined by

$$\begin{array}{c}\hfill {\left(p\right)}_n=\left\{\begin{array}{cc}1\hfill & n=0,\hfill \\ p(p+1)\dots (p+n-1)\hfill & n>0.\hfill \end{array}\right.\end{array}$$

(16)

The requirement that ensures the convergence of the hypergeometric series. It is well-recognized and expressed by

$$\begin{array}{c}\hfill \left|z\right|<1,\end{array}$$

(17)

which, for the case of Equation (13), requires

$$\begin{array}{c}\hfill -1<{\displaystyle \frac{b{\phi }^2\left\{(l+m)(l+2m)\left(a-{\sigma }_1\right)-l(l-m){\sigma }_2\right\}}{l\omega \left\{(l+m+1)(l+2m)\left(a-{\sigma }_1\right)-l(l-m+1){\sigma }_2\right\}}}<1.\end{array}$$

(18)

Finally, Equation (14) establishes the limitation

$$\begin{array}{c}\hfill l\omega \left\{(l+m)(l+2m)\left(a-{\sigma }_1\right)-l(l-m){\sigma }_2\right\}>0,\end{array}$$

(19)

This must also be true for the solutions to be effective.

3.2. Power Law

In nonlinear optics, the transmission of optical solitons (self-sustaining wave packets) across diverse media can be described by nonlinear Schrödinger equations (NLSEs) with power law nonlinearities, among which the Kerr law nonlinearity is a particular case. Power law nonlinearities provide an effective description of media whose nonlinear refractive index deviates from the strictly cubic dependence, for instance, in chalcogenide glasses, semiconductor waveguides, or metamaterials with strong resonance effects, where the response can grow sub- or super-linearly with the intensity. In our model, the power law SPM term captures such non-Kerr behavior in a compact form and allows us to track how the exponent and strength of the nonlinearity impact the existence of quiescent solitons. The resulting implicit solutions and parameter restrictions can be used to estimate intensity thresholds or provide design guidelines for engineered structures where one wishes either to access or to avoid regimes with stalled energy localization.

The Kerr law of SPM is given by

$$\begin{array}{c}\hfill H\left(s\right)=b{s}^m,\end{array}$$

(20)

for non-zero real-valued constant b. This would transform the governing Equation (6) to

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+b{\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(21)

Consequently the ODE for $\phi \left(x\right)$ given by (4) simplifies to

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -b{\phi }^{2m}\left(x\right)\right\}=0.\hfill \end{array}$$

(22)

For (22) to be integrable, one must choose the same relation between m and n as given by (10). This modifies the governing Equation (21) to

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+b{\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(23)

Moreover, the ODE (23) simplifies to

$$\begin{array}{cc}\hfill & \left\{\left(a-{\sigma }_1\right)(l+2m)-{\sigma }_{2}l\right\}{\phi }^{2m+1}\left(x\right){\phi }^{″}\left(x\right)\hfill \\ +& \left\{\left(a-{\sigma }_1\right)(l+2m)(l+2m-1)-(l-1){\sigma }_2\right\}{\phi }^{2m}\left(x\right){\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -b{\phi }^{2m}\left(x\right)\right\}=0.\hfill \end{array}$$

(24)

By using the translational Lie point symmetry, as confirmed by the ODE (24), the implicit solution can be obtained as

$$\begin{array}{c}\hfill x={\displaystyle \frac{1}{m}}\sqrt{{\displaystyle \frac{{(l+2m)}^2\left(a-{\sigma }_1\right)-l^2{\sigma }_2}{b}}}{sin}^{-1}\left[{\phi }^m\sqrt{{\displaystyle \frac{(l+m)(l+2m)\left(a-{\sigma }_1\right)b-l(l-m)b{\sigma }_2}{l{(l+2m)}^2\omega \left(a-{\sigma }_1\right)-l^3\omega {\sigma }_2}}}\right].\end{array}$$

(25)

Equation (25) naturally kicks in two parameter constraints for the existence of the recovered solution. They are

$$\begin{array}{c}\hfill b\left\{{(l+2m)}^2\left(a-{\sigma }_1\right)-l^2{\sigma }_2\right\}>0,\end{array}$$

(26)

and

$$\begin{array}{c}\hfill l\left\{(l+m)(l+2m)\left(a-{\sigma }_1\right)b-l(l-m)b{\sigma }_2\right\}\left\{{(l+2m)}^2\omega \left(a-{\sigma }_1\right)-l^2\omega {\sigma }_2\right\}>0.\end{array}$$

(27)

The integrability conditions derived in the Lie symmetry reduction, such as inequalities (19), (26), (27) and their analogues for the other SPM structures, have a clear physical interpretation. They express the requirement that the effective nonlinear phase accumulation induced by the refractive index modulation must compensate the nonlinear chromatic dispersion in such a way that a stationary, quiescent balance is possible. In other words, only when the dispersion and the intensity-dependent phase shift satisfy these structural relations can the wave amplitude remain spatially localized while its temporal profile is frozen. Outside these parameter domains the same model still describes nonlinear propagation, but the quiescent soliton regime ceases to exist.

3.3. Parabolic (Cubic–Quintic) Law

In nonlinear optics, a cubic–quintic (or parabolic) SPM law describes a medium whose governing equation includes terms proportional to both the cubic and quintic powers of the field so that the nonlinear refractive index embodies a competition between focusing and defocusing contributions. This framework is frequently employed as an effective model for media with saturation or higher-order susceptibility corrections, including fiber and waveguide systems with non-Kerr responses [14]. In such settings the quintic term can regularize the cubic focusing nonlinearity at high intensities. Our quiescent soliton solutions for this law therefore describe stalled pulses in a regime where self-focusing is partially compensated by higher-order defocusing, and the associated constraints on the parameters provide useful information on how strong the quintic correction must be to prevent the formation of immobile localized states.

For the parabolic law of SPM, the following must be complied with:

$$\begin{array}{c}\hfill H\left(s\right)=b_{1}s+b_2{s}^2,\end{array}$$

(28)

for non-zero real-valued constants $b_j$ ($j=1,2$). This would transform the governing Equation (6) to

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left(b_1{\left|\mathrm{\Phi }\right|}^2+b_2{\left|\mathrm{\Phi }\right|}^4\right){\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(29)

Consequently the ODE for $\phi \left(x\right)$ given by (4) simplifies to

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -b_1{\phi }^2\left(x\right)-b_2{\phi }^4\left(x\right)\right\}=0.\hfill \end{array}$$

(30)

For (30) to be integrable, one must select the same relation between m and n as in (10) as well as

$$\begin{array}{c}\hfill l=-2m.\end{array}$$

(31)

These selected values modify the governing Equation (29) to

$$\begin{array}{c}\hfill {{\displaystyle \frac{i}{\left({\mathrm{\Phi }}^{2m}\right)}}}_t+{\displaystyle \frac{b_1{\left|\mathrm{\Phi }\right|}^2+b_2{\left|\mathrm{\Phi }\right|}^4}{{\mathrm{\Phi }}^{2m}}}={\sigma }_2{{\displaystyle \frac{{\left|\mathrm{\Phi }\right|}^{2m}}{\left({\mathrm{\Phi }}^{2m}\right)}}}_{xx}\end{array}$$

(32)

while the ODE for $\phi \left(x\right)$ simplifies to

$$\begin{array}{c}\hfill b_1{\phi }^4\left(x\right)+b_2{\phi }^6\left(x\right)-2m{\sigma }_2{\phi }^{2m}\left(x\right)\left\{(2m+1){\phi }^{\prime }{\left(x\right)}^2-\phi \left(x\right){\phi }^{″}\left(x\right)\right\}+2m\omega {\phi }^2\left(x\right)=0.\end{array}$$

(33)

The implicit solution, using the same translational Lie symmetry established by (33), is expressed in terms of Appell’s hypergeometric function as

$$\begin{array}{c}\hfill x={\phi }^m\sqrt{{\displaystyle \frac{3{\sigma }_2}{m\omega }}}{F}_1\left({\displaystyle \frac{m}{2}};{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}};{\displaystyle \frac{2+m}{2}};A_1,A_2\right),\end{array}$$

(34)

where

$$\begin{array}{c}\hfill A_1=-{\displaystyle \frac{6b_2(3m-1){\phi }^2}{\sqrt{3}\sqrt{(3m-2)\left\{3b_1^2(3m-2)-8b_2{(3m-1)}^2\omega \right\}}+3b_1(3m-2)}},\end{array}$$

