Conservation Laws, Soliton Dynamics, and Stability in a Nonlinear Schrödinger Equation with Second-Order Spatiotemporal Dispersion

Abstract

This paper presents the construction of exact wave solutions for the generalized nonlinear Schrödinger equation (NLSE) with second-order spatiotemporal dispersion using the modified exponential rational function method (mERFM). The NLSE plays a vital role in various fields such as quantum mechanics, oceanography, transmission lines, and optical fiber communications, particularly in modeling pulse dynamics extending beyond the traditional slowly varying envelope estimation. By incorporating higher-order dispersion and nonlinear effects, including cubic–quintic nonlinearities, this generalized model provides a more accurate representation of ultrashort pulse propagation in optical fibers and oceanic environments. A wide range of soliton solutions is obtained, including bright and dark solitons, as well as trigonometric, hyperbolic, rational, exponential, and singular forms. These solutions offer valuable insights into nonlinear wave dynamics and multi-soliton interactions relevant to shallow- and deep-water wave propagation. Conservation laws associated with the model are also derived, reinforcing the physical consistency of the system. The stability of the obtained solutions is investigated through the analysis of modulation instability (MI), confirming their robustness and physical relevance. Graphical representations based on specific parameter selections further illustrate the complex dynamics governed by the model. Overall, the study demonstrates the effectiveness of mERFM in solving higher-order nonlinear evolution equations and highlights its applicability across various domains of physics and engineering.

1. Introduction

Nonlinear evolution equations (NEEs) play a central role in modeling a wide range of physical phenomena, particularly in the context of wave propagation, optical pulse evolution, fluid dynamics, and nonlinear dispersive media [1]. These equations are basic models for examining spatiotemporal wave structures and take into account the combined effects of dispersion and nonlinearity [2]. Because they can describe localized waves, pulse shaping, modulation instability, and soliton interactions in optical fibers, plasma physics, and hydrodynamics, the NLSE and its higher-order generalizations have garnered a lot of attention. Their integrable nature also makes these nonlinear equations highly valuable for describing the dynamics of localized stationary and pulsating wave envelopes [3]. Recognized as universal equations, they provide fundamental insight into wave propagation across a wide range of physical systems [4,5]. These equations are essential for understanding the physical analogies and the differences in the nonlinear behavior of dispersive waves [6,7]. The study of rogue waves in nonlinear dispersive media is particularly intriguing [8]. Nevertheless, the nonlinear complexity of NEEs makes it challenging to derive meaningful analytical solutions and to identify a universal method. As a result, the pursuit of precise results for nonlinear models and the refinement of these methods have garnered significant attention from researchers both domestically and internationally [9,10].

The NLSE is a critical equation in physics and engineering, widely used due to its ability to model nonlinear wave propagation in fields like nonlinear optics, fluid dynamics, and Bose–Einstein condensates [11]. As an exactly integrable NEE, the NLSE can be explored through the inverse scattering transform, providing precise analytical solutions such as solitons, which maintain their form during elastic collisions. The NLSE plays a vital role in modern science, applicable across disciplines including quantum mechanics, condensed matter physics, and phase transition theory [12]. Its significance is highlighted by successful experiments in optical soliton communication, where solitons maintain waveform integrity over long distances, and by the ability to compress and narrow optical pulses using higher-order solitons [13]. Relativistic spacetimes that admit h-almost conformal $\omega$-Ricci–Bourguignon solitons were studied in [14]. The soliton solution, a prominent feature of the NLSE, is central to explaining many nonlinear phenomena.

Advancements in nonlinear science have led to the development and refinement of numerous systematic and efficient techniques for attaining analytical and approximate solutions to NEEs [15,16]. These methods include the direct algebraic technique, the Jacobi elliptic function method, simple equation technique, the modified extended mapping technique, the Kudryashov technique, the semi-inverse variational technique, the Bilinear residual network method, the Sine-Gordon technique, the modified tanh technique, the Darboux technique, the spectral collection technique, expansion methods, the exponential fitting method, the binary bell polynomials, the auxiliary equation methods, and several others [17,18,19,20,21,22,23]. Some researchers also concentrate on the properties and stability of solutions [24,25,26,27].

In [28], the author utilized extended Jacobi elliptic functions, a two-variable $(G^{\prime }/G,1/G)$-expansion and modified tanh–coth techniques to derive soliton solutions for a higher-order NLSE along with cubic–quintic dispersion. In [29], an exact solution to this equation was obtained using the modified hyperbolic auxiliary technique. In [30], the researcher applied the proposed F-expansion method to find traveling wave and soliton results for an NLSE of higher order. Additionally, the second-order spatiotemporal NLSE, along with coefficients for group velocity dispersion and cubic nonlinearity, as discussed in [31], was employed to clarify pulse phenomena beyond the traditional deliberately varying envelope estimate in optical fibers. The authors in [32] analyzed the approximate symmetries and derived new conservation laws for the perturbed generalized BBM equation.

In this work, the mERFM is employed to investigate a higher-order NLSE featuring second-order spatiotemporal dispersion. This method is well known for producing diverse analytical wave structures such as solitary waves, bright and dark solitons, and breather-type solutions, as demonstrated in several related studies [17,33]. Our use of the mERFM produces a wide range of exact analytical solutions, including periodic waves, bright and dark solitons, breather-type structures and generalized soliton profiles. Notably, several of these solutions are novel and, to the best of our knowledge, have not been previously documented in the literature. In addition, fundamental conservation laws, specifically those associated with mass, momentum, and energy, are rigorously derived. A comprehensive stability analysis is also conducted to validate the physical relevance of the obtained solutions. The findings highlight the remarkable capability of the mERFM in unveiling complex and previously inaccessible wave structures within the NLSE framework.

