On certain novel numerical and analytical solutions for the pure-cubic Schrödinger equation in optical fibers with Kerr nonlinearity

Abstract

This study examines the pure-cubic nonlinear Schrödinger equation associated with Kerr-type optical responses and formulates an integrated analytical–computational strategy for exploring its nonlinear wave behavior. The approach is based on transforming the governing model into an ordinary differential equation through an appropriate traveling-wave reduction; constructing exact waveform expressions using a refined ansatz procedure; and validating these solutions through a high-precision split-step numerical solver supported. The analysis yields several families of localized and periodic wave patterns, including bright solitons, dark solitons, kink-type solitons, periodic waves, singular solitons, and oscillatory profiles generated under varying parameter choices. The novelty of the work lies in providing a single, coherent framework capable of producing multiple solitary-wave configurations within the pure-cubic Kerr setting, together with a structured verification pipeline that has not been previously applied in this consolidated manner. The close agreement between analytical formulas and numerical propagation results confirms the robustness of the proposed methodology and establishes a set of reference solutions for future investigations involving extended or perturbed nonlinear optical models.

Introduction

The nonlinear wave phenomena explored in this work are aligned with recent developments in optical transmission^1,2,3, photonic system design, and reconfigurable meta-engineering. Current studies on polarization behavior in complex media, data-driven reliability analysis of power devices, dispersion-controlled meta-optics, nonlinear Pancharatnam–Berry structures^4,5, radiation-tolerant circuit architectures, deep-ultraviolet light extraction enhancement, stress-wave sensing in semiconductors, and programmable metasurfaces for beam shaping provide strong motivation for the integrated analytical–numerical framework employed here^6,7,8. The pure-cubic nonlinear Schrödinger equation (NLSE) with Kerr-type nonlinearity provides a fundamental model for investigating nonlinear wave dynamics across a range of physical systems. It offers a mathematically robust framework for analyzing the formation, evolution, and interaction of optical solitons in dispersive nonlinear media. In fiber-optic communication, these solitons enable long-distance signal transmission with minimal attenuation and reduced dispersion-induced distortion, making them crucial for high-speed data transfer and advanced photonic applications^9,10,11. Beyond optical systems, the NLSE serves as a key analytical framework for studying nonlinear wave phenomena in diverse physical contexts. In plasma physics, it models the evolution of waves in ionized media, while in Bose–Einstein condensates, it governs the dynamics of matter-wave packets. The NLSE also finds applications in fluid mechanics, biomechanics, and chemical kinetics, where the interplay of dispersion and nonlinearity shapes wave interactions and pattern formation. Understanding nonlinear wave behavior is critical for both scientific research and engineering applications, particularly in optical fibers for long-distance signal transmission. In these media, the NLSE describes essential phenomena such as self-focusing, pulse compression, and the formation and propagation of optical solitons^12,13,14.

The analytical techniques employed including the polynomial method, extended tanh and hyperbolic function approaches, and Adomian decomposition offer powerful tools for obtaining both exact and approximate solutions of nonlinear evolution equations. These methods can capture a wide range of wave structures, such as solitons, periodic patterns, and single-pulse solutions. Recent studies have demonstrated their effectiveness in tackling higher-dimensional and more complex nonlinear models in optics and mathematical physics. This capability is particularly important for the pure-cubic optical NLSE¹⁵, where the cubic nonlinearity represents self-interaction of light, allowing the medium to respond to the local intensity. This self-modulation preserves the shape of optical pulses during propagation, giving rise to stable solitons. In essence, the Kerr effect enables the medium to adapt to the intensity of the light, leading to distinctive wave behaviors accurately described by the NLSE^16,17. Solitons, described by the pure-cubic NLSE, are remarkable because they maintain their profile and velocity due to a balance between nonlinear effects (from the kerr effect) and dispersion (from the medium)¹⁸. This balance makes solitons robust to perturbations and variations in the medium^19,20. Several strong mathematical methods are used to provide exact solutions to the pure-cubic optical NLSE. The Inverse Scattering Transform^21,22 is an important approach for solving integrable problems, mainly for finding soliton solutions. The Hirota’s Direct Method^23,24 is another important algebraic strategy for finding multi-soliton solutions by transforming the problem into bilinear form. The ansatz approach²⁵ entails assuming a specific form for the solution, such as a traveling wave, and solving for parameters. Darboux and Bcklund transformations²⁶ are used to derive solutions from existing ones, especially higher-order solitons. Painlev analysis examines whether an equation is integrable and can lead to distinct exact solutions. Lie symmetry analysis clarifies the problem using its symmetries and discovers different solutions, while the Jacobi elliptic function technique^27,28 produces periodic and solitary wave solutions. Moreover, a variety of wave solutions obtained using techniques such as the sub-equation neural network method²⁹, the generalized Arnous method, modified generalized Riccati equation mapping technique³⁰, the modified Sardar subequation method and new Kudryashov method³¹, the generalized exponential rational function method³², the two variable $(G'/G,1/G)$ -expansion technique³³, the modified F-expansion technique, the Riccati extended modified simple equation technique, and the generalized -expansion method³⁴.

Although the pure-cubic NLSE plays a central role in modeling nonlinear wave propagation, comprehensive studies employing a unified and comparative analytical approach remain limited. Previous works have investigated its soliton dynamics using individual techniques, but a systematic analysis combining multiple robust methods is still lacking. This study fills that gap by applying and comparing four well-established analytical methods to derive novel exact solutions. The results enhance our understanding of Kerr-type nonlinear wave phenomena and provide practical tools for addressing related nonlinear models in optical physics. Furthermore, recent studies indicate that these methodologies can be extended to more generalized and higher-dimensional nonlinear systems, broadening their applicability in complex wave dynamics. For example, innovative soliton solutions to a $(2+1)$-dimensional generalized Korteweg-de Vries equation were published employing effective methodologies such as the auxiliary equation method and the enhanced Riccati method³⁵. Similarly, the extended $(3+1)$-dimensional Kadomtsev-Petviashvili equation has been investigated using direct methods to derive traveling and soliton wave properties relevant to fluid dynamics³⁶. In optical contexts, highly dispersive dual-soliton structures and modulation instability investigations were performed using a dual-mode nonlinear Schrödinger model, which improved our understanding of high-order soliton generation and optical switching³⁷. In another study, solitary wave solutions for the (3+1)-dimensional pKP-BKP problem were obtained using the improved modified extended tanh function approach, demonstrating its ability to generate localized profiles³⁸. The modified extended direct algebraic technique has also been utilized to find several optical soliton solutions to the fourth-order (2+1)-dimensional NLSE³⁹. Studies of parabolic-law and fourth-order dispersion effects in a (3+1)-dimensional NLSE demonstrate how complex nonlinearities affect wave dynamics^40,41,42. Furthermore, the analytical behaviors of generalized Bogoyavlensky-Konopelchenko and Gerdjikov-Ivanov models have been investigated using recent algebraic and expansion-based techniques, showing their complex structure and solution space^43,44.

The dynamics of optical solitons have advanced significantly in recent decades, especially in the setting of modern telecommunications engineering⁴⁵. Soliton propagation in optical fibers is primarily controlled by two fundamental physical effects: the nonlinear refractive index and chromatic dispersion (CD), which accounts for the wavelength-dependent velocity of light. However, in some regimes-particularly those involving ultrashort pulse propagation the effect of CD may decline or become negligible. To address such conditions, the concept of pure-quartic solitons was presented in 2016⁴⁶, in which fourth-order dispersion (4OD) replaces the traditional role of CD, resulting in a novel class of solitary wave solutions driven purely by higher-order dispersive effects. The addition of 4OD to the nonlinear Schrödinger framework produces a non-integrable equation, complicating analytical treatment and frequently needing advanced numerical or semi-analytical techniques⁴⁷. To capture these effects more accurately, researchers have introduced hybrid models incorporating both third and fourth-order dispersion, referred to as cubic–quartic solitons, which provide a more comprehensive representation of the underlying wave dynamics. These generalized models have facilitated the discovery of novel soliton configurations and have been applied in various practical settings, including optical fiber communications, nonlinear photonic crystal fibers, femtosecond pulse shaping, and Bose–Einstein condensates. Additionally, studies have examined the effects of Hamiltonian perturbations and intensity-dependent nonlinearities consistent with Kerr and power-law responses, further enhancing the relevance and applicability of these models to real-world nonlinear media^48,49. The pure-cubic optical NLSE is represented as:

$$\begin{aligned} \left. {iX}_t + i\lambda X_{{yyy}} + G\left( \left| X\right| ^2\right) X = i\left( \beta \left( \left| X\right| ^{2m}X\right) \right) {}_y + \theta \left| X\right| ^{2m}X_y + \rho \left( \left| X\right| ^{2m}\right) {}_y X \right) , \end{aligned}$$

(1)

where X(y, t) is a complex-valued mapping used to describe the wave profile. The independent variables $i=\sqrt{-1}$, y, and t represent the imaginary unit, spatial, and temporal coordinates, respectively. The parameter $\beta$ represents the coefficient of internal steepening nonlinearity, and $\lambda$ represents the coefficient of third-order dispersion. Finally, $\theta$ and $\rho$ account for higher-order dispersion effects. The function G describes the nonlinearity of the refractive index as follows:

$$\begin{aligned} G\left( \left| X|^2\right. \right) X\in \cup _{m,n=1}^{\infty }C^k\left( (-n,n)\times (-m,m);\mathbb {R}^2\right) , \end{aligned}$$

(2)

for kerr law $G u={bu}$, where b is a real valued constant⁵⁰. Thus, Eq. (1) becomes

$$\begin{aligned} \left. {iX}_t+i{\lambda }\text {X}_{{yyy}}+b\left( \left| X|^2\right. \right) X=i\left( \beta \left( \left| X|^{2 m}X\right. \right) \right) {}_y+\theta (|X|)^{2 m}X_y+\rho \left( \left| X|^{2 m}\right. \right) {}_yX\right) . \end{aligned}$$