(35)

and

$$\begin{array}{c}\hfill A_2=-{\displaystyle \frac{6b_2(3m-1){\phi }^2}{3b_1(3m-2)-\sqrt{3}\sqrt{(3m-2)\left\{3b_1^2(3m-2)-8b_2{(3m-1)}^2\omega \right\}}}}.\end{array}$$

(36)

This Appell hypergeometric function is defined by the infinite series:

$$\begin{array}{c}\hfill F_1\left(a;b_1,b_2;c;x,y\right)=x^m{y}^n\left(\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(a\right)}_{m+n}{\left(b_1\right)}_m{\left(b_2\right)}_n}{{\left(c\right)}_{m+n}m!n!}}\right),\end{array}$$

(37)

which is convergent within the specified region defined by

$$\begin{array}{c}\hfill max\left(\left|x\right|,\left|y\right|\right)<1,\end{array}$$

(38)

which in this case means

$$\begin{array}{c}\hfill max\left(\left|A_1\right|,\left|A_2\right|\right)<1.\end{array}$$

(39)

An additional constraint that stems from (34) for the existence of the solution is

$$\begin{array}{c}\hfill m\omega {\sigma }_2>0.\end{array}$$

(40)

3.4. Dual Power Law

In nonlinear optics, dual power law nonlinearity refers to a nonlinear optical response in which the refractive index (or, equivalently, the polarization density) depends on the light intensity through two distinct power law terms, rather than a single Kerr-type contribution. The corresponding dual power SPM law, which combines these two power law contributions to the refractive index, provides a simple effective description of media where more than one nonlinear mechanism is active. In composite or otherwise engineered optical media, this can represent, for example, the coexistence of fast electronic and slower orientational contributions or the combined effect of Kerr-like and saturable responses. Closely related multi-exponent refractive index structures—most notably Kudryashov’s quintuple power law and its generalizations—have been employed to model stationary and quiescent optical solitons in systems with nonlinear chromatic dispersion [12,13]. In our framework, the dual power model then allows us to quantify how the interplay between the two exponents shapes the parameter window for quiescent soliton formation. The Lie symmetry reduction shows that only specific combinations of the two powers yield integrable structures, and the resulting implicit solutions can guide the design of multi-component systems where quiescent states are either exploited (for optical storage) or mitigated (for long-haul transmission).

In this case, the SPM is written as

$$\begin{array}{c}\hfill H\left(s\right)=b_1{s}^m+b_2{s}^{m+1}.\end{array}$$

(41)

The governing model therefore takes the form

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left(b_1{\left|\mathrm{\Phi }\right|}^{2m}+b_2{\left|\mathrm{\Phi }\right|}^{2m+2}\right){\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(42)

The corresponding ODE for $\phi \left(x\right)$ now reads

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -b_1{\phi }^{2m}\left(x\right)-b_2{\phi }^{2m+2}\left(x\right)\right\}=0.\hfill \end{array}$$

(43)

For integrability of (43) one must choose

$$\begin{array}{c}\hfill n=-1,\end{array}$$

(44)

and

$$\begin{array}{c}\hfill 2m+1=0.\end{array}$$

(45)

These specific values of the power law parameters modify the governing model (42) to

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\displaystyle \frac{{\mathrm{\Phi }}^l}{\left|\mathrm{\Phi }\right|}}\right)}_{xx}+\left({\displaystyle \frac{b_1}{\left|\mathrm{\Phi }\right|}}+b_2\left|\mathrm{\Phi }\right|\right){\mathrm{\Phi }}^l={\sigma }_1{\left({\displaystyle \frac{{\mathrm{\Phi }}^l}{\left|\mathrm{\Phi }\right|}}\right)}_{xx}+{\sigma }_2{\displaystyle \frac{{\left({\mathrm{\Phi }}^l\right)}_{xx}}{\left|\mathrm{\Phi }\right|}}.\end{array}$$

(46)

Therefore the corresponding ODE for $\phi \left(x\right)$ from (43) simplifies to

$$\begin{array}{cc}\hfill & \left\{(l-1)\left(a-{\sigma }_1\right)-l{\sigma }_2\right\}{\phi }^{″}\left(x\right)\phi \left(x\right)+(l-1)\left\{(l-2)\left(a-{\sigma }_1\right)-l{\sigma }_2\right\}{\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\hfill \\ +& b_1{\phi }^{2m}\left(x\right)+b_2{\phi }^{2m+2}\left(x\right)-l\omega {\phi }^2\left(x\right)=0.\hfill \end{array}$$

(47)

Utilizing translational Lie symmetry, one obtains the implicit solution as

$$\begin{array}{c}\hfill x=\sqrt{-{\displaystyle \frac{{(l-1)}^2\left(a-{\sigma }_1\right)-l^2{\sigma }_2}{b_1}}}ln\left(A_1+A_2+A_3\right),\end{array}$$

(48)

$$\begin{array}{c}\hfill A_1={\displaystyle \frac{2l^{\frac{3}{2}}\omega \left\{-\left(2l^2-1\right)(l-1){\sigma }_2\left(a-{\sigma }_1\right)+{(l-1)}^3{\left(a-{\sigma }_1\right)}^2+l^2(l+1){\sigma }_2^2\right\}}{\sqrt{b_1}\sqrt{-(l-1)\left(4l^2+2l-1\right){\sigma }_2\left(a-{\sigma }_1\right)-{(l-1)}^2(2l-1){\left(a-{\sigma }_1\right)}^2+l(l+1)(2l+1){\sigma }_2^2}}},\end{array}$$

(49)

$$\begin{array}{c}\hfill A_2={\displaystyle \frac{2\sqrt{l{b}_1}\sqrt{-(l-1)\left(4l^2+2l-1\right){\sigma }_2\left(a-{\sigma }_1\right)-{(l-1)}^2(2l-1){\left(a-{\sigma }_1\right)}^2+l(l+1)(2l+1){\sigma }_2^2}}{\phi }},\end{array}$$

(50)

$$\begin{array}{c}\hfill A_3={\displaystyle \frac{2\sqrt{b_1{B}_{3}l+B_1{B}_2\phi }}{\phi }},\end{array}$$

(51)

$$\begin{array}{c}\hfill B_1={(l-1)}^2\left(a-{\sigma }_1\right)-l^2{\sigma }_2,\end{array}$$

(52)

$$\begin{array}{c}\hfill B_2=(l-1)\left(a-{\sigma }_1\right)\left\{b_2(1-2l)\phi +2l^2\omega \right\}+l{\sigma }_2\left\{b_2(2l+1)\phi -2l(l+1)\omega \right\},\end{array}$$

(53)

and

$$\begin{array}{c}\hfill B_3=(l-1)\left(4l^2+2l-1\right){\sigma }_2\left(a-{\sigma }_1\right)-{(l-1)}^2(2l-1){\left(a-{\sigma }_1\right)}^2-l(l+1)(2l+1){\sigma }_2^2\end{array}$$

(54)

The immediate parameter constraints that are readable from (48) for the existence of the solution are

$$\begin{array}{c}\hfill b_1\left\{{(l-1)}^2\left(a-{\sigma }_1\right)-l^2{\sigma }_2\right\}<0,\end{array}$$

(55)

and

$$\begin{array}{c}\hfill A_1+A_2+A_3>0.\end{array}$$

(56)

3.5. Polynomial (Cubic–Quintic–Septic) Law

In nonlinear optics, a polynomial SPM law describes media whose nonlinear response is expressed as a polynomial function of the light intensity (or equivalently the square of the electric field) so that the effective refractive index includes successive nonlinear contributions beyond the basic cubic (Kerr) term. Polynomial SPM laws with cubic–quintic–septic terms model situations in which several higher-order susceptibilities contribute simultaneously to the effective refractive index, leading to a strongly intensity-dependent nonlinearity. Such higher-order nonlinear responses have been used, for example, in models of highly dispersive optical media with cubic–quintic–septic self-phase modulation [14], and, at an effective level, similar behavior can arise in metamaterials with multiple resonances or in complex photonic crystal structures. In our framework, the cubic–quintic–septic polynomial law encapsulates these effects in a compact form, and our analysis reveals how the coefficients of the cubic, quintic, and septic terms must be tuned for quiescent solitons to exist, thereby providing a mapping between device parameters and the emergence of stalled localized patterns in such highly nonlinear environments.