The structure of this article is as follows: Section 2 presents the framework and formulation of the mERFM. In Section 3, the proposed method is applied to derive exact solutions of the higher-order NLSE. Section 4 is dedicated to the derivation of conservation laws associated with the model, focusing on mass, momentum, and energy. Section 5 provides a detailed stability analysis of the obtained solutions. Section 6 examines the physical significance and implications of the results. Finally, Section 7 summarizes the main findings and outlines the key contributions of the study.

2. Methodology Overview

Here, we describe the key steps involved in the mERFM. Consider the general NEE

$$G(u,u_t,u_x,u_{tt},u_{xx},u_{tx},\dots )=0,$$

(1)

where the function $u(x,t)$ is an unknown and G is a polynomial in partial derivatives of u. The key stages are listed as

Step 1:

The traveling wave transformation takes the following form:

$$u(x,t)=\Psi \left(\xi \right),\xi =kx-\omega t,$$

(2)

here, k and $\omega$ are constants representing the wavelength and frequency, respectively. Equation (1) is transformed using the above wave transformation, as follows:

$$Q(\Psi , -\omega {\Psi }^{\prime }, k\Psi , {\omega }^2{\Psi }^{\prime \prime }, k^2{\Psi }^{\prime \prime }, -\omega k{\Psi }^{\prime \prime },\dots )=0,$$

(3)

where Q is a polynomial in $\Psi$ and its ordinary derivatives denote the usual derivatives in terms of variable $\xi .$

Step 2:

Assuming the solution of Equation (3) has the following form:

$$\Psi \left(\xi \right)={\displaystyle \frac{c_0+c_1\phi \left(\xi \right)+\cdots +c_N{\phi }^N\left(\xi \right)}{d_0+d_1\phi \left(\xi \right)+\cdots +d_N{\phi }^N\left(\xi \right)}},$$

(4)

where

$$\phi \left(\xi \right)={\displaystyle \frac{p_1{e}^{q_1\xi }+p_2{e}^{q_2\xi }}{p_3{e}^{q_3\xi }+p_4{e}^{q_4\xi }}},$$

(5)

here, let $c_i,d_i(0\le i\le N)$ be the arbitrary constants, and the values to $p_i,q_i$ are allocated later, where N is a positive integer obtained via balancing principle applied to Equation (3). The parameters p and q are not selected arbitrarily; they are fixed by the polynomial balance between the highest-order derivative and nonlinear terms in the reduced ODE. Only specific $(p,q)$ pairs satisfy this balance and yield solvable forms of $\varphi \left(\xi \right)$, so the method admits only a limited set of valid solution families.

Step 3:

By substituting Equations (4) and (5) into Equation (3) and organizing the polynomial $P(Z_1,Z_2,Z_3,Z_4)$ with $Z_i=e^{q_i\xi }$, $i=1-4$, and then setting each coefficient of $e^{q_i\xi }$ to zero, a system of equations is obtained. The parameter values can be calculated with the help of Mathematica.

Step 4:

Using the parameters determined in step 3, we substitute them into Equation (4) to compute the solutions of Equation (3).

3. Mathematical Model

Consider the generalized NLSE that includes the coefficient of second-order spatio-temporal and group velocity dispersion [30], expressed as follows:

$$i\left({\displaystyle \frac{\partial Q}{\partial x}}+{\alpha }_1{\displaystyle \frac{\partial Q}{\partial t}}\right)+{\alpha }_2{\displaystyle \frac{{\partial }^{2}Q}{\partial t^2}}+{\alpha }_3{\displaystyle \frac{{\partial }^{2}Q}{\partial x^2}}+{\left|Q\right|}^{2}Q=0,$$

(6)

where $Q(x,t)$ represents the macroscopic wave profile, which is complex-valued, and is rescaled to represent the dimensionless envelope of the electric field. The variables t and x denote the temporal and spatial dimensions, respectively. The coefficient ${\alpha }_1$ explains first-order temporal dispersion and correlates to the inverse group velocity. The coefficient ${\alpha }_2$ symbolizes the group velocity dispersion (GVD), which describes the broadening of pulses during propagation, whereas ${\alpha }_3$ is linked to higher-order spatial dispersion effects, which are important in planar waveguides, photonic crystal structures, and optical fibers, where nonparaxial and diffraction effects become important [31,34]. This generalized version of the NLSE aids in investigating the interplay between temporal and spatial dispersion effects and offers a more accurate depiction of ultrashort pulse propagation in nonlinear optical media. The modeling of nonlinear pulse propagation in optic fibers under high-intensity regimes, where the effects of self-phase modulation and higher-order dispersion become prominent, is a particular application of this equation. Solitary wave solutions are constructed using the Fan sub-equation method [34]. Because Equation (6) has a complex form, the transformation is used as follows:

$$Q(x,t)=\Psi \left(\xi \right)e^{i\mu },\mu =\beta x-\nu t+ϵ,\xi =kx-\omega t,$$

(7)

here, $\beta ,k$ and $\omega ,\nu$ are the wavelengths and frequencies of solitons respectively, and the $ϵ$ is a phase constant. The amplitude components of the wave profile are represented by $\Psi \left(\xi \right)$, and $\mu$ denotes the phase factor of solitons. By applying Equations (6) and (7) and converting into an ODE, the real and imaginary parts are separated as follows:

$$\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right){\Psi }^{\prime \prime }+\left({\alpha }_1\nu -\beta -{\alpha }_2\nu -{\alpha }_3{\beta }^2\right)\Psi +{\Psi }^3=0,$$

(8)

with an imaginary part of $\beta ={\displaystyle \frac{{\alpha }_1\omega -k-2{\alpha }_2\nu \omega }{2{\alpha }_{3}k}}$.

