Generalized Modified Unstable Nonlinear Schrödinger’s Equation: Optical Solitons and Modulation Instability

Abstract

This paper proposes the generalized modified unstable nonlinear Schrödinger’s equation with applications in modulated wavetrain instabilities. The extended direct algebra and generalized Ricatti equation methods are applied to find innovative soliton solutions to the equation. The solutions are obtained in the form of elliptic, hyperbolic, and trigonometric functions. Moreover, a Galilean transformation is used to convert the problem into a dynamical system. We use the theory of planar dynamical systems to derive the equilibrium points of the dynamical system and analyze the Hamiltonian polynomial. We further investigate the bifurcation phase portrait of the system and study its chaotic behaviors when an external force is applied to the system. Graphical 2D and 3D plots are explored to support our mathematical analysis. A sensitivity analysis confirms that the variation in initial conditions has no substantial effect on the stability of the solutions. Furthermore, we give the modulation instability gain spectrum of the considered model and graphically indicate its dynamics using 2D plots. The reported results demonstrate not only the dynamics of the analyzed equation but are also conceptually relevant in establishing the temporal development of modest disturbances in stable or unstable media. These disturbances will be critical for anticipating, planning treatments, and creating novel mechanisms for modulated wavetrain instabilities.

1. Introduction

Accurate and convincing solutions to nonlinear partial differential equations (PDEs) are vital in several engineering and scientific domains [1]. These solutions are crucial tools for acquiring basic information and establishing practical applications in disciplines such as fluid dynamics, plasma physics, and optics [2]. The nonlinear Schrödinger’s equation (NLSE) and its variants are widely used in physics subfields, including Langmuir waves in a hot plasma, fiber optics, oceanic rogue waves, telecommunications, optical rogue waves, photonics, deep surface water waves, and laser technology [3,4,5,6,7,8]. They simulate light motion in optical fibers, ensuring reliable data transmission and internet connectivity. In addition, the NLSE defines how light interacts with materials, directing the evolution of optical devices, including lasers [9,10,11]. Furthermore, the NLSE represents the wave function of Bose–Einstein condensates, a state of matter at near-zero temperatures, facilitating research on superfluidity and quantum technology. The NLSE in classical field theory reads as

$$i{u}_t=-{\displaystyle \frac{1}{2}}{u}_{xx}+\kappa {\left|u\right|}^{2}u=0,$$

(1)

where u is a complex field, ${\kappa \left|u\right|}^{2}u$ is a nonlinear term describing the self-interaction [12] with a real constant $\kappa$, and $i=\sqrt{-1}$. Although the unstable nonlinear Schrödinger’s equation (UNLSE) may seem abstract, its practical applications are diverse and impactful. Important concepts such as the analysis of instabilities in two-stream plasma and the baroclinic two-layer can be modeled using the UNLSE; see [13,14]. The UNLSE is

$$i{u}_t+u_{xx}+2\tau {\left|u\right|}^{2}u-2\mu u=0,$$

(2)

where $\tau$ and $\mu$ are real constants [15]. The modified version of UNLSE that determines the instability of modulated wavetrains is as follows:

$$i{u}_t+u_{xx}+2\tau {\left|u\right|}^{2}u-\mu u_{xt}=0.$$

(3)

Various approaches have been developed to extract exact and numerical solutions [16] for different classes of PDEs, including the KdV-type equations, the innovative Biswas–Milovic equations, and the nonlinear Schrödinger’s equation (NLSE). The analytical solutions of these equations are important in understanding the underlying physics behind the UNLSE and its modification. Moreover, analytical solutions to nonlinear PDEs are essential for comprehending the qualitative features and physical interpretation of various phenomena.

Specifically for the UNLSE (2), the equation has been solved analytically using the improved simple equation method [17], the modified Kudryashov method [18], the extended auxiliary equation method [19], the extended direct algebraic method (EDAM) [20], the modified rational function method [21], the extended mapping method [22], the modified mathematical method [23], the improved trial equation method [24], the variational principle method [25], the Jacobi elliptic rational function method, and the classical version of the exponential rational function method [26]. In addition, the generalized version of Equation (2) has been presented and solved using the modified Sardar sub-equation and q-homotopy analysis transform method [27], the Lie symmetry method and its corresponding phase portrait (PP) analysis [28], and the ($G^{\prime }/G^2$) and the polynomial expansion methods [29]. Furthermore, the existence and uniqueness of the $\beta$-time-fractional UNLSE have been proved and solved using the sine-Gordon expansion method [30], the extended Kudryashov method and the Hamiltonian function theorem [31], and the ansatz transformation [32].