In this case the functional F reads:

$$\begin{array}{c}\hfill H\left(s\right)=b_{1}s+b_2{s}^2+b_3{s}^3,\end{array}$$

(57)

for non-zero real-valued constants $b_j$ where $j=1,2,3$. The governing model consequently is represented by

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left(b_1{\left|\mathrm{\Phi }\right|}^2+b_2{\left|\mathrm{\Phi }\right|}^4+b_3{\left|\mathrm{\Phi }\right|}^6\right){\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}\end{array}$$

(58)

Hence upon substituting (2) into (58), the ODE for $\phi \left(x\right)$ is given as

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -b_1{\phi }^2\left(x\right)-b_2{\phi }^4\left(x\right)-b_3{\phi }^6\left(x\right)\right\}=0.\hfill \end{array}$$

(59)

By implementing the translational Lie symmetry supported by (59), one recovers its implicit solution in quadratures as

$$\begin{array}{c}\hfill x=ʃ\sqrt{{\displaystyle \frac{e^{\frac{2\left(A_1+A_2\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}}{2{\int }^{\phi }\frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\tau \left(l\omega -{\tau }^2{b}_1-{\tau }^4{b}_2-{\tau }^6{b}_3\right)}{a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}}d\tau }}}d\phi ,\end{array}$$

(60)

where

$$\begin{array}{c}\hfill A_1={\sigma }_1(l+2m)\left[ln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\phi \right],\end{array}$$

(61)

$$\begin{array}{c}\hfill A_2={\displaystyle \frac{l{\sigma }_2\left[ln\phi \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}},\end{array}$$

(62)

$$\begin{array}{c}\hfill A_3={\sigma }_1(l+2m)\left[ln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\tau \right],\end{array}$$

(63)

and

$$\begin{array}{c}\hfill A_4={\displaystyle \frac{l{\sigma }_2\left[ln\tau \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}}.\end{array}$$

(64)

An essential parameter constraint for the solitons to exist is

$$\begin{array}{c}\hfill {\int }^{\phi }{\displaystyle \frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\tau \left(l\omega -{\tau }^2{b}_1-{\tau }^4{b}_2-{\tau }^6{b}_3\right)}{a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}}}d\tau >0.\end{array}$$

(65)

3.6. Triple Power Law

The triple power SPM law may be viewed as a simplified representation of media where three distinct nonlinear mechanisms operate over different intensity ranges, such as a low-intensity Kerr response, an intermediate saturating regime, and a high-intensity defocusing contribution. In metamaterial platforms, such multi-scale behavior can be engineered by combining several resonant inclusions with different characteristic frequencies. The implicit solutions obtained for the triple power case, which involve elliptic integrals, characterize the resulting quiescent solitons in terms of four effective amplitude roots. These expressions can be used to delineate parameter regions where stationary multi-hump or single-hump profiles are possible and to anticipate transitions between different regimes as the input intensity is varied.

For triple power law of nonlinearity,

$$\begin{array}{c}\hfill H\left(s\right)=b_1{s}^m+b_2{s}^{m+1}+b_3{s}^{m+2},\end{array}$$

(66)

for real-valued non-zero constants $b_j$ with $j=1,2,3$. The corresponding governing model is written as

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left(b_1{\left|\mathrm{\Phi }\right|}^{2m}+b_2{\left|\mathrm{\Phi }\right|}^{2m+2}+b_3{\left|\mathrm{\Phi }\right|}^{2m+4}\right){\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(67)

Upon substituting (2) into (67), the governing equation for $\phi \left(x\right)$ turns out to be

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -b_1{\phi }^{2m}\left(x\right)-b_2{\phi }^{2m+2}\left(x\right)-b_3{\phi }^{2m+4}\left(x\right)\right\}=0.\hfill \end{array}$$

(68)

For integrability, one is compelled to select the same values of m and n as given by (44) and (45) as well as the relation

$$\begin{array}{c}\hfill a-{\sigma }_1=0\end{array}$$

(69)

must hold. With these selected values, the governing model (67) changes to

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+\left({\displaystyle \frac{b_1}{\left|\mathrm{\Phi }\right|}}+b_2\left|\mathrm{\Phi }\right|+b_3{\left|\mathrm{\Phi }\right|}^3\right){\mathrm{\Phi }}^l={\sigma }_2{\displaystyle \frac{{\left({\mathrm{\Phi }}^l\right)}_{xx}}{\left|\mathrm{\Phi }\right|}},\end{array}$$

(70)

while the ODE for $\phi \left(x\right)$ reduces to

$$\begin{array}{c}\hfill b_1{\phi }^2\left(x\right)+b_2{\phi }^4\left(x\right)+b_3{\phi }^6\left(x\right)-l{\sigma }_2{\phi }^{″}\left(x\right)\phi \left(x\right)-l(l-1){\sigma }_2{\left\{{\phi }^{\prime }\left(x\right)\right\}}^2-l\omega {\phi }^3\left(x\right)=0.\end{array}$$

(71)

The translational Lie symmetry supported by (71) would yield its integral in an implicit form as

$$\begin{array}{c}\hfill x=L[lF({sin}^{-1}(\sqrt{{\displaystyle \frac{(\phi -{\kappa }_1)({\kappa }_2-{\kappa }_4)}{(\phi -{\kappa }_2)({\kappa }_1-{\kappa }_4)}}})|{\displaystyle \frac{({\kappa }_2-{\kappa }_3)({\kappa }_1-{\kappa }_4)}{({\kappa }_1-{\kappa }_3)({\kappa }_2-{\kappa }_4)}}){\kappa }_1\\ \hfill +l\mathrm{\Pi }({\displaystyle \frac{{\kappa }_2({\kappa }_1-{\kappa }_4)}{{\kappa }_1({\kappa }_2-{\kappa }_4)}};{sin}^{-1}(\sqrt{{\displaystyle \frac{(\phi -{\kappa }_1)({\kappa }_2-{\kappa }_4)}{(\phi -{\kappa }_2)({\kappa }_1-{\kappa }_4)}}})|{\displaystyle \frac{({\kappa }_2-{\kappa }_3)({\kappa }_1-{\kappa }_4)}{({\kappa }_1-{\kappa }_3)({\kappa }_2-{\kappa }_4)}})\\ \hfill \times (-{\kappa }_1+{\kappa }_2)],\end{array}$$

(72)

where

$$\begin{array}{c}\hfill L=\pm \sqrt{-{\displaystyle \frac{A}{B}}},\end{array}$$

(73)

$$\begin{array}{c}\hfill A=4(1+l)(2+l)(1+2l)\left(\phi -{\kappa }_1\right)\left(\phi -{\kappa }_2\right)\left(\phi -{\kappa }_3\right)\left(\phi -{\kappa }_4\right){\sigma }_2,\end{array}$$

(74)

$$\begin{array}{c}\hfill B=\left[2l^2(1+l)(2+l)\phi \omega +(1+2l)\left(-\left((2+l)\left((1+l)b_1+l{\phi }^2{b}_2\right)\right)-l(1+l){\phi }^4{b}_3\right)\right]C,\end{array}$$

(75)

and

$$\begin{array}{c}\hfill C={\kappa }_1^2{\kappa }_2^2\left({\kappa }_1-{\kappa }_3\right)\left({\kappa }_2-{\kappa }_4\right).\end{array}$$

(76)

Here, ${\kappa }_i$ with $i=1,\dots ,4$ is any solution of

$${\kappa }^4(2b_3{l}^3+3b_3{l}^2+b_{3}l)+{\kappa }^2(2b_2{l}^3+5b_2{l}^2+2b_{2}l)-\kappa (2l^4\omega +6l^3\omega +4l^2\omega )+2b_1{l}^3+7b_1{l}^2+7b_{1}l+2b_1=0.$$

(77)

In (72), $F\left(\psi \right|\mu )$ is elliptic integral of the first kind and is defined by

$$\begin{array}{c}\hfill F\left(\psi \right|\mu )={\int }_0^{\psi }{\displaystyle \frac{1}{\sqrt{1-\mu {sin}^2\left(\theta \right)}}}d\theta ,\end{array}$$

(78)

for

$$\begin{array}{c}\hfill -{\displaystyle \frac{\pi }{2}}<\psi <{\displaystyle \frac{\pi }{2}},\end{array}$$

(79)

and

$$\begin{array}{c}\hfill \mu {sin}^2\psi <1.\end{array}$$

(80)

Also, $\mathrm{\Pi }(\nu ;\psi |m)$ the incomplete elliptic integral and is given by

$$\begin{array}{c}\hfill \mathrm{\Pi }(\nu ;\psi |\mu )={\int }_0^{\psi }{\displaystyle \frac{1}{\left(1-\nu {sin}^2\theta \right)\sqrt{1-\mu {sin}^2\theta }}}d\theta ,\end{array}$$

(81)

where

$$\begin{array}{c}\hfill \nu {sin}^2\psi >1,\end{array}$$

(82)

while (79) and (80) remain valid here too. Again from (73), one must have

$$\begin{array}{c}\hfill AB<0,\end{array}$$

(83)

for the solutions to exist.