4. Formation of Soliton Solutions

This section presents a systematic application of the suggested analytical method to derive the soliton solutions of Equation (6). A general traveling-wave solution expressed in Jacobi elliptic functions is admitted by Equation (6), and its standard degenerate limits are bright, dark, and periodic waves. These elliptic degeneracies exactly match the solution families found in this paper. Because different parameter sets reduce to equivalent elliptic limits, some solutions have similarities. As a result, we only report physically distinct cases that are pertinent under the localized or bounded boundary conditions under consideration. We apply the balancing principle to Equation (8) and assume the solution takes the following form:

$$\Psi \left(\xi \right)={\displaystyle \frac{c_0+c_1\phi }{d_0+d_1\phi }}.$$

(9)

By substituting Equations (5) and (7) into Equation (8), we obtain a polynomial in terms of $e^{q_i\xi }$ and its powers. The coefficients of this polynomial lead to a system of algebraic equations. After solving this system using Mathematica, the following values for the various coefficients are determined.

Family 1:

Taking $p=[-1,-1,1,-1]$ and $q=[1,-1,1,-1]$ in Equation (5), we get

$$\phi \left(\xi \right)=-{\displaystyle \frac{cosh\left(\xi \right)}{sinh\left(\xi \right)}}.$$

(10)

For ${\alpha }_2{\omega }^2+{\alpha }_3{k}^2<0$, we attain the following cases of solutions:

Case 1:

$$\begin{array}{c}\hfill c_0=\pm d_1\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, c_1={\displaystyle \frac{\sqrt{-\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}\left(\left(d_0^2-d_1^2\right)\pm \left(d_0^2+d_1^2\right)\right)}{\sqrt{2}{d}_0}}, \\ \hfill \nu ={\displaystyle \frac{{\alpha }_3{\beta }^2{d}_0^2+\beta d_0^2\pm \left(d_0^2-d_1^2\right)\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)+\left(d_0^2+d_1^2\right)\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}{\left({\alpha }_1-{\alpha }_2\right)d_0^2}}.\end{array}$$

(11)

From Case 1, the exact solution to Equation (6) is obtained as follows:

$$Q_{11}\left(\xi \right)={\displaystyle \frac{\sqrt{-\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}}{\sqrt{2}\left(d_0-d_{1}coth\left(\xi \right)\right)}}\left(2d_{1}p-{\displaystyle \frac{\left(\left(d_0^2+d_1^2\right)p+\left(d_0^2-d_1^2\right)\right)coth\left(\xi \right)}{d_0}}\right)e^{i(\beta x-\nu t+ϵ)}.$$

(12)

The graphical structures of the solution (12) are illustrated in Figure 1.

Case 2:

$$c_0=\pm d_1\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, c_1=0, d_0=0, \nu ={\displaystyle \frac{2{\alpha }_2{\omega }^2+\beta +{\alpha }_3\left({\beta }^2+2k^2\right)}{{\alpha }_1-{\alpha }_2}}.$$

(13)

From Case 2, the exact solution to Equation (6) is obtained as follows:

$$Q_{12}\left(\xi \right)=\mp \sqrt{2}\sqrt{-\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}tanh\left(\xi \right)e^{i(\beta x-\nu t+ϵ)}.$$

(14)

Figure 2 shows the graphical representations of the solution (14).

Case 3:

$$c_0=0, c_1=\mp d_0\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, d_0=d_0, d_1=0, \nu ={\displaystyle \frac{2{\alpha }_2{\omega }^2+\beta +{\alpha }_3\left({\beta }^2+2k^2\right)}{{\alpha }_1-{\alpha }_2}}.$$

(15)

From Case 3, the exact solution to Equation (6) is obtained as follows:

$$Q_{13}\left(\xi \right)=\pm \sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}coth\left(\xi \right)e^{i(\beta x-\nu t+ϵ)}.$$

(16)

Remark 1.

To ensure that the soliton profiles are real and bounded, the parameters are restricted so that the polynomial under the square root in the reduced ODE remains non-negative. Only parameter sets satisfying these reality and boundedness conditions are used, ensuring that all derived traveling-wave solutions remain physically meaningful.

Family 2:

Taking $p=[-1-i,1-i,-1,1]$ and $q=[i,-i,i,-i]$ in Equation (5), we get

$$\phi \left(\xi \right)=1+cot\left(\xi \right).$$

(17)

For ${\alpha }_2{\omega }^2+{\alpha }_3{k}^2<0$, we attain the following cases of results:

Case 1:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill c_0=\mp d_1\sqrt{-8\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, c_1=\pm d_1\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, d_0=0,\\ \hfill \nu ={\displaystyle \frac{\beta \left({\alpha }_3\beta +1\right)-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}{{\alpha }_1-{\alpha }_2}}.\end{array}\end{array}$$

(18)

From Case 1, the exact solution to Equation (6) is obtained as follows:

$$Q_{21}\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}(cot\left(\xi \right)-1)}{cot\left(\xi \right)+1}}{e}^{i(\beta x-\nu t+ϵ)}.$$

(19)

Figure 3 shows the graphical representations of the solution (19).