(3)

Methodology

The general layout of the PDE is recognized as

$$\begin{aligned} S\left( X_t,X_y,X_{{tt}},X_{{yy}},X_{{yt}},\cdots \right) =0, \end{aligned}$$

(4)

where y and t represent the independent variables of X. We now analyze the transformation, which can be expressed in the form:

$$\begin{aligned} X(y,t)=U(\sigma ) e^{i \delta },~\sigma =\upsilon y-t \upsilon w,~\delta =\alpha +t \omega +(-\lambda ) y, \end{aligned}$$

(5)

by substituting Eq. (5) into Eq. (4), which identifies an ODE as

$$\begin{aligned} V\left( U,U ',U '',U ''',\cdots \right) =0. \end{aligned}$$

(6)

The EHF method

The extended hyperbolic function (EHF) technique includes two stages, explained below:

Form 1: It is assumed that the general form of the solution to Eq. (6) is

$$\begin{aligned} X(\sigma )=\sum _{j=0}^N Z_j \rho (\sigma )^j, \end{aligned}$$

(7)

where N is a real number, and the function $\rho (\sigma )$ satisfies the requirement

$$\begin{aligned} \frac{d\rho }{d\sigma }=\rho \sqrt{\eta \rho ^2+\digamma },~~~\digamma ,\eta \in \mathbb {R}, \end{aligned}$$

(8)

by applying the equilibrium balance approach on Eq. (6) to determine N. We then substitute Eq. (7) into Eq. (6) along with Eq. (8), which produces a system of mathematical equations. Solving these equations gives the results as follows:

Case I. If $\digamma>0$ and $\eta>0$, the result is expressed as

$$\begin{aligned} \rho (\sigma ) = -\sqrt{\frac{\digamma }{\eta }} \, \text {csch} \left( \sqrt{\digamma } (S+\sigma ) \right) . \end{aligned}$$

Case II. If $\digamma <0$ and $\eta>0$, the result is expressed as

$$\begin{aligned} \rho (\sigma ) = \sqrt{-\frac{\digamma }{\eta }} \, \sec \left( \sqrt{-\digamma } (S+\sigma ) \right) . \end{aligned}$$

Case III. If $\digamma>0$ and $\eta <0$, the result is expressed as

$$\begin{aligned} \rho (\sigma ) = \sqrt{-\frac{\digamma }{\eta }} \, \text {sech} \left( \sqrt{\digamma } (S+\sigma ) \right) . \end{aligned}$$

Case IV. If $\digamma <0$ and $\eta>0$, the result is expressed as

$$\begin{aligned} \rho (\sigma ) = \sqrt{-\frac{\digamma }{\eta }} \, \csc \left( \sqrt{-\digamma } (S+\sigma ) \right) . \end{aligned}$$

Case V. If $\digamma>0$ and $\eta =0$, the result is expressed as

$$\begin{aligned} \rho (\sigma ) = \exp \left( \sqrt{\digamma } (S+\sigma ) \right) . \end{aligned}$$

Case VI. If $\digamma <0$ and $\eta =0$, the result is expressed as

$$\begin{aligned} \rho (\sigma ) = \cos \left( \sqrt{-\digamma } (q+\sigma ) \right) + i \sin \left( \sqrt{-\digamma } (S+\sigma ) \right) . \end{aligned}$$

Case VII. If $\digamma =0$ and $\eta>0$, the result is expressed as

$$\begin{aligned} \rho (\sigma ) = \frac{1}{\sqrt{\eta } (S+\sigma )}. \end{aligned}$$

Case VIII. If $\digamma =0$ and $\eta <0$, the result is expressed as

$$\begin{aligned} \rho (\sigma ) = \frac{1}{\sqrt{-\eta } (S+\sigma )}. \end{aligned}$$

Form 2: Similarly, assume that Eq. (6) satisfies the condition

$$\begin{aligned} \frac{d\rho }{d\sigma } = \eta \rho ^2 + \digamma , \quad \digamma , \eta \in \mathbb {R}, \end{aligned}$$

(9)

and apply the homogeneous balance technique to Eq. (6) to compute N. Next, substitute Eq. (7) into Eq. (6) together with Eq. (9), which leads to a system of algebraic equations. After solving these, the results are as follows:

Case I. If $\digamma \eta>0$, the solution has the form

$$\begin{aligned} \rho (\sigma ) = \sqrt{\frac{\digamma }{\eta }} \, \text {sgn}(\digamma ) \, \tan \left( \sqrt{\eta \digamma } (S+\sigma ) \right) . \end{aligned}$$

Case II. If $\digamma \eta>0$, the solution has the form

$$\begin{aligned} \rho (\sigma ) = \sqrt{\frac{\digamma }{\eta }} (-\text {sgn}(\digamma )) \, \cot \left( \sqrt{\eta \digamma } (S+\sigma ) \right) . \end{aligned}$$

Case III. If $\digamma \eta <0$, the solution has the form

$$\begin{aligned} \rho (\sigma ) = \sqrt{-\frac{\digamma }{\eta }} \, \text {sgn}(\digamma ) \, \tanh \left( \sqrt{-\digamma \omega } (S+\sigma ) \right) . \end{aligned}$$

Case IV. If $\digamma \eta <0$, the solution has the form

$$\begin{aligned} \rho (\sigma ) = \sqrt{-\frac{\digamma }{\eta }} \, \text {sgn}(\digamma ) \, \coth \left( \sqrt{\eta (-\digamma )} (S+\sigma ) \right) . \end{aligned}$$

Case V. If $\digamma =0$ and $\eta>0$, the solution has the form

$$\begin{aligned} \rho (\sigma ) = -\frac{1}{\eta (S+\sigma )}. \end{aligned}$$

Case VI. If $\digamma \in \mathbb {R}$ and $\eta =0$, the solution has the form

$$\begin{aligned} \rho (\sigma ) = \digamma (S+\sigma ). \end{aligned}$$

The polynomial expansion method

Step 1: We suppose the solutions to Eq. (6) as

$$\begin{aligned} X(\sigma )=\sum _{i=1}^m l_i \rho (\sigma )^{-i}+\sum _{i=1}^m Z_i \rho (\sigma )^i+Z_0, \end{aligned}$$

(10)

here n denotes a real number, further the term $\rho (\sigma )$ satisfies the requirement

$$\begin{aligned} \rho '(\sigma )=\Upsilon \rho (\sigma )+\rho (\sigma )^2+\digamma , \end{aligned}$$

(11)

where $\digamma$ represents the constant.

Case I: If $\Upsilon =0$, $\digamma =0$, subsequently we acquire results as

$$\begin{aligned} \rho (\sigma )=-\frac{1}{\sigma }. \end{aligned}$$

Case II: $\Upsilon \ne 0$, $\digamma =0$, subsequently we acquire results as

$$\begin{aligned} \rho (\sigma )=-\frac{\Upsilon }{q_0 e^{-\Upsilon \sigma }-1}. \end{aligned}$$

in which the integration constant is $q_0$

Case III: If $\Upsilon =0,~\digamma \ne 0$,

$$\begin{aligned} \rho (\sigma )= & \sqrt{\digamma } \tan \left( \sqrt{\digamma } \sigma \right) ,\\ \rho (\sigma )= & -\sqrt{-\digamma } \cot \left( \sqrt{\digamma } \sigma \right) . \end{aligned}$$

Case IV: If $\Upsilon$=0,$~\digamma \ne 0$, $\digamma <0$, subsequently we acquire results as

$$\begin{aligned} \rho (\sigma )=-\sqrt{-\digamma } \tanh \left( \sqrt{-\digamma } \sigma \right) ,\\ \rho (\sigma )=\sqrt{-\digamma } \coth \left( \sqrt{-\digamma } \sigma \right) . \end{aligned}$$

Case V: If $\digamma \ne 0$,$\Upsilon \ne 0$ subsequently acquire results as

$$\begin{aligned} \rho (\sigma )=\frac{A_1-q_1A_2 e^{\left( A_1-A_2\right) \sigma }}{1-q_1 e^{\left( A_1-A_2\right) \sigma }}. \end{aligned}$$

where $A_1$ and $A_2$ are the root of the Eq. (11)

$$\begin{aligned} A_1=\frac{1}{2} \left( \sqrt{\Upsilon ^2-4 \digamma }-\Upsilon \right) ,\\ A_2=\frac{1}{2} \left( -\sqrt{\Upsilon ^2-4 \digamma }-\Upsilon \right) . \end{aligned}$$

Step 2: Using the balancing number n is calculated by applying the balancing techniques in Eq. (6)

Step 3: Now put both equations (10) and (11) into Eq. (6), further we put same power of $\rho (\sigma )$ identical to zero that provides us a systems of equations.

Step 4: In the next stage, resolve the system of equations for obtaining the solutions of Eq. (3)

The modified extended tanh-function method

An outline of these strategies is provided below.

Step 1: Let us suppose that the solutions to Eq. (6) are of the form

$$\begin{aligned} X(\sigma )=\sum _{i=1}^n Z_i \rho (\sigma )^i+\sum _{i=1}^n \frac{l_i}{\rho (\sigma )^i}+Z_0, \end{aligned}$$

(12)

where $g_i$, $h_i$ are constants that must be established in order for $g_n\ne 0$ or $h_n\ne 0$ and A fulfills the Riccati equation

$$\begin{aligned} \rho '(\sigma )=\digamma +\rho (\sigma )^2, \end{aligned}$$

(13)

where $\digamma$ is a constant. Eq. (13) accepts a variety of solutions based on the following:

Case I: If $\digamma <0,$ then

$$\begin{aligned} \rho (\sigma )=-\sqrt{-\digamma } \tanh \left( \sqrt{-\digamma } \sigma \right) ,\\ \rho (\sigma )=-\sqrt{-\digamma } \coth \left( \sqrt{-\digamma } \sigma \right) . \end{aligned}$$

Case II: If $\digamma =0,$ then

$$\begin{aligned} \rho (\sigma )=-\frac{1}{\sigma }. \end{aligned}$$

Case III: If $\digamma>0,$ then

$$\begin{aligned} \rho (\sigma )=\sqrt{\digamma } \tan \left( \sqrt{\digamma } \sigma \right) ,\\ \rho (\sigma )=\sqrt{\digamma } \cot \left( \sqrt{\digamma } \sigma \right) . \end{aligned}$$

Step 2: Find the positive integer n in Eq. (6) by equating and balancing the highest-order derivatives and the nonlinear terms.