3.7. Cubic–Quintic–Septic–Nonic Law

The cubic–quintic–septic–nonic SPM law extends the previous polynomial model by including yet another high-order contribution, thereby mimicking media with very strong nonlinearity or metamaterials designed to approximate an underlying saturable or multi-resonant response by a high-degree polynomial. While such a model is primarily of theoretical interest, it also serves as a useful effective description for numerical design of optical components where the refractive index must remain bounded at high powers. The corresponding quiescent soliton solutions, expressed in quadratures, indicate how successive higher-order terms progressively narrow the admissible parameter window for stalled propagation and thus offer guidance for engineering “quiescence-free” operating regimes.

The law of SPM in this case is given by

$$\begin{array}{c}\hfill H\left(s\right)=b_{1}s+b_2{s}^2+b_3{s}^3+b_4{s}^4,\end{array}$$

(84)

for real-valued non-zero constants $b_j$ with $1\le j\le 4$. Thus, the governing model equation would be given by

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left(b_1{\left|\mathrm{\Phi }\right|}^2+b_2{\left|\mathrm{\Phi }\right|}^4+b_3{\left|\mathrm{\Phi }\right|}^6+b_4{\left|\mathrm{\Phi }\right|}^8\right){\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(85)

Substituting (2) into (85) would give the ODE for $\phi \left(x\right)$ as

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -b_1{\phi }^2\left(x\right)-b_2{\phi }^4\left(x\right)-b_3{\phi }^6\left(x\right)-b_4{\phi }^8\left(x\right)\right\}=0.\hfill \end{array}$$

(86)

Next, utilizing the translational Lie symmetry supported by (86) leads its implicit integral to be

$$\begin{array}{c}\hfill x=ʃ\sqrt{{\displaystyle \frac{e^{\frac{2\left(A_1+A_2\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}}{2{\int }^{\phi }\frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\tau \left(l\omega -{\tau }^2{b}_1-{\tau }^4{b}_2-{\tau }^6{b}_3-{\tau }^8{b}_4\right)}{a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}}d\tau }}}d\phi ,\end{array}$$

(87)

where

$$\begin{array}{c}\hfill A_1={\sigma }_1(l+2m)\left[log\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\phi \right],\end{array}$$

(88)

$$\begin{array}{c}\hfill A_2={\displaystyle \frac{l{\sigma }_2\left[ln\phi \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}},\end{array}$$

(89)

$$\begin{array}{c}\hfill A_3={\sigma }_1(l+2m)\left[ln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\tau \right],\end{array}$$

(90)

and

$$\begin{array}{c}\hfill A_4={\displaystyle \frac{l{\sigma }_2\left[ln\tau \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}}\end{array}$$

(91)

From (87), the parameter constraint that must hold for the solitons to exist is

$$\begin{array}{c}\hfill {\int }^{\phi }{\displaystyle \frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\tau \left(l\omega -{\tau }^2{b}_1-{\tau }^4{b}_2-{\tau }^6{b}_3-{\tau }^8{b}_4\right)}{a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}}}d\tau >0.\end{array}$$

(92)

3.8. Quadrupled Power Law

The quadrupled power SPM structure provides another example of how multiple nonlinear mechanisms can be combined in a single effective law. This situation is relevant, for instance, in graded-index metamaterials where the local composition varies across the structure, giving rise to spatially dependent intensity exponents. The quadrature representation of the corresponding quiescent soliton solutions allows one to map directly from material parameters to existence domains, thereby helping to identify fabrication tolerances that either enhance or suppress quiescent localization in such multi-power media.

To express the quadrupled power law of the SPM structure, one may formulate H as

$$\begin{array}{c}\hfill H\left(s\right)=b_1{s}^m+b_2{s}^{m+1}+b_3{s}^{m+2}+b_4{s}^{m+3}.\end{array}$$

(93)

Hence, the governing model can be written as

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left(b_1{\left|\mathrm{\Phi }\right|}^{2m}+b_2{\left|\mathrm{\Phi }\right|}^{2m+2}+b_3{\left|\mathrm{\Phi }\right|}^{2m+4}+b_4{\left|\mathrm{\Phi }\right|}^{2m+6}\right){\mathrm{\Phi }}^l\\ \hfill ={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(94)

Substituting (2) into (94) leads the ODE for $\phi \left(x\right)$ to be

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -b_1{\phi }^{2m}\left(x\right)-b_2{\phi }^{2m+2}\left(x\right)-b_3{\phi }^{2m+4}\left(x\right)-b_4{\phi }^{2m+6}\left(x\right)\right\}=0.\hfill \end{array}$$

(95)

By the applicable translational Lie symmetry, supported by (95), this gives the implicit solution in terms of quadratures as

$$\begin{array}{c}\hfill x=ʃ\sqrt{{\displaystyle \frac{e^{\frac{2\left(A_1+A_2\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}}{2{\int }^{\phi }\frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\tau \left\{l\omega -{\tau }^{2m}\left(b_1+{\tau }^2{b}_2+{\tau }^4{b}_3+{\tau }^6{b}_4\right)\right\}}{a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}}d\tau }}}\end{array}$$

(96)

where

$$\begin{array}{c}\hfill A_1={\sigma }_1(l+2m)\left[ln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\phi \right],\end{array}$$

(97)

$$\begin{array}{c}\hfill A_2={\displaystyle \frac{l{\sigma }_2\left[ln\phi \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}},\end{array}$$

(98)

$$\begin{array}{c}\hfill A_3={\sigma }_1(l+2m)\left[ln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\tau \right],\end{array}$$

(99)

and

$$\begin{array}{c}\hfill A_4={\displaystyle \frac{l{\sigma }_2\left[ln\tau \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}}.\end{array}$$

(100)

The parameter constraint that naturally enters the picture from (96) for the solution to exist is

$$\begin{array}{c}\hfill {\int }^{\phi }{\displaystyle \frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\tau \left\{l\omega -{\tau }^{2m}\left(b_1+{\tau }^2{b}_2+{\tau }^4{b}_3+{\tau }^6{b}_4\right)\right\}}{a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}}}d\tau >0.\end{array}$$

(101)

3.9. Anti-Cubic Law

Anti-cubic SPM laws involve inverse powers of the intensity and appear in effective models of nonlocal or resonant media where the nonlinear index correction weakens at low intensities but grows more rapidly at intermediate levels. In optical metamaterials this can arise from specific resonant inclusions or from nonlocal coupling between unit cells. Our quiescent soliton solutions for the anti-cubic case illuminate how such unusual nonlinearities can still support stalled structures, and the corresponding parameter constraints may be exploited to design devices in which nonlocal effects are strong enough to prevent unwanted quiescent trapping.

The form of SPM for the anti-cubic law is given by

$$\begin{array}{c}\hfill H\left(s\right)={\displaystyle \frac{b_1}{s^2}}+b_{2}s+b_3{s}^2,\end{array}$$

(102)

for non-zero real-valued constants $b_j$ with $1\le j\le 3$. The corresponding governing model therefore is

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left({\displaystyle \frac{b_1}{{\left|\mathrm{\Phi }\right|}^4}}+b_2{\left|\mathrm{\Phi }\right|}^2+b_3{\left|\mathrm{\Phi }\right|}^4\right){\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx},\end{array}$$

(103)

and the ODE for $\phi \left(x\right)$ consequently takes the form

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -{\displaystyle \frac{b_1}{{\phi }^4\left(x\right)}}-b_2{\phi }^2\left(x\right)-b_3{\phi }^4\left(x\right)\right\}=0.\hfill \end{array}$$

(104)

This ODE for $\phi \left(x\right)$ supports the translational Lie symmetry, which, when applied, would invert it to

$$\begin{array}{c}\hfill x=ʃ\sqrt{{\displaystyle \frac{e^{\frac{2\left(A_1+A_2\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}}{2{\int }^{\phi }\frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\left(l{\tau }^4\omega -b_1-{\tau }^6{b}_2-{\tau }^8{b}_3\right)}{a(l+n){\tau }^{n+3}-{\tau }^{2m+3}\left\{(2m+l){\sigma }_1+l{\sigma }_2\right\}}d\tau }}}d\phi ,\end{array}$$