Case 2:

$$\begin{array}{c}\hfill c_0=0, c_1=\mp {\displaystyle \frac{d_0\sqrt{-\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}}{\sqrt{2}}}, d_1=-{\displaystyle \frac{d_0}{2}}, \nu ={\displaystyle \frac{\beta \left({\alpha }_3\beta +1\right)-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}{{\alpha }_1-{\alpha }_2}}.\end{array}$$

(20)

From Case 2, the exact solution to Equation (6) is obtained as follows:

$$Q_{22}\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}(cot\left(\xi \right)+1)}{cot\left(\xi \right)-1}}{e}^{i(\beta x-\nu t+ϵ)}.$$

(21)

Case 3:

$$\begin{array}{c}\hfill c_0=-\left(\sqrt{2}+2\right)\sqrt{-d_0^2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, c_1=-\left(\sqrt{2}+1\right)\sqrt{-d_0^2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, \\ \hfill d_1={\displaystyle \frac{d_0}{\sqrt{2}}}, \nu ={\displaystyle \frac{{\alpha }_3{\beta }^2+\beta +\left(4\sqrt{2}+6\right)\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}{{\alpha }_1-{\alpha }_2}}.\end{array}$$

(22)

From Case 3, the exact result to Equation (6) is obtained as follows:

$$Q_{23}\left(\xi \right)=-{\displaystyle \frac{2\sqrt{-d_0^2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}\left(\left(\sqrt{2}+1\right)cot\left(\xi \right)+2\sqrt{2}+3\right)}{d_0\left(\sqrt{2}cot\left(\xi \right)+\sqrt{2}+2\right)}}{e}^{i(\beta x-\nu t+ϵ)}.$$

(23)

Case 4:

$$\begin{array}{c}\hfill c_0=-\left(\sqrt{2}+2\right)\sqrt{-d_0^2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, c_1=\left(\sqrt{2}+1\right)\sqrt{-d_0^2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, \\ \hfill d_1={\displaystyle \frac{d_0}{\sqrt{2}}}, \nu ={\displaystyle \frac{\beta \left({\alpha }_3\beta +1\right)-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}{{\alpha }_1-{\alpha }_2}}.\end{array}$$

(24)

From Case 4, the exact solution to Equation (6) is obtained as follows:

$$Q_{24}\left(\xi \right)={\displaystyle \frac{2\sqrt{-d_0^2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}\left(\left(\sqrt{2}+1\right)cot\left(\xi \right)-1\right)}{d_0\left(\sqrt{2}cot\left(\xi \right)+\sqrt{2}+2\right)}}{e}^{i(\beta x-\nu t+ϵ)}.$$

(25)

Figure 4 shows the graphical representations of the solution (25).

Family 3:

Taking $p=[2,0,1,1]$ and $q=[-1,0,1,-1]$ in Equation (5), we get

$$\phi \left(\xi \right)={\displaystyle \frac{cosh\left(\xi \right)-sinh\left(\xi \right)}{cosh\left(\xi \right)}}.$$

(26)

For ${\alpha }_2{\omega }^2+{\alpha }_3{k}^2<0$, we attain the following cases of results:

Case 1:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill c_0=\pm d_0\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, c_1=\mp \left(d_0+d_1\right)\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)},\\ \hfill d_0=d_0, d_1=d_1, \nu ={\displaystyle \frac{2{\alpha }_2{\omega }^2+\beta +{\alpha }_3\left({\beta }^2+2k^2\right)}{{\alpha }_1-{\alpha }_2}}.\end{array}\end{array}$$

(27)

From Case 1, the exact result to Equation (6) is obtained as follows:

$$Q_{31}\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{-\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}\left(\left(d_0+2d_1\right)(tanh\left(\xi \right)-1)+d_0(tanh\left(\xi \right)+1)\right)}{\sqrt{2}\left(d_0-d_1(tanh\left(\xi \right)-1)\right)}}{e}^{i(\beta x-\nu t+ϵ)}.$$

(28)

Case 2:

$$\begin{array}{c}\hfill c_0=\mp d_0\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, c_1=\mp d_1\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)},\\ \hfill d_0=d_0, d_1=d_1, \nu ={\displaystyle \frac{2{\alpha }_2{\omega }^2+\beta +{\alpha }_3\left({\beta }^2+2k^2\right)}{{\alpha }_1-{\alpha }_2}}.\end{array}$$

(29)

From Case 2, the exact result to Equation (6) is obtained as follows:

$$Q_{32}\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{-\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}\left(\left(d_0+2d_1\right)(tanh\left(\xi \right)-1)-d_0(tanh\left(\xi \right)+1)\right)}{\sqrt{2}\left(d_0-d_1(tanh\left(\xi \right)-1)\right)}}{e}^{i(\beta x-\nu t+ϵ)}.$$

(30)

Family 4: Taking $p=[-3,-2,1,1]$ and $q=[0,1,0,1]$ in Equation (5), we get

$$\phi \left(\xi \right)={\displaystyle \frac{-2e^{\xi }-3}{e^{\xi }+1}}.$$

(31)

For ${\alpha }_2{\omega }^2+{\alpha }_3{k}^2<0$, we attain the following cases of results:

Case 1:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill c_0=\mp 6d_1\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, c_1=\mp {\displaystyle \frac{5d_1\sqrt{-\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}}{\sqrt{2}}},d_0=0,\\ \hfill \nu ={\displaystyle \frac{2\beta \left({\alpha }_3\beta +1\right)+\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}{2\left({\alpha }_1-{\alpha }_2\right)}}.\end{array}\end{array}$$

(32)

From Case 1, the exact result to Equation (6) is obtained as follows:

$$Q_{41}\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{-\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}\left(2e^{\xi }-3\right)}{\sqrt{2}\left(2e^{\xi }+3\right)}}{e}^{i(\beta x-\nu t+ϵ)}.$$

(33)

The graphical representations of the solution (33) are illustrated in Figure 5.