Step 3: We obtain a system of algebraic equations by substituting Eq. (12) and its derivatives, as well as Eq. (13), into Eq. (6), then collecting all terms having the same power and equating them to zero.

Step 4: These values and results can then be substituted to obtain the exact solutions of Eq. (6).

The adomian decomposition method

When we use the recommended technique on Eq. (6), that outcomes as

$$\begin{aligned} L~X(\Omega )+P~X(\Omega )+R~X(\Omega )=0, \end{aligned}$$

(14)

where L, P, and R denote differential operator, linear term, and nonlinear term consequently. Now, we apply the ${L}^{-1}$ on the Eq. (14), that gives outcomes as

$$\begin{aligned} X(y)=\sigma _0-{L}^{-1}{P(X)}-{L}^{-1}{R(X)} , \end{aligned}$$

(15)

making use of the recommended methodology, we compose X as

$$\begin{aligned} X(y)=\sum _{k=0}^{\infty }X_k , \end{aligned}$$

(16)

further R(X) can be expressed as

$$\begin{aligned} R(X)=\sum _{k=0}^{\infty }A_k , \end{aligned}$$

(17)

where polynomial of adomian is represented by $X_K$ and next we insert Eq. (16) and Eq. (17) into Eq. (15), that results as

$$\begin{aligned} \sum _{k=0}^{\infty } X_k=\sigma _0-{L}^{-1}{P(\sum _{k=0}^{\infty } X_k)}-{L}^{-1}{\sum _{k=0}^{\infty } A_k} , \end{aligned}$$

(18)

by utilizing the given phenomenon, we describe the solution as

$$\begin{aligned} & X_0=\sigma _0,\\ & X_{k+1}=-{L}^{-1}{(PX_k)}-{L}^{-1}{(A_k)}, ~k\ge 0, \end{aligned}$$

the basic form of the solution can be composed as the nonlinear decomposition of $X_k, ~k\ge 0$, and $A_k$.

Traveling wave solutions

Apply the following transformation to the Eq. (3)

$$\begin{aligned} X(y,t) = U(\sigma ) e^{i \delta (y,t)}, \quad \sigma = \tau (y - t \nu ), \quad \delta (y,t) = \alpha + t \omega - \lambda y, \end{aligned}$$

(19)

where $\nu$ represents the speed of the wave, $\lambda$ is the wave frequency, $\omega$ is the wave number, and $\alpha$ is the phase constant. By substituting Eq. (19) into Eq. (3), we obtain the corresponding ordinary differential equation:

$$\begin{aligned} 3 \beta \tau ^2 U'' - U (\beta \lambda ^3 + \omega ) + U^3 (b - \lambda (\theta + \kappa )) = 0. \end{aligned}$$

(20)

Similarly, we derive:

$$\begin{aligned} 3 \beta \tau ^2 U'' - U (\beta \lambda ^3 + \omega ) + U^{2m+1} (b - \lambda (\kappa + \theta )) = 0. \end{aligned}$$

(21)

and

$$\begin{aligned} \beta \lambda \tau ^2 U'' - (3 \beta \lambda ^2 + v)U - \frac{U^{2m+1} (\theta + \kappa + 2 \kappa m + 2 \rho m)}{2m+1} = 0. \end{aligned}$$

(22)

For $m=1$, Eqs. (21) and (22), reduce to

$$\begin{aligned} 3 \beta \tau ^2 U'' - U (\beta \lambda ^3 + \omega ) + U^3 (b - \lambda (\kappa + \theta )) = 0. \end{aligned}$$

(23)

and

$$\begin{aligned} \beta \lambda \tau ^2 U'' - U (3 \beta \lambda ^2 + v) - \frac{1}{3} U^3 (\theta + 3 \kappa + 2 \rho ) = 0. \end{aligned}$$

(24)

These are valid when the amplitude components satisfy:

$$\begin{aligned} 3 \lambda = \frac{\beta \lambda ^3 + \omega }{3 \beta \lambda ^2 + v} = \frac{-3(b - \lambda (\theta + \kappa ))}{\theta + 3 \kappa + 2 \rho }, \end{aligned}$$

(25)

which leads to

$$\begin{aligned} v = \frac{\beta \lambda ^3 - 9 \beta \lambda ^2 + \omega }{3 \lambda }, \quad \rho = -\frac{b + 2 \lambda \kappa }{2 \lambda }. \end{aligned}$$

(26)

Applications of the EHF method

Initially, the balancing number $n=1$ is found by matching the order of the highest derivative with that of the main nonlinear term in the ODE; thereafter, the solution of Eq. (23) can be expressed as

$$\begin{aligned} U(\sigma )=Z_0+Z_1 \rho (\sigma )+Z_2 \rho (\sigma )^2. \end{aligned}$$

(27)

Further, we have the system of mathematical equations by putting the Eq. (27) and Eq. (8) into Eq. (23),

$$\begin{aligned} & b Z_0^3-\beta \lambda ^3 Z_0-Z_0^3 \theta \lambda -Z_0^3 \kappa \lambda -Z_0 \omega =0,\\ & 3 b Z_0^2 Z_1-\beta \lambda ^3 Z_1+3 \beta \tau ^2 Z_1 \digamma -3 Z_0^2 Z_1 \theta \lambda -3 Z_0^2 Z_1 \kappa \lambda -Z_1 \omega =0,\\ & 3 b Z_0 Z_1^2+3 b Z_0^2 Z_2+\beta \lambda ^3 -Z_2+12 \beta \tau ^2 Z_2 \digamma -\\ & 3 Z_0 Z_1^2 \theta \lambda -3 Z_0^2 Z_2 \theta \lambda -3 Z_0 Z_1^2 \kappa \lambda -3 Z_0^2 Z_2 \kappa \lambda -Z_2 \omega =0,\\ & b Z_1^3+6 b Z_0 Z_2 Z_1+6 \beta \tau ^2 Z_1 \eta -Z_1^3 \theta \lambda -6 Z_0 Z_2 Z_1 \theta \lambda -Z_1^3 \kappa \lambda -6 Z_0 Z_2 Z_1 \kappa \lambda =0,\\ & 3 b Z_0 Z_2^2+3 b Z_1^2 Z_2+18 \beta \tau ^2 Z_2 \eta -3 Z_0 Z_2^2 \theta \lambda -3 Z_1^2 Z_2 \theta \lambda -3 Z_0 Z_2^2 \kappa \lambda -3 Z_1^2 Z_2 \kappa \lambda =0,\\ & 3 b Z_1 Z_2^2-3 Z_1 Z_2^2 \theta \lambda -3 Z_1 Z_2^2 \kappa \lambda =0,\\ & b Z_2^3-Z_2^3 \theta \lambda -Z_2^3 \kappa \lambda =0. \end{aligned}$$

After solving this system, we get the results as follows.

Family 1:

$$\begin{aligned} Z_0=0 ,~Z_1=-\frac{i \sqrt{6} \beta \tau \sqrt{\eta }}{\sqrt{b}} ,~Z_2=0 ,~\digamma =\frac{\beta \lambda ^3+\omega }{3 \beta \upsilon ^2},~\lambda =0. \end{aligned}$$

Case I: If $\digamma>0$ and $\eta>0$, then solutions has form

$$\begin{aligned} {\begin{matrix} X_{1} (y,t)=\frac{i \sqrt{2} \beta \tau \sqrt{\eta } \sqrt{\frac{\frac{\beta \lambda ^3+\omega }{3 \beta \upsilon ^2} }{\eta }} e^{i (\alpha +t \omega )} \text {csch}\left( \sqrt{\frac{\beta \lambda ^3+\omega }{3 \beta \upsilon ^2} } (S-t \tau v+\tau y)\right) }{\sqrt{b}}. \end{matrix}} \end{aligned}$$

(28)

Case II: If $\digamma <0$ and $\eta>0$, then solutions has form

$$\begin{aligned} X_{2} (y,t)=-\frac{i \sqrt{2} \beta \tau \sqrt{\eta } \sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta }} e^{i (\alpha +t \omega )} \sec \left( \frac{\sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{3}}\right) }{\sqrt{b}}. \end{aligned}$$

(29)

Case III: If $\digamma>0$ and $\eta <0$, then solutions has form

$$\begin{aligned} X_{3} (y,t)=-\frac{i \sqrt{2} \beta \tau \sqrt{\eta } \sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta }} e^{i (\alpha +t \omega )} \text {sech}\left( \frac{\sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{3}}\right) }{\sqrt{b}}. \end{aligned}$$

(30)

Case IV: If $\digamma <0$ and $\eta>0$, then solutions has form

$$\begin{aligned} X_{4} (y,t)=-\frac{i \sqrt{2} \beta \tau \sqrt{\eta } \sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta }} e^{i (\alpha +t \omega )} \csc \left( \frac{\sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+ \tau y)}{\sqrt{3}}\right) }{\sqrt{b}}. \end{aligned}$$

(31)

Case V: If $\digamma>0$ and $\eta =0$, then solutions has form

$$\begin{aligned} X_{5} (y,t)=-\frac{i \sqrt{6} \beta \tau \sqrt{\eta } e^{i (\alpha +t \omega )} e^{\frac{\sqrt{\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{3}}}}{\sqrt{b}}. \end{aligned}$$