(105)

where

$$\begin{array}{c}\hfill A_1={\sigma }_1(l+2m)\left[ln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\phi \right],\end{array}$$

(106)

$$\begin{array}{c}\hfill A_2={\displaystyle \frac{l{\sigma }_2\left[ln\phi \left\{2m(l+n-1)-n(l-1)\right\}-nln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}},\end{array}$$

(107)

$$\begin{array}{c}\hfill A_3={\sigma }_1(l+2m)\left[ln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\tau \right],\end{array}$$

(108)

and

$$\begin{array}{c}\hfill A_4={\displaystyle \frac{l{\sigma }_2\left[ln\tau \left\{2m(l+n-1)-n(l-1)\right\}-nln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}}.\end{array}$$

(109)

The parameter constraint for the existence of the solution is written as

$$\begin{array}{c}\hfill {\int }^{\phi }{\displaystyle \frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\left(l{\tau }^4\omega -b_1-{\tau }^6{b}_2-{\tau }^8{b}_3\right)}{a(l+n){\tau }^{n+3}-{\tau }^{2m+3}\left\{(2m+l){\sigma }_1+l{\sigma }_2\right\}}}d\tau >0.\end{array}$$

(110)

3.10. Generalized Anti-Cubic Law

The generalized anti-cubic law extends this idea by combining inverse and positive powers of the intensity so that the medium behaves effectively as a mixture of nonlocal and Kerr-type responses. This type of model is relevant for hybrid structures in which a local nonlinear component is embedded within a nonlocal host material. The implicit soliton solutions obtained here therefore provide a theoretical description of quiescent states in such hybrid configurations and highlight how the competition between the inverse and direct power terms controls the existence and shape of the localized profiles.

The SPM structure for such a law of nonlinearity is

$$\begin{array}{c}\hfill H\left(s\right)={\displaystyle \frac{b_1}{s^{m+1}}}+b_2{s}^m+b_3{s}^{m+1},\end{array}$$

(111)

which leads the governing model to read

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left({\displaystyle \frac{b_1}{{\left|\mathrm{\Phi }\right|}^{2(m+1)}}}+b_2{\left|\mathrm{\Phi }\right|}^{2m}+b_3{\left|\mathrm{\Phi }\right|}^{2(m+1)}\right){\mathrm{\Phi }}^l\\ \hfill ={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(112)

Upon substituting (2) into (112), the ODE for $\phi \left(x\right)$ is

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -{\displaystyle \frac{b_1}{{\phi }^{2m+2}\left(x\right)}}-b_2{\phi }^{2m}\left(x\right)-b_3{\phi }^{2m+2}\left(x\right)\right\}=0.\hfill \end{array}$$

(113)

After utilizing the translational Lie symmetry to (113), which it supports, the implicit integral is in quadratures given by

$$\begin{array}{c}\hfill x=ʃ\sqrt{{\displaystyle \frac{e^{\frac{2\left(A_1+A_2\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}}{2{\int }^{\phi }\frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\left[b_1-{\tau }^{2m+2}\left\{l\omega -{\tau }^{2m}\left(b_2+{\tau }^2{b}_3\right)\right\}\right]}{a(l+n){\tau }^{2m+n+1}-{\tau }^{4m+1}\left\{(2m+l){\sigma }_1+l{\sigma }_2\right\}}d\tau }}}d\phi ,\end{array}$$

(114)

where

$$\begin{array}{c}\hfill A_1={\sigma }_1(l+2m)\left[ln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(2m+l)+l{\sigma }_2\right\}\right|+(l-1)ln\phi \right],\end{array}$$

(115)

$$\begin{array}{c}\hfill A_2={\displaystyle \frac{l{\sigma }_2\left[ln\phi \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(2m+l)+l{\sigma }_2\right\}\right|\right]}{2m-n}},\end{array}$$

(116)

$$\begin{array}{c}\hfill A_3={\sigma }_1(l+2m)\left[ln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\tau \right],\end{array}$$

(117)

and

$$\begin{array}{c}\hfill A_4={\displaystyle \frac{l{\sigma }_2\left[ln\tau \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}}.\end{array}$$

(118)

The parameter constraint that must hold for the implicit solution to exist is given as

$$\begin{array}{c}\hfill {\int }^{\phi }{\displaystyle \frac{e^{\frac{2\left(A_3+A_4\right)}{(2m+l){\sigma }_1+l{\sigma }_2}}\left[b_1-{\tau }^{2m+2}\left\{l\omega -{\tau }^{2m}\left(b_2+{\tau }^2{b}_3\right)\right\}\right]}{a(l+n){\tau }^{2m+n+1}-{\tau }^{4m+1}\left\{(2m+l){\sigma }_1+l{\sigma }_2\right\}}}d\tau >0.\end{array}$$

(119)

3.11. Quadratic–Cubic Law

The quadratic–cubic SPM law is motivated by media in which both second- and third-order susceptibilities play a role, for example, in non-centrosymmetric crystals or in metamaterials where an effective quadratic response can be engineered. In this case the square root term in $H\left(s\right)$ accounts for a contribution proportional to the field amplitude, while the cubic term models the usual Kerr nonlinearity. The quiescent soliton solutions thus correspond to stalled states in which second-harmonic-like and Kerr-type mechanisms act together, and the associated constraints indicate how the relative strength of these effects must be tuned to reach or avoid such operating points.

The SPM structure for the quadratic–cubic law is

$$\begin{array}{c}\hfill H\left(s\right)=b_1\sqrt{s}+b_{2}s\end{array}$$

(120)

so that the governing model takes the form

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left(b_1\left|\mathrm{\Phi }\right|+b_2{\left|\mathrm{\Phi }\right|}^2\right){\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(121)

Substituting (2) into (121) leads the structure of the ODE for $\phi \left(x\right)$ to be

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -b_1\phi \left(x\right)-b_2{\phi }^2\left(x\right)\right\}=0.\hfill \end{array}$$

(122)

Next using the translational Lie symmetry, supported by (122), leads its implicit integral to be

$$\begin{array}{c}\hfill x=\pm ʃ\sqrt{{\displaystyle \frac{e^{\frac{2\left(A_1+A_2\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}}{2{\int }^{\phi }\frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\tau \left\{l\omega -\tau \left(b_1+\tau b_2\right)\right\}}{a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}}d\tau }}}d\phi ,\end{array}$$

(123)

where

$$\begin{array}{c}\hfill A_1={\sigma }_1(l+2m)\left[ln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\phi \right],\end{array}$$

(124)

$$\begin{array}{c}\hfill A_2={\displaystyle \frac{l{\sigma }_2\left[ln\phi (2m(l+n-1)-ln+n)-nln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}},\end{array}$$

(125)

$$\begin{array}{c}\hfill A_3={\sigma }_1(l+2m)\left[ln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\tau \right],\end{array}$$

(126)

and

$$\begin{array}{c}\hfill A_4={\displaystyle \frac{l{\sigma }_2\left[ln\tau \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}}\end{array}$$

(127)

The parameter constraint that is needed for the solution to exist is

$$\begin{array}{c}\hfill {\int }^{\phi }{\displaystyle \frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\tau \left\{l\omega -\tau \left(b_1+\tau b_2\right)\right\}}{a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}}}d\tau >0.\end{array}$$

(128)

3.12. Generalized Quadratic–Cubic Law

The generalized quadratic–cubic law raises the intensity dependence of the quadratic contribution to an arbitrary power, which permits modeling of media where the effective quadratic response itself varies with intensity, as may occur in cascaded ${\chi }^{\left(2\right)}$ processes or in strongly driven metamaterial resonators. In this setting the recovered quiescent soliton solutions give a convenient analytical handle on how such generalized quadratic contributions reshape the stationary profiles, and they provide criteria for the parameter choices that keep the device away from undesired quiescent trapping regimes.