Case 2:

$$\begin{array}{c}\hfill c_0=0, c_1=\mp {\displaystyle \frac{d_0\sqrt{-\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}}{12\sqrt{2}}}, d_1={\displaystyle \frac{5d_0}{12}}, \nu ={\displaystyle \frac{2\beta \left({\alpha }_3\beta +1\right)+\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}{2\left({\alpha }_1-{\alpha }_2\right)}}.\end{array}$$

(34)

From Case 2, the exact solution to Equation (6) is obtained as follows:

$$Q_{42}\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{-\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}\left(2e^{\xi }+3\right)}{\sqrt{2}\left(2e^{\xi }-3\right)}}{e}^{i(\beta x-\nu t+ϵ)}.$$

(35)

Case 3:

$$\begin{array}{c}\hfill c_0={\displaystyle \frac{\left(2\sqrt{6}-5\right)\sqrt{-d_0^2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}}{\sqrt{2}}}, c_1={\displaystyle \frac{\left(2\sqrt{6}-5\right)\sqrt{-d_0^2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}}{2\sqrt{3}}},\\ \hfill d_1={\displaystyle \frac{d_0}{\sqrt{6}}}, \nu ={\displaystyle \frac{2\beta \left({\alpha }_3\beta +1\right)+\left(49-20\sqrt{6}\right)\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}{2\left({\alpha }_1-{\alpha }_2\right)}}.\end{array}$$

(36)

From Case 3, the exact result to Equation (6) is attained as follows:

$$Q_{43}\left(\xi \right)=-{\displaystyle \frac{\left(2\sqrt{6}-5\right)\sqrt{-d_0^2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}\left(\left(3\sqrt{2}-2\sqrt{3}\right)e^{\xi }+3\left(\sqrt{2}-\sqrt{3}\right)\right)}{d_0\left(2\left(\sqrt{6}-3\right)e^{\xi }+3\left(\sqrt{6}-2\right)\right)}}{e}^{i(\beta x-\nu t+ϵ)}.$$

(37)

Family 5: Taking $p=[-2-i,-2+i,1,1]$ and $q=[i,-i,i,-i]$ in Equation (5), we get

$$\phi \left(\xi \right)={\displaystyle \frac{sin\left(\xi \right)-2cos\left(\xi \right)}{cos\left(\xi \right)}}.$$

(38)

For ${\alpha }_2{\omega }^2+{\alpha }_3{k}^2<0$, we attain the following cases of results:

Case 1:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill c_0=\mp 5d_1\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, c_1=\mp 2d_1\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, d_0=0,\\ \hfill \nu ={\displaystyle \frac{\beta \left({\alpha }_3\beta +1\right)-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}{{\alpha }_1-{\alpha }_2}}.\end{array}\end{array}$$

(39)

From Case 1, the exact solution to Equation (6) is obtained as follows:

$$Q_{51}\left(\xi \right)=\mp {\displaystyle \frac{\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}\left(2tan\left(\xi \right)+1\right)}{tan\left(\xi \right)-2}}{e}^{i(\beta x-\nu t+ϵ)}.$$

(40)

Case 2:

$$\begin{array}{c}\hfill c_0=0, c_1=\mp {\displaystyle \frac{d_0\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}}{5}}, d_1={\displaystyle \frac{2d_0}{5}}, \nu ={\displaystyle \frac{\beta \left({\alpha }_3\beta +1\right)-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}{{\alpha }_1-{\alpha }_2}}.\end{array}$$

(41)

From Case 2, the exact solution to Equation (6) is obtained as follows:

$$Q_{52}\left(\xi \right)=\mp {\displaystyle \frac{\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}(tan\left(\xi \right)-2)}{2tan\left(\xi \right)+1}}{e}^{i(\beta x-\nu t+ϵ)}.$$

(42)

Case 3:

$$\begin{array}{c}\hfill c_0=-\left(\sqrt{5}+2\right)d_0\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, c_1=-\sqrt{{\displaystyle \frac{2\left(4\sqrt{5}+9\right)}{5}}}\sqrt{-d_0^2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, \\ \hfill d_1=-{\displaystyle \frac{d_0}{\sqrt{5}}}, \nu ={\displaystyle \frac{\beta \left({\alpha }_3\beta +1\right)-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}{{\alpha }_1-{\alpha }_2}}.\end{array}$$

(43)

From Case 3, the exact solution to Equation (6) is obtained as follows:

$$Q_{53}\left(\xi \right)={\displaystyle \frac{\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}\left(\sqrt{5\left(4\sqrt{5}+9\right)}\left(tan\left(\xi \right)-2\right)+5\left(\sqrt{5}+2\right)\right)}{\sqrt{5}tan\left(\xi \right)-2\sqrt{5}-5}}{e}^{i(\beta x-\nu t+ϵ)}.$$

(44)

Case 4:

$$\begin{array}{c}\hfill c_0=-\left(\sqrt{5}+2\right)d_0\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}, c_1=\sqrt{{\displaystyle \frac{2\left(4\sqrt{5}+9\right)}{5}}}\sqrt{-d_0^2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)},\\ \hfill d_1=-{\displaystyle \frac{d_0}{\sqrt{5}}}, \nu ={\displaystyle \frac{{\alpha }_3{\beta }^2+\beta +2\left(4\sqrt{5}+9\right)\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}{{\alpha }_1-{\alpha }_2}}.\end{array}$$

(45)

From Case 4, the exact result to Equation (6) is attained as follows:

$$Q_{54}\left(\xi \right)={\displaystyle \frac{\sqrt{-2\left({\alpha }_2{\omega }^2+{\alpha }_3{k}^2\right)}\left(\sqrt{5\left(4\sqrt{5}+9\right)}\left(2-tan\left(\xi \right)\right)+5\left(\sqrt{5}+2\right)\right)}{\sqrt{5}tan\left(\xi \right)-2\sqrt{5}-5}}{e}^{i(\beta x-\nu t+ϵ)}.$$

(46)

5. Conservation Laws

The conservation laws (CLs) used here are constructed using standard multiplier and variational-identity methods that are frequently applied to NEEs [32,35,36]. Here, the conservation laws (CLs) associated with the higher-order NLSE presented in Equation (6) are examined in detail. Usually, the CLs for Equation (6) can be found when the equation can be written as follows.