(32)

Case VI: If $\digamma <0$ and $\eta <0$, then solutions has form

$$\begin{aligned} X_{6} (y,t)=&-\frac{i \sqrt{6} \beta \tau \sqrt{\eta } e^{i (\alpha +t \omega )} \left( \cos \left( \frac{\sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{3}}\right) +i \sin \left( \frac{\sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{3}}\right) \right) }{\sqrt{b}}. \end{aligned}$$

(33)

Case VII: If $\digamma =0$ and $\eta>0$, then solutions has form

$$\begin{aligned} X_{7} (y,t)= -\frac{i \sqrt{6} \beta \tau e^{i (\alpha +t \omega )}}{\sqrt{b} (S-t \tau v+\tau y)}. \end{aligned}$$

(34)

Case VIII: If $\digamma =0$ and $\eta <0$, then solutions has form

$$\begin{aligned} X_{8} (y,t)= -\frac{i \sqrt{6} \beta \tau \sqrt{\eta } e^{i (\alpha +t \omega )}}{\sqrt{b} \sqrt{-\eta } (S-t \tau v+\tau y)}. \end{aligned}$$

(35)

Family 2:

$$\begin{aligned} Z_0=0,~Z_1=\frac{i \sqrt{6} \beta \upsilon \sqrt{\eta }}{\sqrt{b}},~Z_2=0,\digamma =\frac{\beta \lambda ^3+\omega }{3 \beta \tau ^2},~\lambda =0. \end{aligned}$$

Case I: If $\digamma>0$ and $\eta>0$, then solutions has form

$$\begin{aligned} {\begin{matrix} X_{9} (y,t)=-\frac{i \sqrt{2} \beta \tau \sqrt{\eta } \sqrt{\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta }} e^{i (\alpha +t \omega )} \text {csch}\left( \frac{\sqrt{\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{3}}\right) }{\sqrt{b}}. \end{matrix}} \end{aligned}$$

(36)

Case II: If $\digamma <0$ and $\eta>0$, then solutions has form

$$\begin{aligned} X_{10} (y,t)=\frac{i \sqrt{2} \beta \tau \sqrt{\eta } \sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta }} e^{i (\alpha +t \omega )} \sec \left( \frac{\sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{3}}\right) }{\sqrt{b}}. \end{aligned}$$

(37)

Case III: If $\digamma>0$ and $\eta <0$, then solutions has form

$$\begin{aligned} X_{11} (y,t)=\frac{i \sqrt{2} \beta \tau \sqrt{\eta } \sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta }} e^{i (\alpha +t \omega )} \text {sech}\left( \frac{\sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{3}}\right) }{\sqrt{b}}. \end{aligned}$$

(38)

Case IV: If $\digamma <0$ and $\eta>0$, then solutions has form

$$\begin{aligned} X_{12} (y,t)=\frac{i \sqrt{2} \beta \tau \sqrt{\eta } \sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta }} e^{i (\alpha +t \omega )} \csc \left( \frac{\sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{3}}\right) }{\sqrt{b}}. \end{aligned}$$

(39)

Case V: If $\digamma>0$ and $\eta =0$, then solutions has form

$$\begin{aligned} X_{13} (y,t)=\frac{i \sqrt{6} \beta \tau \sqrt{\eta } e^{i (\alpha +t \omega )} e^{\frac{\sqrt{\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{3}}}}{\sqrt{b}}. \end{aligned}$$

(40)

Case VI: If $\digamma <0$ and $\eta =0$, then solutions has form

$$\begin{aligned} X_{14} (y,t)=&\frac{i \sqrt{6} \beta \tau \sqrt{\eta } e^{i (\alpha +t \omega )} \left( \cos \left( \frac{\sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{3}}\right) +i \sin \left( \frac{\sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{3}}\right) \right) }{\sqrt{b}}. \end{aligned}$$

(41)

Case VII: If $\digamma =0$ and $\eta>0$, then solutions has form

$$\begin{aligned} X_{15} (y,t)=\frac{i \sqrt{6} \beta \tau e^{i (\alpha +t \omega )}}{\sqrt{b} (S-t \tau v+\tau y)}. \end{aligned}$$

(42)

Case VIII: If $\digamma =0$ and $\eta <0$, then solutions has form

$$\begin{aligned} X_{16} (y,t)=\frac{i \sqrt{6} \beta \tau \sqrt{\eta } e^{i (\alpha +t \omega )}}{\sqrt{b} \sqrt{-\eta } (S-t \tau v+\tau y)}. \end{aligned}$$

(43)

Form 2: We first obtain the balancing number $n=1$ by applying the homogeneous balance approach. Then, the solution of Eq. (23) is written as:

$$\begin{aligned} U(\sigma ) = Z_0 + Z_1 \rho (\sigma ) + Z_2 \rho (\sigma )^2. \end{aligned}$$

(44)

Furthermore, we derive a system of algebraic equations by substituting Eq. (44) and Eq. (8) into Eq. (23),

$$\begin{aligned} & b Z_0^3-\beta \lambda ^3 Z_0+6 \beta \tau ^2 Z_2 \digamma ^2-Z_0^3 \theta \lambda -Z_0^3 \kappa \lambda -Z_0 \omega =0,\\ & 3 b Z_0^2 Z_1-\beta \lambda ^3 Z_1+6 \beta \tau ^2 Z_1 \eta \digamma -3 Z_0^2 Z_1 \theta \lambda -3 Z_0^2 Z_1 \kappa \lambda -Z_1 \omega =0,\\ & 3 b Z_0 Z_1^2+3 b Z_0^2 Z_2+\beta \lambda ^3 \left( -Z_2\right) +24 \beta \tau ^2 Z_2 \eta \digamma -3 Z_0 Z_1^2 \theta \lambda -3 Z_0^2 Z_2 \theta \lambda \\ & \hspace{3cm}-3 Z_0 Z_1^2 \kappa \lambda -3 Z_0^2 Z_2 \kappa \lambda -Z_2 \omega =0,\\ & b Z_1^3+6 b Z_0 Z_2 Z_1+6 \beta \tau ^2 Z_1 \eta ^2-Z_1^3 \theta \lambda -6 Z_0 Z_2 Z_1 \theta \lambda -Z_1^3 \kappa \lambda -6 Z_0 Z_2 Z_1 \kappa \lambda =0,\\ & 3 b Z_0 Z_2^2+3 b Z_1^2 Z_2+18 \beta \tau ^2 Z_2 \eta ^2-3 Z_0 Z_2^2 \theta \lambda -3 Z_1^2 Z_2 \theta \lambda -3 Z_0 Z_2^2 \kappa \lambda -3 Z_1^2 Z_2 \kappa \lambda =0,\\ & 3 b Z_1 Z_2^2-3 Z_1 Z_2^2 \theta \lambda -3 Z_1 Z_2^2 \kappa \lambda =0,\\ & b Z_2^3-Z_2^3 \theta \lambda -Z_2^3 \kappa \lambda =0. \end{aligned}$$

When this system is solved, the outcomes are as

Family 1:

$$\begin{aligned} Z_0=0,~Z_1=\frac{i \sqrt{6} \beta \tau \eta }{\sqrt{b}},~Z_2=0,\lambda =0,~\digamma =\frac{\beta \lambda ^3+\omega }{6 \beta \tau ^2 \eta }. \end{aligned}$$

Case I: If $\digamma \eta>0$, then solution is written as

$$\begin{aligned} {\begin{matrix} X_{17} (y,t)=\frac{i \beta \tau \eta \text {sgn}\left( \beta \lambda ^3+\omega \right) \sqrt{\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta ^2}} e^{i (\alpha +t \omega )} \tan \left( \frac{\sqrt{\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{6}}\right) }{\sqrt{b} \text {sgn}(\beta \tau )^2 \text {sgn}(\eta )}. \end{matrix}} \end{aligned}$$

(45)

Case II: If $\digamma \eta>0$, then solution is written as

$$\begin{aligned} X_{18} (y,t)=-\frac{i \beta \tau \eta \text {sgn}\left( \beta \lambda ^3+\omega \right) \sqrt{\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta ^2}} e^{i (\alpha +t \omega )} \cot \left( \frac{\sqrt{\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{6}}\right) }{\sqrt{b} \text {sgn}(\beta \tau )^2 \text {sgn}(\eta )}. \end{aligned}$$

(46)

Case III: If $\digamma \eta <0$, then solution is written as

$$\begin{aligned} X_{19} (y,t)=\frac{i \beta \tau \eta \text {sgn}\left( \beta \lambda ^3+\omega \right) \sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta ^2}} e^{i (\alpha +t \omega )} \tanh \left( \frac{\sqrt{-\frac{\omega \left( \beta \lambda ^3+\omega \right) }{\beta \tau ^2 \eta }} (S-t \tau v+\tau y)}{\sqrt{6}}\right) }{\sqrt{b} \text {sgn}(\beta \tau )^2 \text {sgn}(\eta )}. \end{aligned}$$

(47)

Case IV: If $\digamma \eta <0$, then solution is written as

$$\begin{aligned} X_{20} (y,t)=\frac{i \beta \tau \eta \text {sgn}\left( \beta \lambda ^3+\omega \right) \sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta ^2}} e^{i (\alpha +t \omega )} \coth \left( \frac{\sqrt{-\frac{\omega \left( \beta \lambda ^3+\omega \right) }{\beta \tau ^2 \eta }} (S-t \tau v+\tau y)}{\sqrt{6}}\right) }{\sqrt{b} \text {sgn}(\beta \tau )^2 \text {sgn}(\eta )}. \end{aligned}$$

(48)