The law of SPM can be expressed as a generalized form of the quadratic–cubic law. That is,

$$\begin{array}{c}\hfill H\left(s\right)=b_1{s}^{{\displaystyle \frac{m}{2}}}+b_2{s}^{{\displaystyle \frac{m+1}{2}}},\end{array}$$

(129)

so that the governing model would read

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left(b_1{\left|\mathrm{\Phi }\right|}^m+b_2{\left|\mathrm{\Phi }\right|}^{m+1}\right){\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(130)

Substituting (2) into (130) the ODE for $\phi \left(x\right)$ is structured as

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -b_1{\phi }^m\left(x\right)-b_2{\phi }^{m+1}\left(x\right)\right\}=0.\hfill \end{array}$$

(131)

With the aid of the translational Lie symmetry supported by (131), its implicit integral reads

$$\begin{array}{c}\hfill x=ʃ\sqrt{{\displaystyle \frac{e^{\frac{2\left(A_1+A_2\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}}{2{\int }^{\phi }\frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\tau \left\{l\omega -{\tau }^m\left(b_1+\tau b_2\right)\right\}}{a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}}d\tau }}}d\phi ,\end{array}$$

(132)

where

$$\begin{array}{c}\hfill A_1={\sigma }_1(l+2m)\left[ln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\phi \right],\end{array}$$

(133)

$$\begin{array}{c}\hfill A_2={\displaystyle \frac{l{\sigma }_2\left[ln\phi \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}},\end{array}$$

(134)

$$\begin{array}{c}\hfill A_3={\sigma }_1(l+2m)\left[ln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\tau \right],\end{array}$$

(135)

and

$$\begin{array}{c}\hfill A_4={\displaystyle \frac{l{\sigma }_2\left[ln\tau \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}}.\end{array}$$

(136)

The parameter constraint for the existence of the solution is therefore given by

$$\begin{array}{c}\hfill {\int }^{\phi }{\displaystyle \frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\tau \left\{l\omega -{\tau }^m\left(b_1+\tau b_2\right)\right\}}{a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}}}d\tau >0.\end{array}$$

(137)

3.13. Quadratic–Cubic–Quartic Law

When quadratic, cubic, and quartic contributions are all present, the SPM law can capture richer competition between different orders of nonlinearity, which is particularly relevant for composite or stratified media. Here the quartic term can either enhance or counteract the lower-order contributions depending on its sign. The implicit quiescent soliton solutions derived for this model can therefore be interpreted as stationary states in a medium with three competing nonlinear mechanisms, and the resulting parameter conditions may be used to identify operating windows in which the combined response remains favorable for mobile, rather than quiescent, propagation.

For the quadratic–cubic–quartic law of SPM, the functional H is expressed as

$$\begin{array}{c}\hfill H\left(s\right)=b_1\sqrt{s}+b_{2}s+b_{3}s\sqrt{s}.\end{array}$$

(138)

The governing NLSE can therefore be written as

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left(b_1\left|\mathrm{\Phi }\right|+b_2{\left|\mathrm{\Phi }\right|}^2+b_3{\left|\mathrm{\Phi }\right|}^3\right){\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(139)

Substituting (2) into (139) leads the ODE for $\phi \left(x\right)$ to be

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -b_1\phi \left(x\right)-b_2{\phi }^2\left(x\right)-b_3{\phi }^3\left(x\right)\right\}=0.\hfill \end{array}$$

(140)

The translational Lie point symmetry leads its integral to be

$$\begin{array}{c}\hfill x=\int \sqrt{{\displaystyle \frac{A}{B}}}d\tau ,\end{array}$$

(141)

where

$$\begin{array}{c}\hfill A=exp\left[{\displaystyle \frac{2\left(A_1+\frac{A_2}{2m-n}\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\right],\end{array}$$

(142)

$$\begin{array}{c}\hfill B=2{\displaystyle {ʃ}^{\phi }}{\displaystyle \frac{exp\left(\frac{2\left(A_3+\frac{A_4}{2m-n}\right)}{(l+2m){\sigma }_1+l{\sigma }_2}\right)\tau \left[l\omega -\tau \left\{b_1+\tau \left(b_2+\tau b_3\right)\right\}\right]}{a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}}}d\tau ,\end{array}$$

(143)

$$\begin{array}{c}\hfill A_1={\sigma }_1(l+2m)\left[ln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\phi \right],\end{array}$$

(144)

$$\begin{array}{c}\hfill A_2=l{\sigma }_2\left[ln\phi \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]\end{array}$$

(145)

$$\begin{array}{c}\hfill A_3={\sigma }_1(l+2m)\left[log\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\tau \right]\end{array}$$

(146)

and

$$\begin{array}{c}\hfill A_4=l{\sigma }_2\left[ln\tau \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right].\end{array}$$

(147)

The parametric condition that must remain valid for the existence of the solution is given by

$$\begin{array}{c}\hfill {ʃ}^{\phi }{\displaystyle \frac{exp\left(\frac{2\left(A_3+\frac{A_4}{2m-n}\right)}{(l+2m){\sigma }_1+l{\sigma }_2}\right)\tau \left[l\omega -\tau \left\{b_1+\tau \left(b_2+\tau b_3\right)\right\}\right]}{a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}}}d\tau >0.\end{array}$$

(148)

3.14. Generalized Quadratic–Cubic–Quartic Law

In the generalized quadratic–cubic–quartic case, the exponents of the intensity are allowed to vary, which provides an effective description of metamaterials whose nonlinear response has been tailored by design, for example, via spatially varying inclusions or multi-frequency excitation. Such a model emphasizes the flexibility of the Lie symmetry approach: once the exponents and coefficients are specified, the same reduction yields implicit quiescent soliton solutions and parameter constraints. These results may be viewed as a catalog from which one can extract specific scenarios corresponding to particular engineered devices.

For the generalized form of the quadratic–cubic–quartic form of SPM, the functional form of H is

$$\begin{array}{c}\hfill H\left(s\right)=b_1{s}^{{\displaystyle \frac{m}{2}}}+b_2{s}^{{\displaystyle \frac{m+1}{2}}}+b_3{s}^{{\displaystyle \frac{2m+1}{2}}},\end{array}$$

(149)

so that the governing model appears as

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left(b_1{\left|\mathrm{\Phi }\right|}^m+b_2{\left|\mathrm{\Phi }\right|}^{m+1}+b_3{\left|\mathrm{\Phi }\right|}^{2m+1}\right){\mathrm{\Phi }}^l\\ \hfill ={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(150)

Substituting (2) into (150) leads the ODE for $\phi \left(x\right)$ to be

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -b_1{\phi }^m\left(x\right)-b_2{\phi }^{m+1}\left(x\right)-b_3{\phi }^{2m+1}\left(x\right)\right\}=0.\hfill \end{array}$$

(151)

Finally, implementing the translational Lie symmetry, one arrives at the implicit solution to the ODE as

$$x=ʃ\sqrt{{\displaystyle \frac{e^{\frac{2(A_1+A_2)}{A_3}}}{2{\int }^{\phi }{B}_{2}d\tau }}}d\phi$$

(152)

$$A_1={\sigma }_1(l+2m)[ln|a(l+n){\phi }^n-{\phi }^{2m}\{{\sigma }_1(l+2m)+l({\sigma }_2+{\sigma }_3)\}|+(l-1)ln\phi ]$$

(153)

$$A_2={\displaystyle \frac{l({\sigma }_2+{\sigma }_3)[ln\phi \{2m(l+n-1)-ln+n\}-nln|a(l+n){\phi }^n-{\phi }^{2m}\{{\sigma }_1(l+2m)+l({\sigma }_2+{\sigma }_3)\}\left|\right]}{2m-n}}$$

(154)

$$A_3={\sigma }_1(l+2m)+l({\sigma }_2+{\sigma }_3)$$

(155)

$$\begin{array}{cc}\hfill B_1& =2\left[{\sigma }_1(l+2m)\left\{ln\left|a(l+n){\tau }^n-A_3{\tau }^{2m}\right|+(l-1)ln\tau \right\}\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{l\left({\sigma }_2+{\sigma }_3\right)\left[ln\tau \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\tau }^n-A_3{\tau }^{2m}\right|\right]}{2m-n}}\right]\hfill \end{array}$$

(156)

and

$$B_2={\displaystyle \frac{\tau e^{\frac{B_1}{A_3}}[{\tau }^m\{\tau (b_3{\tau }^m+b_2)+b_1\}-l\omega ]}{A_3{\tau }^{2m}-a(l+n){\tau }^n}}.$$

(157)

The parametric constraint that guarantees the existence of the solution is given as

$$\begin{array}{c}\hfill {\int }^{\phi }{B}_{2}d\tau >0.\end{array}$$

(158)

3.15. Parabolic Nonlocal Law

The parabolic nonlocal SPM law incorporates a weakly nonlocal contribution that is proportional to the second spatial derivative of the intensity and thus models media in which the refractive index at a given point depends not only on the local intensity but also on its nearby spatial distribution [39]. Such behavior is relevant for thermal nonlinearities, diffusion-dominated processes, and certain metamaterial or magneto-optic structures with spatially extended unit cells and parabolic–nonlocal refractive index laws. Our quiescent soliton solutions for this law describe stalled waveforms shaped jointly by nonlinear chromatic dispersion and nonlocal self-phase modulation, and the associated constraints show how strong the nonlocal term must be to permit or suppress quiescent localization.