$${\partial }_t{\mathcal{T}}_i+{\partial }_x{\mathcal{X}}_i=0,i=1,2,3,$$

(47)

where the following formulation produces conserved quantities derived from Equation (6) through the density terms, represented by $\mathcal{T}$, and the associated fluxes, represented by $\mathcal{X}$, as

$${\mathcal{I}}_i={\int }_{-\infty }^{-\infty }{\mathcal{T}}_{i}dx=Constant.$$

(48)

5.1. First Conservation Law

Direct computations are carried out to determine the first CL that corresponds to Equation (6). Equation (6) is reformulated as a system of equations for convenience as

$$i\left(Q_x+{\alpha }_1{Q}_t\right)+{\alpha }_2{Q}_{tt}+{\alpha }_3{Q}_{xx}+{\left|Q\right|}^{2}Q=0.$$

(49)

Its conjugate is as follows:

$$-i\left(Q_x^{\ast }+{\alpha }_1{Q}_t^{\ast }\right)+{\alpha }_2{Q}_{tt}^{\ast }+{\alpha }_3{Q}_{xx}^{\ast }+{\left|Q\right|}^2{Q}^{\ast }=0.$$

(50)

Multiplying Equation (49) by $Q^{\ast }$ and Equation (50) by Q and subtracting the resulting equations yields the following

$$\begin{array}{c}\hfill i\left(Q_x^{\ast }Q+Q_x{Q}^{\ast }\right)+i{\alpha }_1\left(Q_t^{\ast }Q+Q_t{Q}^{\ast }\right)+{\alpha }_2\left(Q_{tt}{Q}^{\ast }-Q_{tt}^{\ast }Q\right)+{\alpha }_3\left(Q_{xx}{Q}^{\ast }-Q_{xx}^{\ast }Q\right)=0.\end{array}$$

(51)

We utilized identities

$$\begin{array}{cc}\hfill Q_x^{\ast }Q+Q_x{Q}^{\ast }& ={\partial }_x\left({\left|Q\right|}^2\right).\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill Q_t^{\ast }Q+Q_t{Q}^{\ast }& ={\partial }_t\left({\left|Q\right|}^2\right).\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill Q_{tt}{Q}^{\ast }-Q_{tt}^{\ast }Q& ={\partial }_t\left(Q_t{Q}^{\ast }-Q_t^{\ast }Q\right).\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill Q_{xx}{Q}^{\ast }-Q_{xx}^{\ast }Q& ={\partial }_x\left(Q_x{Q}^{\ast }-Q_x^{\ast }Q\right).\hfill \end{array}$$

(55)

Substituting these identities into Equation (51) yields

$$\begin{array}{c}\hfill i{\partial }_x\left({\left|Q\right|}^2\right)+i{\alpha }_1{\partial }_t\left({\left|Q\right|}^2\right)+{\alpha }_2{\partial }_t\left(Q_t{Q}^{\ast }-Q_t^{\ast }Q\right)+{\alpha }_3{\partial }_x\left(Q_x{Q}^{\ast }-Q_x^{\ast }Q\right)=0.\end{array}$$

(56)

Multiplying the above equation by $-i$ gives a real CL, as follows:

$${\partial }_t\left[{\alpha }_1{\left|Q\right|}^2+2{\alpha }_2\Im \left(Q_t{Q}^{\ast }\right)\right]+{\partial }_x\left[{\left|Q\right|}^2+2{\alpha }_3\Im \left(Q_x{Q}^{\ast }\right)\right]=0.$$

(57)

Thus, the above equation can be voiced in CL form as

$${\partial }_t{\mathcal{T}}_1+{\partial }_x{\mathcal{X}}_1=0,$$

(58)

having the real density and flux as

$$\begin{array}{cc}\hfill {\mathcal{T}}_1& ={\alpha }_1{\left|Q\right|}^2+2{\alpha }_2\Im \left(Q_t{Q}^{\ast }\right),\hfill \\ \hfill {\mathcal{X}}_1& ={\left|Q\right|}^2+2{\alpha }_3\Im \left(Q_x{Q}^{\ast }\right).\hfill \end{array}$$

Therefore, the conserved quantity (CQ) can be obtained from Equation (58) and written as follows:

$$\begin{array}{cc}\hfill {\mathcal{I}}_1& ={\int }_{-\infty }^{+\infty }\left({\alpha }_1{\left|Q\right|}^2+2{\alpha }_2\Im \left(Q_t{Q}^{\ast }\right)\right)dx.\hfill \end{array}$$

(59)

This is a generalized power (mass) for the model.

5.2. Second Conservation Law

The second CL for Equation (6) is obtained by multiplying Equation (49) by $Q_x^{\ast }$ and Equation (50) by $Q_x$, and subtracting the two resulting expressions using the identities yields the following:

$$\begin{array}{c}\hfill {\partial }_t\left(\Im \left(Q_x{Q}^{\ast }\right)+{\alpha }_1\Im \left(Q_t{Q}^{\ast }\right)+{\alpha }_2\Re \left(Q_x{Q}_t^{\ast }\right)\right)+{\partial }_x\left({\alpha }_1\Im \left(Q_x{Q}^{\ast }\right)+{\alpha }_2\Re \left(Q_x^{\ast }{Q}_t\right)\right.\\ \hfill +{\alpha }_3\left(|Q_x{|}^2-\Re \left(Q_{xx}{Q}^{\ast }\right)\right)-{\displaystyle \frac{{\left|Q\right|}^4}{2}})=0.\end{array}$$

(60)