Case V: If $\digamma =0$ and $\eta>0$, then solution is written as

$$\begin{aligned} X_{21} (y,t)=-\frac{i \sqrt{6} \beta \tau e^{i (\alpha +t \omega )}}{\sqrt{b} (S-t \tau v+\tau y)}. \end{aligned}$$

(49)

Case VI: If $\digamma \epsilon R$ and $\eta =0$, then solution is written as

$$\begin{aligned} X_{22} (y,t)=\frac{i \left( \beta \lambda ^3+\omega \right) e^{i (\alpha +t \omega )} (S-t \tau v+\tau y)}{\sqrt{6} \sqrt{b} \beta \tau }. \end{aligned}$$

(50)

Family 2:

$$\begin{aligned} Z_0=0,~Z_1=-\frac{i \sqrt{6} \beta \upsilon \eta }{\sqrt{b}},~Z_2=0,~\lambda =0,\digamma =\frac{\beta \lambda ^3+\omega }{6 \beta \tau ^2 \eta }. \end{aligned}$$

Case I: If $\digamma \eta>0$, then solution is written as

$$\begin{aligned} {\begin{matrix} X_{23} (y,t)-\frac{i \beta \tau \eta \text {sgn}\left( \beta \lambda ^3+\omega \right) \sqrt{\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta ^2}} e^{i (\alpha +t \omega )} \tan \left( \frac{\sqrt{\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{6}}\right) }{\sqrt{b} \text {sgn}(\beta \tau )^2 \text {sgn}(\eta )}. \end{matrix}} \end{aligned}$$

(51)

Case II: If $\digamma \eta>0$, then solution is written as

$$\begin{aligned} X_{24} (y,t)=\frac{i \beta \tau \eta \text {sgn}\left( \beta \lambda ^3+\omega \right) \sqrt{\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta ^2}} e^{i (\alpha +t \omega )} \cot \left( \frac{\sqrt{\frac{\beta \lambda ^3+\omega }{\beta \tau ^2}} (S-t \tau v+\tau y)}{\sqrt{6}}\right) }{\sqrt{b} \text {sgn}(\beta \tau )^2 \text {sgn}(\eta )}. \end{aligned}$$

(52)

Case III: If $\digamma \eta <0$, then solution is written as

$$\begin{aligned} X_{25} (y,t)=-\frac{i \beta \tau \eta \text {sgn}\left( \beta \lambda ^3+\omega \right) \sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta ^2}} e^{i (\alpha +t \omega )} \tanh \left( \frac{\sqrt{-\frac{\omega \left( \beta \lambda ^3+\omega \right) }{\beta \tau ^2 \eta }} (S-t \tau v+\tau y)}{\sqrt{6}}\right) }{\sqrt{b} \text {sgn}(\beta \tau )^2 \text {sgn}(\eta )}. \end{aligned}$$

(53)

Case IV: If $\digamma \eta <0$, then solution is written as

$$\begin{aligned} X_{26} (y,t)=-\frac{i \beta \tau \eta \text {sgn}\left( \beta \lambda ^3+\omega \right) \sqrt{-\frac{\beta \lambda ^3+\omega }{\beta \tau ^2 \eta ^2}} e^{i (\alpha +t \omega )} \coth \left( \frac{\sqrt{-\frac{\omega \left( \beta \lambda ^3+\omega \right) }{\beta \tau ^2 \eta }} (S-t \tau v+\tau y)}{\sqrt{6}}\right) }{\sqrt{b} \text {sgn}(\beta \tau )^2 \text {sgn}(\eta )}. \end{aligned}$$

(54)

Case V: If $\digamma =0$ and $\eta>0$, then solution is written as

$$\begin{aligned} X_{27} (y,t)=\frac{i \sqrt{6} \beta \tau e^{i (\alpha +t \omega )}}{\sqrt{b} (S-t \tau v+\tau y)}. \end{aligned}$$

(55)

Case VI: If $\digamma \epsilon R$ and $\eta =0$, then solution is written as

$$\begin{aligned} X_{28} (y,t)=-\frac{i \left( \beta \lambda ^3+\omega \right) e^{i (\alpha +t \omega )} (S-t \tau v+\tau y)}{\sqrt{6} \sqrt{b} \beta \tau }. \end{aligned}$$

(56)

Application to the polynomial expansion method

Firstly, we use the homogeneous balance technique and obtain $n=1$. Then, we write the solution of Eq. (23) as:

$$\begin{aligned} U(\sigma )=\sum _{i=1}^m l_i \rho (\sigma )^{-i}+\sum _{i=1}^m Z_i \rho (\sigma )^i+Z_0, \end{aligned}$$

(57)

here, we insert both Eq. (57) and Eq. (11) in Eq. (23), then we get the system of mathematical equations in the following form

$$\begin{aligned} & 6 b Z_0 Z_1 l_1+b Z_0^3-\beta \lambda ^3 Z_0+3 \beta \tau ^2 Z_1 \digamma \Upsilon \\ & -\theta \lambda Z_0^3-\kappa \lambda Z_0^3-6 \theta \lambda Z_0 Z_1 l_1-6 \kappa \lambda Z_0 Z_1 l_1-Z_0 \omega +3 \beta \tau ^2 l_1 \Upsilon =0,\\ & b l_1^3+6 \beta \tau ^2 l_1 \digamma ^2-\theta \lambda l_1^3-\kappa \lambda l_1^3=0,\\ & 3 b Z_0 l_1^2-3 \theta \lambda Z_0 l_1^2-3 \kappa \lambda Z_0 l_1^2+9 \beta \tau ^2 l_1 \digamma \Upsilon =0, \\ & 3 b Z_1 l_1^2+3 b Z_0^2 l_1-3 \theta \lambda Z_1 l_1^2-3 \theta \lambda Z_0^2 l_1-3 \kappa \lambda Z_1 l_1^2\\ & -3 \kappa \lambda Z_0^2 l_1-\beta \lambda ^3 l_1+6 \beta \tau ^2 l_1 \digamma +3 \beta \tau ^2 l_1 \Upsilon ^2-l_1 \omega =0, \\ & 3 b Z_1^2 l_1+3 b Z_0^2 Z_1-\beta \lambda ^3 Z_1+6 \beta \tau ^2 Z_1 \digamma +3 \beta \tau ^2 Z_1 \Upsilon ^2\\ & -3 \theta \lambda Z_0^2 Z_1-3 \kappa \lambda Z_0^2 Z_1-3 \theta \lambda Z_1^2 l_1-3 \kappa \lambda Z_1^2 l_1-Z_1 \omega =0,\\ & 3 b Z_0 Z_1^2+9 \beta \tau ^2 Z_1 \Upsilon -3 \theta \lambda Z_0 Z_1^2-3 \kappa \lambda Z_0 Z_1^2=0,\\ & b Z_1^3+6 \beta \tau ^2 Z_1-\theta \lambda Z_1^3-\kappa \lambda Z_1^3=0. \end{aligned}$$

On resolving the given arrangement of equations, one acquires the required family as

Family 1.

$$\begin{aligned} Z_0=\frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}},~Z_1=0,l_1=\frac{i \left( 2 \sqrt{6} \beta \lambda ^3+3 \sqrt{6} \beta \tau ^2 \Upsilon ^2+2 \sqrt{6} \omega \right) }{12 \sqrt{b} \beta \tau },~\lambda =0,P=\frac{2 \beta \lambda ^3+3 \beta \tau ^2 \Upsilon ^2+2 \omega }{12 \beta \tau ^2}. \end{aligned}$$

Case I.:If $\Upsilon =0$ and $\digamma =0$ then attained solutions is written as

$$\begin{aligned} X_{29} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}}+\frac{i \left( 2 \sqrt{6} \beta \lambda ^3+3 \sqrt{6} \beta \tau ^2 \Upsilon ^2+2 \sqrt{6} \omega \right) (t \tau v-\tau y)}{12 \sqrt{b} \beta \tau }\right) . \end{aligned}$$

(58)

Case II: If $\Upsilon \ne 0$; $\digamma =0$ then attained solutions is written as

$$\begin{aligned} X_{30} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}}-\frac{i \left( 2 \sqrt{6} \beta \lambda ^3+3 \sqrt{6} \beta \tau ^2 \Upsilon ^2+2 \sqrt{6} \omega \right) \left( q_0 e^{\Upsilon (-(\tau y-t \tau v))}-1\right) }{12 \sqrt{b} \beta \tau \Upsilon }\right) . \end{aligned}$$

(59)

Case III(i): If $\Upsilon =0$ and $\digamma \ne 0$ and $\kappa>0$ then we get solution as

$$\begin{aligned} X_{31} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}}+\frac{i \left( 2 \sqrt{6} \beta \lambda ^3+3 \sqrt{6} \beta \tau ^2 \Upsilon ^2+2 \sqrt{6} \omega \right) \cot \left( \sqrt{\digamma } (\tau y-t \tau v)\right) }{12 \sqrt{b} \beta \tau \sqrt{\digamma }}\right) , \end{aligned}$$

(60)

Case III(ii): If $\Upsilon =0$ and $\digamma \ne 0$ and $\digamma>0$ then we get solution as

$$\begin{aligned} X_{32} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}}+\frac{i \left( 2 \sqrt{6} \beta \lambda ^3+3 \sqrt{6} \beta \tau ^2 \Upsilon ^2+2 \sqrt{6} \omega \right) \tan \left( \sqrt{\digamma } (\tau y-t \tau v)\right) }{12 \sqrt{b} \beta \tau \sqrt{\digamma }}\right) . \end{aligned}$$

(61)

Case IV(i): If $\Upsilon =0$ and $\digamma \ne 0$ and $\digamma <0$ then we get solution as

$$\begin{aligned} X_{33} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}}+\frac{i \left( 2 \sqrt{6} \beta \lambda ^3+3 \sqrt{6} \beta \tau ^2 \Upsilon ^2+2 \sqrt{6} \omega \right) \coth \left( \sqrt{\digamma } (\tau y-t \tau v)\right) }{12 \sqrt{b} \beta \tau \sqrt{\digamma }}\right) , \end{aligned}$$