For parabolic nonlocal law of nonlinear refractive index, the SPM structure stems from

$$\begin{array}{c}\hfill H\left(s\right)=b_{1}s+b_2{s}^2+b_3{s}^{″}.\end{array}$$

(159)

Hence the governing model takes shape as

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left[b_1{\left|\mathrm{\Phi }\right|}^2+b_2{\left|\mathrm{\Phi }\right|}^4+b_3{\left({\left|\mathrm{\Phi }\right|}^2\right)}_{xx}\right]{\mathrm{\Phi }}^l\\ \hfill ={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(160)

Substituting (2) into (160) gives the ODE for $\phi \left(x\right)$ to be

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left[l\omega -b_1{\phi }^2\left(x\right)-b_2{\phi }^4\left(x\right)-2b_3{\left\{{\phi }^2\left(x\right)\right\}}^{″}\right]=0.\hfill \end{array}$$

(161)

For (161) to be integrable one must choose

$$\begin{array}{c}\hfill n=2,\end{array}$$

(162)

and

$$\begin{array}{c}\hfill m=1.\end{array}$$

(163)

Consequently, the governing model is adjusted to

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^2{\mathrm{\Phi }}^l\right)}_{xx}+\left[b_1{\left|\mathrm{\Phi }\right|}^2+b_2{\left|\mathrm{\Phi }\right|}^4+b_3{\left({\left|\mathrm{\Phi }\right|}^2\right)}_{xx}\right]{\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^2{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^2{\left({\mathrm{\Phi }}^l\right)}_{xx},\end{array}$$

(164)

and consequently the ODE for $\phi \left(x\right)$ shrinks to

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left\{(l+2)\left(a-{\sigma }_1\right)+2b_3-l{\sigma }_2\right\}+{\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left\{(l+1)(l+2)\left(a-{\sigma }_1\right)+2b_3-(l-1)l{\sigma }_2\right\}\hfill \\ +& b_2{\phi }^4\left(x\right)+b_1{\phi }^2\left(x\right)-l\omega =0.\hfill \end{array}$$

(165)

The implicit integral derived from the translational Lie point symmetry indicated by Equation (165) is as follows:

$$\begin{array}{c}\hfill x=\sqrt{{\displaystyle \frac{2}{A_2+\sqrt{A_2^2+4A_1{A}_3}}}}F\left(i{sinh}^{-1}\left(\sqrt{2}\phi \sqrt{{\displaystyle \frac{A_3}{A_2-\sqrt{A_2^2+4A_1{A}_3}}}}\right)|{\displaystyle \frac{A_2-\sqrt{A_2^2+4A_1{A}_3}}{A_2+\sqrt{A_2^2+4A_1{A}_3}}}\right),\end{array}$$

(166)

$$\begin{array}{c}\hfill A_1={\displaystyle \frac{l\omega }{(l+1)(l+2)\left(a-{\sigma }_1\right)+2b_3-(l-1)l{\sigma }_2}},\end{array}$$

(167)

$$\begin{array}{c}\hfill A_2={\displaystyle \frac{b_1}{{(l+2)}^2\left(a-{\sigma }_1\right)+4b_3-l^2{\sigma }_2}},\end{array}$$

(168)

$$\begin{array}{c}\hfill A_3={\displaystyle \frac{b_2}{(l+2)(l+3)\left(a-{\sigma }_1\right)+6b_3-l(l+1){\sigma }_2}}.\end{array}$$

(169)

The parameter constraints that result from Equation (166) for the existence of the implicit solution are

$$\begin{array}{c}\hfill A_2\pm \sqrt{A_2^2+4A_1{A}_3}\ne 0.\end{array}$$

(170)

3.16. Saturating Law

Saturating SPM laws model media whose nonlinear refractive index tends to a finite value at large intensities, such as semiconductor waveguides, organic materials, and certain metamaterial structures with saturable inclusions. In these systems the saturation parameter controls the transition from a linear to a strongly nonlinear response. The quiescent soliton solutions derived for the saturating law therefore provide a theoretical description of stalled pulses in such media and indicate how the saturation level and characteristic intensity should be chosen in order to avoid trapping effects in high-power telecommunication applications.

The saturating law of SPM expresses the functional H as follows:

$$\begin{array}{c}\hfill H\left(s\right)={\displaystyle \frac{b_{1}s}{b_2+b_{3}s}}\end{array}$$

(171)

The corresponding NLSE with such a form of the SPM structure takes the form

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left({\displaystyle \frac{b_1{\left|\mathrm{\Phi }\right|}^2}{b_2+b_3{\left|\mathrm{\Phi }\right|}^2}}\right){\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(172)

Next, substituting (2) into (172) leads the the ODE for $\phi \left(x\right)$ to be

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -{\displaystyle \frac{b_1{\left|\mathrm{\Phi }\right|}^2}{b_2+b_3{\left|\mathrm{\Phi }\right|}^2}}\right\}=0.\hfill \end{array}$$

(173)

Finally, implementing the translational Lie symmetry yields the implicit solution to the ODE as

$$\begin{array}{c}\hfill x=ʃ\sqrt{-{\displaystyle \frac{e^{\frac{2\left(A_1+A_2\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}}{2{\int }^{\phi }\frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\tau \left({\tau }^2{b}_1-l\omega \left(b_2+{\tau }^2{b}_3\right)\right)}{\left(b_2+{\tau }^2{b}_3\right)\left[a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}\right]}d\tau }}}d\phi ,\end{array}$$

(174)

where

$$\begin{array}{c}\hfill A_1={\sigma }_1(l+2m)\left[ln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\phi \right],\end{array}$$

(175)

$$\begin{array}{c}\hfill A_2={\displaystyle \frac{l{\sigma }_2\left[ln\phi \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}},\end{array}$$

(176)

$$\begin{array}{c}\hfill A_3={\sigma }_1(l+2m)\left[ln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\tau \right],\end{array}$$

(177)

and

$$\begin{array}{c}\hfill A_4={\displaystyle \frac{l{\sigma }_2\left[ln\tau \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}}\end{array}$$

(178)

The parametric constraint that guarantees the existence of the solution is

$$\begin{array}{c}\hfill {\int }^{\phi }{\displaystyle \frac{e^{\frac{2\left(A_3+A_4\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}\tau \left\{{\tau }^2{b}_1-l\omega \left(b_2+{\tau }^2{b}_3\right)\right\}}{\left(b_2+{\tau }^2{b}_3\right)\left[a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}\right]}}d\tau <0.\end{array}$$

(179)

3.17. Exponential Law

The exponential SPM response can be viewed as a convenient analytical model for strongly saturating or soft-threshold nonlinearities, where the refractive index change grows rapidly at low intensities and then saturates. Such behavior has been reported in various engineered metamaterials and nonlinear optical composites. The implicit quiescent soliton solutions in this case, involving exponential factors, show how the saturation parameter b filters the range of admissible amplitudes and thus provides guidance on how to tune the material design so that stationary stalled structures either appear (for optical buffering) or are avoided (for long-distance transmission).