Thus, the above equation can be voiced in CL form as

$${\partial }_t{\mathcal{T}}_2+{\partial }_x{\mathcal{X}}_2=0,$$

(61)

having the density and flux as

$$\begin{array}{cc}\hfill {\mathcal{T}}_2& =\Im \left(Q_x{Q}^{\ast }\right)+{\alpha }_1\Im \left(Q_t{Q}^{\ast }\right)+{\alpha }_2\Re \left(Q_x{Q}_t^{\ast }\right),\hfill \\ \hfill {\mathcal{X}}_2& ={\alpha }_1\Im \left(Q_x{Q}^{\ast }\right)+{\alpha }_2\Re \left(Q_x^{\ast }{Q}_t\right)+{\alpha }_3\left(|Q_x{|}^2-\Re \left(Q_{xx}{Q}^{\ast }\right)\right)-{\displaystyle \frac{{\left|Q\right|}^4}{2}}.\hfill \end{array}$$

Therefore, the CQ can be obtained from Equation (61) and written as follows:

$${\mathcal{I}}_2=M={\int }_{-\infty }^{+\infty }{\mathcal{T}}_{2}dx={\int }_{-\infty }^{+\infty }\left(\Im \left(Q_x{Q}^{\ast }\right)+{\alpha }_1\Im \left(Q_t{Q}^{\ast }\right)+{\alpha }_2\Re \left(Q_x{Q}_t^{\ast }\right)\right)dx,$$

(62)

this represents the momentum-like invariant for Equation (6).

5.3. Third Conservation Law Derivation

The third CL for Equation (6) is obtained by multiplying Equation (49) by $Q_t^{\ast }$ and Equation (50) by $Q_t$, and adding the two resulting expressions using the identities yields the following:

$$\begin{array}{c}\hfill {\partial }_t\left({\alpha }_1\Im \left(Q_t{Q}^{\ast }\right)+{\alpha }_2|Q_t{|}^2+{\alpha }_3{\left|Q_x\right|}^2-{\displaystyle \frac{{\left|Q\right|}^4}{2}}\right)+{\partial }_x\left({\alpha }_1\Im \left(Q_x{Q}^{\ast }+\right){\alpha }_2\Re \left(Q_x^{\ast }{Q}_t\right)\right.\\ \hfill \left.-{\alpha }_3\Re \left(Q_{xx}^{\ast }{Q}_x\right)+\Im \left(Q_x{Q}^{\ast }\right)\right)=0.\end{array}$$

(63)

Thus, the above equation can be voiced in CL form as

$${\partial }_t{\mathcal{T}}_3+{\partial }_x{\mathcal{X}}_3=0,$$

(64)

having the density and flux as

$$\begin{array}{cc}\hfill {\mathcal{T}}_3& ={\alpha }_1\Im \left(Q_t{Q}^{\ast }\right)+{\alpha }_2|Q_t{|}^2+{\alpha }_3{\left|Q_x\right|}^2-{\displaystyle \frac{{\left|Q\right|}^4}{2}},\hfill \\ \hfill {\mathcal{X}}_3& ={\alpha }_1\Im \left(Q_x{Q}^{\ast }+\right){\alpha }_2\Re \left(Q_x^{\ast }{Q}_t\right)-{\alpha }_3\Re \left(Q_{xx}^{\ast }{Q}_x\right)+\Im \left(Q_x{Q}^{\ast }\right).\hfill \end{array}$$

Therefore, the CQ is obtained from Equation (64) and written as

$${\mathcal{I}}_3=H={\int }_{-\infty }^{+\infty }{\mathcal{T}}_{3}dx={\int }_{-\infty }^{+\infty }\left({\alpha }_1\Im \left(Q_t{Q}^{\ast }\right)+{\alpha }_2|Q_t{|}^2+{\alpha }_3{\left|Q_x\right|}^2-{\displaystyle \frac{{\left|Q\right|}^4}{2}}\right)dx.$$

(65)

This is equivalent to the Hamiltonian, which represents the energy conservation of the soliton solution of Equation (6).

6. Stability Analysis

Many higher-order nonlinear models have instabilities resulting from the interaction of dispersive and nonlinear effects, which require steady-state modulation study. Using typical linear stability analysis, the MI of the higher-order NLSE is examined [2,12]. The evolution and growth of minor, time-dependent disturbances throughout the propagation distance are assessed using this method. The shape of the higher-order NLSE’s steady-state solution is as follows:

$$q(t,x)=\left(\sqrt{S}+\phi (t,x)\right)e^{i\mu },\mu =S\lambda ϵt,$$

(66)

The normalized form of the optical power S is taken into consideration. A linear stability analysis is carried out to look at how perturbations $\phi (t,x)$ behave. The equations determining the stability characteristics are obtained by applying linearization and substituting Equation (66) into Equation (6) as

$$iA{\displaystyle \frac{\partial \phi }{\partial t}}+i{\displaystyle \frac{\partial \phi }{\partial x}}+{\alpha }_2{\displaystyle \frac{{\partial }^2\phi }{\partial t^2}}+{\alpha }_3{\displaystyle \frac{{\partial }^2\phi }{\partial x^2}}+SB\phi +S{\phi }^{\ast }=0,$$

(67)

here, $A={\alpha }_1+2{\alpha }_2\lambda Sϵ$, $B=2-{\alpha }_1\lambda ϵ-S{\alpha }_2{\lambda }^2{ϵ}^2$, and the complex conjugate is denoted by *. Suppose that the solution to Equation (67) is of the following form:

$$\phi (t,x)={\gamma }_1{e}^{i(\rho x-\tau t)}+{\gamma }_2{e}^{-i(\rho x-\tau t)},$$

(68)

here, the normalized wave number and frequency of the disturbance are denoted by $\rho$ and $\tau$, respectively. A linear evolution equation with constant coefficients and the dispersion relation $\tau =\tau \left(\rho \right)$ characterize the relationship between temporal oscillations $e^{i\rho x}$ and spatial oscillations $e^{i\tau t}$. Substituting the perturbation ansatz Equation (68) into Equation (67) and eliminating the amplitudes leads to the dispersion relation (DR).

$${\left(SB-{\alpha }_2{\tau }^2-{\alpha }_3{\rho }^2\right)}^2-{(A\tau -\rho )}^2-S^2=0.$$

(69)

The above DR can also be written explicitly as a quartic polynomial in $\tau$, as follows:

$${\alpha }_2^2{\tau }^4+\left(-2{\alpha }_{2}C-A^2\right){\tau }^2+2A\rho \tau +\left(C^2-{\rho }^2-S^2\right)=0,$$

(70)

where $C=SB-{\alpha }_3{\rho }^2$. For fixed system parameters, this quartic is solved numerically for each real $\rho$; modulational instability occurs when at least one solution $\tau \left(\rho \right)$ has $Im\left(\tau \right)>0$. The instability growth rate plotted in Figure 6 is $g\left(\rho \right)={max}_{\mathrm{roots}\tau }Im\left(\tau \right)$.