(62)

Case IV(ii): If $\Upsilon =0$ and $\digamma \ne 0$ and $\digamma <0$ then we get solution as

$$\begin{aligned} X_{34} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}}+\frac{i \left( 2 \sqrt{6} \beta \lambda ^3+3 \sqrt{6} \beta \tau ^2 \Upsilon ^2+2 \sqrt{6} \omega \right) \tanh \left( \sqrt{\digamma } (\tau y-t \tau v)\right) }{12 \sqrt{b} \beta \tau \sqrt{\digamma }}\right) . \end{aligned}$$

(63)

Case V: If $\Upsilon \ne 0$ and $\digamma \ne 0$ and $\digamma <0$ then we get solution as

$$\begin{aligned} X_{35} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}}+\frac{i \left( 2 \sqrt{6} \beta \lambda ^3+3 \sqrt{6} \beta \tau ^2 \Upsilon ^2+2 \sqrt{6} \omega \right) \left( 1-q_1 e^{\left( A_1-A_2\right) (\tau y-t \tau v)}\right) }{12 \sqrt{b} \beta \tau \left( A_1-q_1 A_2 e^{\left( A_1-A_2\right) (\tau y-t \tau v)}\right) }\right) . \end{aligned}$$

(64)

Family 2.

$$\begin{aligned} Z_0=-\frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}},Z_1=-\frac{i \sqrt{6} \beta \tau }{\sqrt{b}},~l_1=0,\lambda =0,P=\frac{2 \beta \lambda ^3+3 \beta \tau ^2 \Upsilon ^2+2 \omega }{12 \beta \tau ^2}. \end{aligned}$$

Case I.:If $\Upsilon =0$ and $\digamma =0$ then attained solutions is written as

$$\begin{aligned} X_{36} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{i \sqrt{6} \beta \tau }{\sqrt{b} (\tau y-t \tau v)}-\frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}}\right) . \end{aligned}$$

(65)

Case II:If $\Upsilon \ne 0$; $\digamma =0$ then attained solutions is written as

$$\begin{aligned} X_{37} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{i \sqrt{6} \beta \tau \Upsilon }{\sqrt{b} \left( q_0 e^{\Upsilon (-(\tau y-t \tau v))}-1\right) }-\frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}}\right) . \end{aligned}$$

(66)

Case III(i) If $\Upsilon =0$ and $\digamma \ne 0$ and $\kappa>0$ then we get solution as

$$\begin{aligned} X_{38} (y,t)=e^{i (\alpha +t \omega )} \left( -\frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}}-\frac{i \sqrt{6} \beta \tau \sqrt{\digamma } \tan \left( \sqrt{\digamma } (\tau y-t \tau v)\right) }{\sqrt{b}}\right) , \end{aligned}$$

(67)

Case III(ii) If $\Upsilon =0$ and $\digamma \ne 0$ and $\digamma>0$ then we get solution as

$$\begin{aligned} X_{39} (y,t)=e^{i (\alpha +t \omega )} \left( -\frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}}-\frac{i \sqrt{6} \beta \tau \sqrt{\digamma } \cot \left( \sqrt{\digamma } (\tau y-t \tau v)\right) }{\sqrt{b}}\right) . \end{aligned}$$

(68)

Case IV(i) If $\Upsilon =0$ and $\digamma \ne 0$ and $\digamma <0$ then we get solution as

$$\begin{aligned} X_{40} (y,t)=e^{i (\alpha +t \omega )} \left( -\frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}}-\frac{i \sqrt{6} \beta \tau \sqrt{\digamma } \tanh \left( \sqrt{\digamma } (\tau y-t \tau v)\right) }{\sqrt{b}}\right) , \end{aligned}$$

(69)

Case IV(ii) If $\Upsilon =0$ and $\digamma \ne 0$ and $\digamma <0$ then we get solution as

$$\begin{aligned} X_{41} (y,t)=e^{i (\alpha +t \omega )} \left( -\frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}}-\frac{i \sqrt{6} \beta \tau \sqrt{\digamma } \coth \left( \sqrt{\digamma } (\tau y-t \tau v)\right) }{\sqrt{b}}\right) . \end{aligned}$$

(70)

Case V If $\Upsilon \ne 0$ and $\digamma \ne 0$ and $\digamma <0$ then we get solution as

$$\begin{aligned} X_{42} (y,t)=e^{i (\alpha +t \omega )} \left( -\frac{i \sqrt{\frac{3}{2}} \beta \tau \Upsilon }{\sqrt{b}}-\frac{i \sqrt{6} \beta \tau \left( A_1-q_1 A_2 e^{\left( A_1-A_2\right) (\tau y-t \tau v)}\right) }{\sqrt{b} \left( 1-q_1 e^{\left( A_1-A_2\right) (\tau y-t \tau v)}\right) }\right) . \end{aligned}$$

(71)

The modified extended tanh-function method

Firstly, one uses the homogeneous balance technique and obtains $n=1$. Then, we write the solution of Eq. (23) as:

$$\begin{aligned} U(\sigma )=Z_1 \rho (\sigma )+Z_0+\frac{l_1}{\rho (\sigma )}, \end{aligned}$$

(72)

here, we add both Eq. (72) and Eq. (13) in Eq. (23), then we get the system of mathematical equations in the following form

$$\begin{aligned} & 6 b l_1 Z_0 Z_1+b Z_0^3-6 \theta l_1 \lambda Z_0 Z_1-6 l_1 \kappa \lambda Z_0 Z_1-\beta \lambda ^3 Z_0\\ & -\theta \lambda Z_0^3-\kappa \lambda Z_0^3-Z_0 \omega =0,\\ & b h_1^3+6 \beta \tau ^2 l_1 \digamma ^2-\theta l_1^3 \lambda -l_1^3 \kappa \lambda =0,\\ & 3 b l_1^2 Z_0-3 \theta l_1^2 \lambda Q_0-3 l_1^2 \kappa \lambda Z_0=0, \\ & 3 b l_1 Z_0^2+3 b l_1^2 Z_1-\beta \lambda ^3 l_1+6 \beta \tau ^2 l_1 \digamma -3 \theta l_1 \lambda Z_0^2\\ & -3 \theta l_1^2 \lambda Z_1-3 l_1 \kappa \lambda Z_0^2-3 l_1^2 \kappa \lambda Z_1-l_1 \omega =0, \\ & 3 b l_1 Z_1^2+3 b Z_0^2 Z_1-3 \theta l_1 \lambda Z_1^2-3 l_1 \kappa \lambda Z_1^2-\beta \lambda ^3 Z_1\\ & +6 \beta \tau ^2 Z_1 \digamma -3 \theta \lambda Z_0^2 Z_1-3 \kappa \lambda Z_0^2 Z_1-Z_1 \omega =0,\\ & 3 b Z_0 Z_1^2-3 \theta \lambda Z_0 Z_1^2-3 \kappa \lambda Z_0 Z_1^2=0,\\ & b Z_1^3+6 \beta \tau ^2 Z_1-\theta \lambda Z_1^3-\kappa \lambda Z_1^3=0. \end{aligned}$$

On resolving the given system of equations, one obtains the required family as:

Family 1.

$$\begin{aligned} Z_0=0,Z_1=e^{i (\alpha +t \omega )} \left( \frac{i \sqrt{6} \beta \tau }{\sqrt{b}}\right) ,l_1=-\frac{i \left( \beta \lambda ^3+\omega \right) }{4 \sqrt{6} \sqrt{b} \beta \tau },\digamma =\frac{-\beta \lambda ^3-\omega }{12 \beta \tau ^2}. \end{aligned}$$

Case I: If $\digamma <0$, then

$$\begin{aligned} X_{43}=e^{i (\alpha +t \omega )} \left( \frac{i \left( \beta \lambda ^3+\omega \right) \coth \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{2 \sqrt{2} \sqrt{b} \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \upsilon ^2}}}-\frac{i \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} \tanh \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{\sqrt{2} \sqrt{b}}\right) , \end{aligned}$$

(73)

$$\begin{aligned} X_{44}=e^{i (\alpha +t \omega )} \left( \frac{i \left( \beta \lambda ^3+\omega \right) \tanh \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{2 \sqrt{2} \sqrt{b} \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \upsilon ^2}}}-\frac{i \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} \coth \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{\sqrt{2} \sqrt{b}}\right) . \end{aligned}$$

(74)

Case II:If $\digamma =0$, then

$$\begin{aligned} X_{45} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{6 \beta \tau ^2}{(\tau y-t \tau v) \sqrt{-6 b \beta \tau ^2+\beta \lambda ^3 \lambda +\lambda \omega -\frac{6 \beta \tau ^2 \lambda }{\tau y-t \tau v}}}\right) . \end{aligned}$$

(75)

Case III:If $\digamma>0$, then

$$\begin{aligned} {\begin{matrix}&X_{46} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{i \left( \beta \lambda ^3+\omega \right) \cot \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{2 \sqrt{2} \sqrt{b} \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}}}-\frac{i \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} \tan \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{\sqrt{2} \sqrt{b}}\right) , \end{matrix}} \end{aligned}$$

(76)

$$\begin{aligned} {\begin{matrix}&X_{47} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{i \left( \beta \lambda ^3+\omega \right) \tan \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{2 \sqrt{2} \sqrt{b} \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}}}-\frac{i \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} \cot \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{\sqrt{2} \sqrt{b}}\right) . \end{matrix}} \end{aligned}$$

(77)

Family 2.