The exponential form of the SPM structure defines the functional H as follows:

$$\begin{array}{c}\hfill H\left(s\right)={\displaystyle \frac{1}{b}}\left(1-e^{-bs}\right),\end{array}$$

(180)

for non-zero real-valued constant b as long as

$$\begin{array}{c}\hfill b\ne 0.\end{array}$$

(181)

Thus, the governing NLSE reads

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+\left\{{\displaystyle \frac{1}{b}}\left(1-e^{-b{\left|\mathrm{\Phi }\right|}^2}\right)\right\}{\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(182)

Substituting (2) into (182) leads to the ODE for $\phi \left(x\right)$ taking the form

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\displaystyle \frac{bl\omega {\phi }^2\left(x\right)+e^{-b{\phi }^2\left(x\right)}-1}{b}}=0.\hfill \end{array}$$

(183)

The translational Lie symmetry applied to (183) leads to its integral

$$\begin{array}{c}\hfill x=ʃ\sqrt{{\displaystyle \frac{e^{\frac{2\left(A_1+A_2\right)}{(l+2m){\sigma }_1+l{\sigma }_2}}}{2{\int }^{\phi }\frac{exp\left(-b{\tau }^2+\frac{A_3+A_4}{(2m-n)\left((l+2m){\sigma }_1+l{\sigma }_2\right)}\right)\tau \left\{1-e^{b{\tau }^2}(1-bl\omega )\right\}}{b\left[a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}\right]}d\tau }}}d\phi ,\end{array}$$

(184)

where

$$\begin{array}{c}\hfill A_1={\sigma }_1(l+2m)\left[ln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\phi \right],\end{array}$$

(185)

$$\begin{array}{c}\hfill A_2={\displaystyle \frac{l{\sigma }_2\left[ln\phi \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\phi }^n-{\phi }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right]}{2m-n}},\end{array}$$

(186)

$$\begin{array}{c}\hfill A_3=2{\sigma }_1(l+2m)(2m-n)\left[ln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|+(l-1)ln\tau \right],\end{array}$$

(187)

and

$$\begin{array}{c}\hfill A_4=2l{\sigma }_2\left[ln\tau \left\{2m(l+n-1)-ln+n\right\}-nln\left|a(l+n){\tau }^n-{\tau }^{2m}\left\{{\sigma }_1(l+2m)+l{\sigma }_2\right\}\right|\right].\end{array}$$

(188)

The parameter constraint that must hold for the existence of the solution is

$$\begin{array}{c}\hfill {ʃ}^{\phi }{\displaystyle \frac{exp\left[-b{\tau }^2+\frac{A_3+A_4}{(2m-n)\left((l+2m){\sigma }_1+l{\sigma }_2\right)}\right]\tau \left\{1-e^{b{\tau }^2}(1-bl\omega )\right\}}{b\left[a(l+n){\tau }^n-{\tau }^{2m}\left\{(l+2m){\sigma }_1+l{\sigma }_2\right\}\right]}}d\tau >0.\end{array}$$

(189)

3.18. Logarithmic Law

Finally, the logarithmic SPM law is motivated by models of media with slow, logarithmic growth of the nonlinear index at high intensities, which may arise as effective descriptions of complex saturation processes. Although such a dependence is less common experimentally, it is useful for capturing gradual saturation in theoretical studies of nonlinear wave propagation. In this framework our quiescent soliton solutions, expressed in terms of the error function, characterize stalled pulses in logarithmic media and highlight how the logarithmic coefficient controls both the existence domain and the qualitative shape of the localized structures.

For logarithmic law of SPM,

$$\begin{array}{c}\hfill H\left(s\right)=blns,\end{array}$$

(190)

for non-zero real-valued constant b. The corresponding governing model therefore reads:

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^n{\mathrm{\Phi }}^l\right)}_{xx}+bln\left({\left|\mathrm{\Phi }\right|}^2\right){\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(191)

The corresponding ODE for $\phi \left(x\right)$ with (2) being plugged into (191) is

$$\begin{array}{cc}\hfill & \phi \left(x\right){\phi }^{″}\left(x\right)\left[a(l+n){\phi }^n\left(x\right)-\left\{{\sigma }_1\left(l+2m\right)+{\sigma }_{2}l\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ +& {\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left[a(l+n)(l+n-1){\phi }^n\left(x\right)-\left\{{\sigma }_1(l+2m)(l+2m-1)+{\sigma }_2(l-1)\right\}{\phi }^{2m}\left(x\right)\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -2bln\phi \left(x\right)\right\}=0.\hfill \end{array}$$

(192)

For (192) to be integrable, one needs to have the relation as given in (10). This changes the governing model (191) to

$$\begin{array}{c}\hfill i{\left({\mathrm{\Phi }}^l\right)}_t+a{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+bln{\left|\mathrm{\Phi }\right|}^2{\mathrm{\Phi }}^l={\sigma }_1{\left({\left|\mathrm{\Phi }\right|}^{2m}{\mathrm{\Phi }}^l\right)}_{xx}+{\sigma }_2{\left|\mathrm{\Phi }\right|}^{2m}{\left({\mathrm{\Phi }}^l\right)}_{xx}.\end{array}$$

(193)

The ODE for $\phi \left(x\right)$ thus simplifies to

$$\begin{array}{cc}\hfill & {\phi }^{2m+1}\left(x\right){\phi }^{″}\left(x\right)\left\{\left(a-{\sigma }_1\right)(l+2m)-l{\sigma }_2\right\}\hfill \\ +& \left[{\phi }^{2m}\left(x\right){\left\{{\phi }^{\prime }\left(x\right)\right\}}^2\left\{\left(a-{\sigma }_1\right)(l+2m)(l+2m-1)-l(l-1){\sigma }_2\right\}\right]\hfill \\ -& {\phi }^2\left(x\right)\left\{l\omega -2bln\phi \left(x\right)\right\}=0.\hfill \end{array}$$

(194)

By virtue of the translational Lie symmetry supported by (194) one arrives at its integral as

$$\begin{array}{c}\hfill x=ABC,\end{array}$$

(195)

where

$$\begin{array}{c}\hfill A=exp\left[{\displaystyle \frac{m(l+2m)\left\{b+l(l+m)\omega \right\}\left(a-{\sigma }_1\right)-lm\left\{b+l(l-m)\omega \right\}{\sigma }_2}{2\left\{b(l+m)(l+2m)\left(a-{\sigma }_1\right)-bl(l-m){\sigma }_2\right\}}}\right],\end{array}$$

(196)

$$\begin{array}{c}\hfill B=\sqrt{{\displaystyle \frac{(l+m)(l+2m)\pi \left(a-{\sigma }_1\right)-l(l-m)\pi {\sigma }_2}{2bm}}},\end{array}$$

(197)

$$\begin{array}{c}\hfill C=\mathrm{erf}\left[\sqrt{{\displaystyle \frac{L}{2b\left\{(l+m)(l+2m)\left(a-{\sigma }_1\right)-l(l-m){\sigma }_2\right\}}}}\right],\end{array}$$

(198)

and

$$\begin{array}{c}\hfill L=m(l+2m)\left\{b+l(l+m)\omega -2b(l+m)ln\phi \right\}\left(a-{\sigma }_1\right)-lm\left\{b+l(l-m)\omega -2b(l-m)ln\phi \right\}{\sigma }_2,\end{array}$$

(199)

with the error function being defined as

$$\begin{array}{c}\hfill \mathrm{erf}\left(u\right)={\displaystyle \frac{2}{\sqrt{\pi }}}{\int }_0^u{e}^{-t^2}dt.\end{array}$$

(200)

Taken together, the parameter constraints obtained for each SPM law delimit the regions in which the balance between nonlinear chromatic dispersion and self-phase modulation supports implicit quiescent solitons. They therefore provide design guidelines for tailoring optical metamaterials so as either to favor or to suppress the formation of such stalled structures.

4. Conclusions

The study identified quiescent solitons in optical metamaterials characterized by nonlinear chromatic dispersion and generalized temporal evolution. The model was adopted with arbitrary intensity. The Lie symmetry analysis served as the integration tool utilized to access quiescent optical solitons. The results are presented for the array of SPM structures. The solutions are implicit and expressed in terms of special functions, including Gauss’ hypergeometric functions, Appell’s hypergeometric functions, elliptic functions, and others. The other solutions are expressed in terms of quadratures. The results show that telecommunication engineers must avoid accidentally creating quiet optical solitons as this could lead to serious problems in an internet-based setting. The CD must be rendered linearly, and the temporal evolution needs to remain linear to ensure the mobility of the solitons.

These findings will subsequently be used to produce more optoelectronic devices. Examples of such devices include optical couplers, magneto-optic waveguides, Bragg gratings, polarization mode dispersion, and dispersion-flattening fibers. The final results of these study activities will be gradually and systematically disclosed to the curious and engaged community over time. Therefore, we advise these readers to cultivate patience and steadiness. Persevere in following our updates as each revelation will contribute to a more profound understanding of the advancements in optoelectronic technology. We appreciate the interest and enthusiasm of the community, which drives innovation and collaboration in this exciting field.

From a physical standpoint, several of the SPM responses analyzed here, most notably the Kerr, power law, cubic–quintic (parabolic), dual power, polynomial, and saturating laws, together with their weakly nonlocal variants, serve as effective models for real optical fibers, semiconductor and chalcogenide waveguides, photonic–crystal fibers, and metamaterial structures and have already been employed to interpret or guide experiments. By contrast, the more exotic higher-order, multi-power, anti-cubic, exponential, and logarithmic laws should at present be viewed primarily as flexible phenomenological descriptions or design tools for complex engineered media, with direct implementations largely confined to proof-of-concept or laboratory-scale configurations.