7. Results Analysis and Interpretation from a Physical Perspective

Since the hypothetical results (4) and (5) derived using the improved technique deviate from those obtained by existing methods, the solutions produced through this approach are fundamentally distinct from those reported by other researchers employing conventional schemes. In particular, Equation (5) provides specific parameter conditions that characterize various families of solutions. For instance, the authors in [34] applied the extended fan sub-equation scheme to derive dark, bright, and optical soliton solutions of the dynamical model (6), while another work [30] utilized the F-expansion scheme to solve this equation, obtaining optical soliton results. In contrast, the results obtained via the improved mERFM approach represent novel solutions that have not been previously reported.

Graphical representations serve as an effective means to analyze and visualize solutions, providing a clear illustration of both quantitative and qualitative data for easier comparison. To enhance the explanation and application of the model, we present structural solutions through two- and three-dimensional graphs, accompanied by suitable parameters to offer deeper insights into these solutions. The structures in Figure 1 demonstrate that a localized and stable soliton wave is represented by Solution (12); the multi-peak and breather patterns in Figure 1a–c indicate periodic compression and expansion of the wave energy during propagation, while the 2D views in Figure 1d–f confirm that the soliton remains localized without spreading. This illustrates a crucial aspect of soliton dynamics: the solution’s ability to hold its shape and energy balance over time.

Figure 2 shows that dark and dark–bright periodic waves are represented by Solution (14). While the combined dark–bright behavior exhibits periodic energy redistribution between components, the dark soliton profile in Figure 2a–c manifests as a localized intensity dip on a continuous background. These recurring oscillations and the wave’s stable localization are verified by the 2D plots in Figure 2d–f. This shows that the solution keeps nonlinearity and dispersion in check, enabling the wave to travel without altering its general structure.

The periodic multi-peak and breather-type soliton behavior of Solution (19) is demonstrated in Figure 3. Repeated peaks with different amplitudes can be seen in the wave profiles in Figure 3a–c, suggesting energy localization that progressively becomes stronger and weaker as it propagates. This illustrates how nonlinearity and dispersion interact to create oscillatory (breathing) dynamics. The multi-peak structure and periodic oscillations are confirmed by the corresponding 2D views in Figure 3d–f, which show that the soliton retains its overall shape while its amplitude changes rhythmically.

Figure 4 shows that singular breather-type wave patterns and periodic singular dark solitary waves are produced by Solution (25). The profiles in Figure 4a–c demonstrate stable dark wave propagation with periodic recurrence of localized dips on a non-zero background. The balance between dispersion and nonlinearity causes the breather-like modulation to reflect an oscillatory energy exchange during propagation. These localized and periodic structures are confirmed by the 2D views in Figure 4d–f, which demonstrate that the wave retains its shape despite varying in amplitude rhythmically.

The different wave structures produced by Solution (33) are depicted in Figure 5. The profiles in Figure 5a–c show the evolution of dark soliton and W-type solitary wave patterns. The dark soliton appears as a localized intensity dip on a continuous background, whereas the W-type solitary wave exhibits an oscillatory double-peak structure arising from higher-order nonlinear effects. These waveforms result from the balance between dispersion and nonlinearity, enabling stable and periodic propagation. The corresponding 2D views in Figure 5d–f further confirm these localized and oscillatory characteristics, demonstrating that the soliton maintains its overall shape during evolution despite periodic amplitude modulation.

Figure 6 illustrates the dispersion relation $\tau =\tau \left(\rho \right)$, which describes how the wave number varies with frequency in the context of MI. The nonlinear curvature of the curve indicates regions where dispersion and nonlinearity strongly interact. In these regions, small perturbations can grow, leading to instability and the possible formation of localized wave structures such as solitons or breathers.

8. Concluding Remarks

In this study, we efficaciously applied the mERFM to the generalized NLSE with second-order spatiotemporal dispersion. This approach produced a wide range of exact optical soliton solutions, including bright and dark solitons, as well as solutions expressed through trigonometric, hyperbolic, rational, exponential, and singular functions. Several of these solutions are novel, as confirmed through comparisons with the existing literature. By assigning appropriate parameter values, we uncovered various wave structures such as kink and anti-kink solitons, multi-peak solitons, and breather-type waves with unique profiles, as illustrated in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6. These solutions provide meaningful insight into the complex dynamics of nonlinear wave proliferation in optical fibers and oceanic environments. Furthermore, we derived conservation laws related to the model and calculated conserved quantities such as power, momentum, and energy associated with the soliton solutions. The stability of the model was also analyzed using MI, confirming the robustness and physical relevance of the obtained solutions. The demonstrated effectiveness and versatility of mERFM highlight its potential for broader application in solving higher-order NEEs across multiple disciplines in physics and engineering. Future research will focus on extending this methodology to coupled and fractional NLSEs using analytical, semi-analytical, and numerical techniques. These advancements will contribute to a deeper understanding of nonlinear wave phenomena and their applications in various scientific and technological domains.