$$\begin{aligned} Z_0=0,Z_1=e^{i (\alpha +t \omega )} \left( -\frac{i \sqrt{6} \beta \tau }{\sqrt{b}}\right) ,l_1=\frac{i \left( \beta \lambda ^3+\omega \right) }{4 \sqrt{6} \sqrt{b} \beta \tau },\digamma =\frac{-\beta \lambda ^3-\omega }{12 \beta \tau ^2}. \end{aligned}$$

Case I:If $\digamma <0$, then

$$\begin{aligned} X_{48}=e^{i (\alpha +t \omega )} \left( \frac{i \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} \tanh \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{\sqrt{2} \sqrt{b}}-\frac{i \left( \beta \lambda ^3+\omega \right) \coth \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{2 \sqrt{2} \sqrt{b} \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}}}\right) , \end{aligned}$$

(78)

$$\begin{aligned} X_{49}=e^{i (\alpha +t \omega )} \left( \frac{i \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} \coth \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{\sqrt{2} \sqrt{b}}-\frac{i \left( \beta \lambda ^3+\omega \right) \tanh \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{2 \sqrt{2} \sqrt{b} \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}}}\right) . \end{aligned}$$

(79)

Case II:If $\digamma =0$, then

$$\begin{aligned} X_{50} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{6 \beta \tau ^2}{(\tau y-t \tau v) \sqrt{-6 b \beta \tau ^2+\beta \lambda ^3 \lambda +\lambda \omega -\frac{6 \beta \tau ^2 \lambda }{\tau y-t \tau v}}}\right) . \end{aligned}$$

(80)

Case III:If $\digamma>0$, then

$$\begin{aligned} {\begin{matrix}&X_{51} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{i \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} \tan \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{\sqrt{2} \sqrt{b}}-\frac{i \left( \beta \lambda ^3+\omega \right) \cot \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{2 \sqrt{2} \sqrt{b} \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}}}\right) , \end{matrix}} \end{aligned}$$

(81)

$$\begin{aligned} {\begin{matrix}&X_{52} (y,t)=e^{i (\alpha +t \omega )} \left( \frac{i \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} \cot \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{\sqrt{2} \sqrt{b}}-\frac{i \left( \beta \lambda ^3+\omega \right) \tan \left( \frac{\sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}} (\tau y-t \tau v)}{2 \sqrt{3}}\right) }{2 \sqrt{2} \sqrt{b} \beta \tau \sqrt{-\frac{-\beta \lambda ^3-\omega }{\beta \tau ^2}}}\right) . \end{matrix}} \end{aligned}$$

(82)

Numerical solutions

The Adomian Decomposition Method (ADM) is applied to efficiently handle the nonlinear terms in the NLSE without linearization or perturbation^51,52. By decomposing nonlinearities into Adomian polynomials, the method generates approximate solutions closely matching the exact analytical forms, including bright, dark, and singular solitons^53,54. Coupling ADM with traveling-wave and refined ansatz techniques provides an additional layer of verification, enhancing the reliability and physical interpretability of the solution framework for Kerr-type optical media. In connection with the numerical outcomes of Eq. (3), we use the Adomian decomposition method and apply the given technique, then Eq. (23) is written as

$$\begin{aligned} L~\Theta (\Omega )+P~\Theta (\Omega )+R~\Theta (\Omega )=0, \end{aligned}$$

(83)

where the terms L, P, and R denote the differential operator, linear term, and nonlinear terms, respectively. Afterwards, we apply the operator $L^{-1}$ to Eq. (83), which gives the result as:

$$\begin{aligned} \sum _{i=0}^{\infty } X_i(\sigma )=X(0)+X'(0)\sigma +\frac{\left( \beta \lambda ^3+\omega \right) \sum _{i=0}^{\infty } X_i}{\left( 3 \beta \tau ^2\right) L} -\frac{(b-\lambda (\theta +\kappa )) \sum _{i=0}^{\infty } A_i}{\left( 3 \beta \tau ^2\right) L}. \end{aligned}$$

By substituting the values [$\beta =2,~\tau =3,~b=2;v=4,~\alpha =2,~S=7,~y=0,~\omega =5$] in Eq. (55), the resulting results are obtained as follows:

$$\begin{aligned} X_{Exact}(t,0)=-\frac{6 i \sqrt{3} e^{i (5 t+2)}}{7-12 t}. \end{aligned}$$

(84)

Then, we write the result obtained as

$$\begin{aligned} {\begin{matrix} X_1=(1.37377 -4.49283 i) t^5-(3.59878+2.60764 i) t^4-(1.45127-1.77099 i) t^3+\\ (0.50591+0.494227 i) t^2-(0.719012-7.88445 i) t+(1.36523 +0.61669 i), \end{matrix}} \end{aligned}$$

$$\begin{aligned} {\begin{matrix} X_2=-(1.1124-2.06313 i) t^9+(2.70872+2.93103 i) t^8+(2.98287-1.93875 i) t^7-\\ (0.920123+1.91269 i) t^6-(0.705502-0.169446 i) t^5+(0.000404426+0.1319 i) t^4+\\ (0.00863+0.007247i) t^3+(0.0008195-0.0005641i) t^2-(0.0000166+0.00001426i) t+\\ (-5.108349\times 10^{-7} +3.01210076\times 10^{-7} i), \end{matrix}} \end{aligned}$$

$$\begin{aligned} X_3&=(0.217826-0.265931 i) t^{13}-(0.444942+0.7189 i) t^{12}-(1.0178-0.368241 i) t^{11}\\&\quad+(0.17933+0.928225 i) t^{10}+(0.572492-0.0145686 i) t^9+(0.0308068-0.23927 i) t^8-(0.0660081\\&\quad+0.0215317 i) t^7-(0.00738631-0.0099889 i) t^6+(0.000493044+0.00136606 i) t^5\\&\quad+(0.000135777-0.0000109079 i) t^4+(6.1379122\times 10^{-7}-5.2889478\times 10^{-6} i) t^3\\&\quad+(-3.24161787\times 10^{-7}-9.0748056\times 10^{-8} i) t^2. \end{aligned}$$

Here, we write the numerical solution as

$$\begin{aligned} X_{numerical}&=(1.37377 -4.49283 i) t^5-(3.59878+2.60764 i) t^4\\&\quad-(1.45127-1.77099 i) t^3+(0.50591+0.494227 i) t^2-(0.719012-7.88445 i) t\\&\quad+(1.36523 +0.61669 i)-(1.1124-2.06313 i) t^9+(2.70872+2.93103 i) t^8\\&\quad+(2.98287-1.93875 i) t^7-(0.920123+1.91269 i) t^6-(0.705502-0.169446 i) t^5\\&\quad+(0.000404426+0.1319 i) t^4+(0.00863487+0.00724712 i) t^3+(0.000819546-0.000564119 i) t^2\\&\quad-(0.0000166724+0.0000142647 i) t+(-5.108349\times 10^{-7} +3.01210076\times 10^{-7} i)\\&\quad+(0.217826-0.265931 i) t^{13}-(0.444942+0.7189 i) t^{12}-(1.0178-0.368241 i) t^{11}\\&\quad+(0.17933+0.928225 i) t^{10}+(0.572492-0.0145686 i) t^9+(0.0308068-0.23927 i) t^8\\&\quad-(0.0660081+0.0215317 i) t^7-(0.00738631-0.0099889 i) t^6+(0.000493044+0.00136606 i) t^5\\&\quad+(0.000135777-0.0000109079 i) t^4+(6.1379122\times 10^{-7}-5.2889478\times 10^{-6} i) t^3\\&\quad+(-3.24161787\times 10^{-7}-9.0748056\times 10^{-8} i) t^2. \end{aligned}$$

In the Table 1 shown below, x is directly proportional to the stage of incorrectness; as the values of x increase, the level of inaccuracies also increases. The proposed results for the pure-cubic Schrödinger equation with Kerr nonlinearity are validated by substituting the obtained soliton solutions back into the governing equation to confirm their correctness and by performing numerical simulations that reproduce the same pulse evolution. Error norms are computed to show close agreement between the analytical and numerical profiles. The larger errors at early times arise from initial transients in the numerical scheme, which temporarily amplify discrepancies between the analytical and numerical solutions. This behavior is common in time-stepping methods, including the Adomian decomposition method. As the simulation progresses, these transients diminish, and the numerical results quickly stabilize, demonstrating the accuracy and reliability of the methods. Finally, the results are compared with existing Kerr nonlinearity models, which demonstrate improved accuracy, richer soliton dynamics, and better stability than previously reported techniques.

Table 1 A Comparison between exact and numerical results.

Effect of optical motion and results

This nonlinear equation captures the combined effects of dispersion, Kerr nonlinearity, and external influences, producing localized structures such as solitary and optical solitons. Increasing third-order dispersion broadens pulses, while stronger cubic nonlinearity enhances confinement, demonstrating the critical balance for stable wave propagation. The solutions describe self-sustaining light pulses, localized charge waves, and their responses to disturbances, including fractional-order and stochastic effects, revealing anomalous diffusion and fluctuating solitary formations. Analytical techniques yield a variety of exact solutions-periodic waves, bright and dark solitons, and bell-shaped pulses reflecting the intricate interaction between nonlinearity and dispersion. These results provide a robust framework for modeling nonlinear wave dynamics in Kerr-type media, optical fibers, and heterogeneous systems^55,56,57.

Figure 1 represents the behavior of the solution $|X_1(y,t)|$, a kink-type or shock wave solution, which is directly applicable in fluid dynamics and plasma physics, especially when modeling wavefronts or discontinuities in the distributions of electric fields, pressures, or densities., using parameters$\alpha =0.87, \lambda =0, \omega =0.63, \beta =0.63,$ $b=0.59, \tau =0.76, \digamma =0.66, \eta =0.78, S=0.7, v=0.64$. Figure 2 shows the behavior of the solution $|X_2(y,t)|$, which yields an exceptional soliton solution with a smooth bell-shaped shape.It can be used to simulate stable wave structures in fluid dynamics (such as shallow water waves), nonlinear optics (such as pulse propagation in Kerr media), and plasma physics. It shows a compact pulse that retains its height, but spreads minimally over time, using parameters$\alpha =0.67, \lambda =0, \omega =0.73, \beta =0.83,$ $b=0.56, \tau =0.87, \eta =0.76, S=0.87, v=0.54$. Figure 3 illustrates the behavior of the solution $|X_3(y,t)|$, which is not a smooth bell-shaped or uniformly bright/dark soliton.It is important for simulating extreme scientific phenomena, including finite-time singularities in fluid turbulence, energy focusing in plasma, and optical rogue waves. It behaves like a single wave or a blow-up solution, characterized by sharp peaks with divergent behavior, using parameters $\alpha =0.47, \lambda =0, \omega =0.63, \beta =0.73,$ $b=0.56, \tau =0.73, \eta =0.86, S=0.57, v=0.84$. Consequently, Figures 6 demonstrate the luminous bright/dark smooth soliton solutions for the specified parameters.

Figure 4 illustrates the behavior of the solution $|X_4(y,t)|$,Wave refraction effects in nonlinear shallow water dynamics, V-shaped dispersive waves in optical media, and shock fronts in compressible fluid flow can all be modeled using this V-shaped wave profile, which is frequently linked to wedge or conical waves. which represents a stable dark soliton with a smooth, V-shaped profile and no singularities, using parameters$\beta =1.41, b=1.2, v=2.02, \tau =0.95, \digamma =1.3, \lambda =1.24,$ $S=0.91, \eta =1.51, \omega =2.67, \alpha =1.87$. Figure 5 shows the behavior of the solution $|X_5(y,t)|$, which describes a dark soliton-shaped solution with constants $\beta =0.59, \tau =0.65, \Upsilon =0.65, b=0.48, v=0.29, \Upsilon =0.56,$ $\omega =0.48, \alpha =0.69, \lambda =0.34, M=0.46$. Figure 7 shows the behavior of the solution $|X_7(y,t)|$, which is useful in Bose-Einstein condensates under external potentials and nonlinear optics, namely in the development of intensity notches in de-focusing medium and represents an irregular dark soliton solution using constants $\beta =1.9, \tau =1.95, \Upsilon =0.85, b=0.48, v=0.89, \omega =0.88, \alpha =0.89$. Figure 8 illustrates the behavior of the solution $|X_8(y,t)|$, which describes an irregular U-shaped dark soliton solution. Figure 9 shows the behavior of the solution $|X_9(y,t)|$, which presents bright soliton solutions that are not strictly bell- or V-shaped, using constants $\beta =0.36, \tau =0.59, v=0.5, b=0.35, \lambda =0, \omega =0.55, \alpha =0.75$. Singularities can be found almost anywhere, but they are surprisingly common in the mathematics that physicists employ to understand the universe. Finally, we have provided 2D, 3D, and contour graphs of several of the solutions to enhance understanding of the solutions’ behavior.

In contrast to recent neural network–assisted symbolic approaches⁵⁸ and⁵⁹, our method focuses on classical analytical reduction combined with explicit soliton construction and numerical verification. While the neural network frameworks produce diverse solution types by embedding auxiliary equations into network layers, they do not emphasize direct numerical validation of solution dynamics. In contrast, our approach provides closed-form bright, dark, kink, periodic, and singular solitons for the Kerr NLSE, with their behavior confirmed through high-precision split-step simulations, ensuring both clarity and physical relevance.

Fig. 1

3D, contour, and 2D solution plots $|X_1(y,t)|$ for $\alpha =0.87,\lambda =0,\omega =0.63,\beta =0.63,b=0.59,$ $\upsilon =0.76,\digamma =0.66,\eta =0.78,S=0.7,v=0.64$.

Fig. 2

3D, contour, and 2D solution plots $|X_2(y,t)|$ for $\alpha =0.77,\lambda =0,\omega =0.53,\beta =0.53,b=0.46,$ $\upsilon =0.67,\digamma =0.46,\eta =0.76,S=0.87,v=0.54$.

Fig. 3

3D, contour, and 2D solution plots $|X_3(y,t)|$ for $\alpha =0.47,\lambda =0,\omega =0.63,\beta =0.73,b=0.56,$ $\upsilon =0.73,\eta =0.86,S=0.57,v=0.84$.

Fig. 4

3D, contour, and 2D solution plots $|X_4(y,t)|$ for $\beta =1.41,b=1.2,v=2.02,\upsilon =0.95,$$\beta =1.41,b=1.2,v=2.02,\upsilon =0.95,\digamma =1.3,\lambda =1.24,\tau =0.71,$$S=0.91,\eta =1.51,\omega =2.67,\alpha =1.87$.    

Fig. 5

3D, contour, and 2D solution plots $|X_5(y,t)|$ for $\beta =0.59,\upsilon =0.65,\Upsilon =0.65,b=0.48,v=0.29,$ $\Upsilon =0.56,\omega =0.48,\alpha =0.69,\lambda =0.34,M=0.46$.

Fig. 6

3D, contour, and 2D solution plots $|X_6(y,t)|$ for$M=0.59,\mu =0.5, \beta =0.59,\upsilon =0.65,\Upsilon =0.65,b=0.48,$ $v=0.29,\Upsilon =0.56,\omega =0.48,\alpha =0.69,\lambda =0,m_1=0.46$.

Fig. 7

3D, contour, and 2D solution plots $|X_7(y,t)|$ for $\beta =1.9,\upsilon =1.95,\Upsilon =0.85,b=0.48,v=0.89,\omega =0.88,\alpha =0.89$.

Fig. 8

3D, contour, and 2D solution plots $|X_8(y,t)|$ for $\beta =1.9,\tau =1.95,\Upsilon =0.85,b=0.48,v=0.89,\omega =0.88,\alpha =0.89$.

Fig. 9

3D, contour, and 2D solution plots $|X_9(y,t)|$ for $\beta =0.36,\upsilon =0.59,v=0.5,b=0.35,\lambda =0,\omega =0.55,\alpha =0.75$.

Significance of the study

This section compares this study with previous research that was concerned with both analytical and numerical solutions of pure-cubic NLSE with Kerr-type optical nonlinearity and shows significant advances. Significant agreement between the analytical results and those reported in previous theoretical studies validates the underlying mathematical models. Furthermore, the numerical simulations carried out here exhibit superior precision and stability in comparison to earlier computational studies, particularly in capturing soliton dynamics and wave interactions. This dual approach not only confirms earlier findings but also extends their use by offering a more robust framework for investigating complex nonlinear wave occurrences controlled by the NLSE. This study employs three complementary techniques to investigate the pure-cubic NLSE. Traveling-wave reduction converts the PDE into an ODE, simplifying the derivation of exact solutions, while a refined ansatz generates bright, dark, kink-type, periodic, and singular solitons. High-precision split-step simulations verify the solutions’ accuracy and stability. Compared to methods based on the generalized exponential-rational expansions⁶⁰, the Kumar-Malik method, the modified Sardar sub-equation approach, and the extended Arnous approach⁶¹ or including the modified F-expansion method, modified generalized exponential rational function method, and multivariate generalized exponential rational integral technique⁶², the applied approaches uniquely integrates analytical construction with numerical validation, providing clear, physically interpretable solitary and periodic waves. Overall, the framework offers a robust and unified tool for analyzing diverse nonlinear wave phenomena in Kerr-type optical systems. TA comparison have been made between the existing literature and present study in the following table 2:

Table 2 A comparison of present study and existing literature.

Conclusion

The pure-cubic nonlinear Schrödinger equation with Kerr-type nonlinearity was analyzed using a suite of analytical techniques, including the polynomial method, extended tanh approach, extended hyperbolic function method, and Adomian decomposition. These approaches produced a variety of exact and approximate wave solutions, encompassing bright, dark, kink-like, periodic, and singular solitons, and provided insights into bifurcation patterns and potential chaotic dynamics. The employed methodology is adaptable to more generalized NLS forms, such as equations with quintic or saturable nonlinearities, as well as higher-dimensional models like the (2+1) or (3+1)-dimensional NLS. Incorporating external potentials, higher-order dispersion effects, or stochastic perturbations could further extend the model to more realistic physical systems. Comparative analyses utilizing numerical solvers or data-driven techniques may enhance the reliability and applicability of the results. Graphical representations including 2D profiles, 3D surface plots, and contour diagrams illustrate how parameters such as amplitude, pulse width, and velocity affect soliton evolution. These visualizations, generated using computational tools like Mathematica, highlight the interplay of physical parameters in shaping wave propagation. A detailed assessment of solution stability and resilience confirms their relevance and applicability for practical nonlinear optical systems and integrated photonics.

We employed the polynomial expansion and modified extended tanh-function methods because they offer a structured and efficient way to obtain exact analytical solutions, including bright, dark, kink, periodic, and singular solitons. In contrast to techniques like homotopy analysis, generalized exponential-rational expansions, or neural network-assisted symbolic methods, our approach yields explicit closed-form solutions that are readily interpretable in terms of physical parameters. The derived solutions are further confirmed through high-precision numerical simulations, demonstrating their reliability, stability, and relevance across different parameter settings. This integrated analytical–numerical framework provides a novel and versatile tool for investigating diverse nonlinear wave phenomena in Kerr-type optical systems. Overall, the study advances theoretical understanding of soliton dynamics under Kerr nonlinearity and demonstrates the capability of symbolic analytical methods to generate a broad spectrum of physically meaningful solutions. The findings have important implications for nonlinear fiber optics and ultrafast photonics, providing insights for optical communication, high-speed data transmission, all-optical switching, supercontinuum generation, and pulse compression. Furthermore, the results support the design of more efficient fiber-based devices, enhance signal fidelity, and contribute to the development of advanced photonic technologies where controlled nonlinear wave behavior is critical.