Moreover, for the modified UNLSE (3), the equation has also been analytically solved via various techniques, such as the extended simple equation method [17], the modified Kudryashov method [18], some groups of classical ansatz methods [33], the improved modified Sardar sub-equation method [34], the modified extended mapping method and its modulation instability [35], and the auxiliary equation method [36]. The modified UNLSE with space-fractional derivatives has been studied using the extended tangent hyperbolic function method [37]. Studies on the modified UNLSE are very scarce. Also, motivated by the above discussions and the vast number of applications of UNLSE in wave instabilities, two-stream plasma, modulated wavetrains, and the baroclinic two-layer, this work will study a generalized modified UNLSE. The generalized modified UNLSE is given by

$$i{u}_t+u_{xx}+g{\left|u\right|}^{2}u+b{u}_{xt}=0,$$

(4)

where g and b are real constants. It is worth noticing that when $g=2\tau$ and $b=-\mu$, the generalized modified UNLSE (4) reduces to the classical modified UNLSE (3). We are going to apply the EDAM and the generalized Riccati equation method (GREM) [38] to propose abundant solutions to this generalized equation. We will also study this equation using PP analysis and bifurcation analysis with the help of the Galilean transformation. Lastly, this study will provide the modulation instability analysis of the proposed equation.

The description of the methodologies and their applications to the generalized modified UNLSE is presented in Section 2. Section 3 contains a graphical illustration of the reported solutions and their comparison with existing studies, bifurcation analysis, chaotic properties, and modulation instability of the considered model. Finally, Section 4 concludes the study.

2. The Mathematical Analysis

2.1. The Main Procedure of the Methods

The main algorithms of the extended direct algebra method (EDAM) and the generalized Riccati equation method (GREM) are described in this section.

2.1.1. The EDAM

This section briefly describes the major steps of the EDAM to discover wave solutions of a nonlinear PDE as follows.

1.

Given a nonlinear PDE

$$P^{\mathit{PDE}}(u,u_x,u_t,u_{xx},u_{tt},\dots )=0,$$

(5)

where $u=u(x,t)$ is an unknown function, $u_{(.)}$ represents partial derivatives of u, and P is a polynomial in $u(x,t)$ and its partial derivatives.

2.

We deduce the equivalent ordinary differential equation for Equation (5) using

$$u(x,t)=U\left(\zeta \right),\zeta ={\lambda }_{1}x-{\lambda }_{2}t,$$

(6)

where ${\lambda }_1$ and ${\lambda }_2$ are constants. Substituting Equation (6) into Equation (5), we get

$$P^{\mathit{ODE}}\left(U\left(\zeta \right),U^{\prime }\left(\zeta \right),U^{\prime \prime }\left(\zeta \right),\dots \right)=0.$$

(7)

3.

The EDAM assumes the solution of Equation (7) to be

$$U\left(\zeta \right)=\sum _{j=0}^N{b}_j{B}^j\left(\zeta \right),$$

(8)

where N is a positive integer to be obtained by balancing the nonlinear term with the highest derivative term in Equation (7), and $b_j$ for $j=0,1,2,\dots ,N$ are constants to be determined with $b_N\ne 0$. The function $B\left(\zeta \right)$ satisfies the auxiliary equation

$$B^{\prime }\left(\zeta \right)=ln\left(A\right)\left(m_1+m_{2}B\left(\zeta \right)+m_{3}B{\left(\zeta \right)}^2\right),A>0,\mathrm{and}A\ne 1,$$

(9)

where $m_1$, $m_2$, and $m_3$ are unknown constants.

4.

Substituting Equation (8) into Equation (7) and using the different solutions $B\left(\zeta \right)$ of Equation (9) that correspond to various values of $m_1$, $m_2$, and $m_3$, we get various solutions for the nonlinear PDE in Equation (5). About thirty-seven solutions $B\left(\zeta \right)$, satisfying the auxiliary Equation (9) and corresponding to different values of $m_1$, $m_2$, and $m_3$, were reported in [39]. Replacing Equation (8) into Equation (7) along with Equation (9) and equating the coefficients of $B^j\left(\zeta \right)$ to zero, we obtain a system of algebraic equations that can be solved using Maple for the unknown constants $b_j$ ($j=0,1,2,\dots ,N$), $m_1$, $m_2$, and $m_3$. Finally, we use the unknown values to obtain various solutions for Equation (7) and, by extension, Equation (5).

2.1.2. The GREM

Steps 1 and 2 of the GREM are the same as those in the EDAM.

3.

The GREM assumes the solution of Equation (7) to be:

$$U\left(\zeta \right)=\sum _{j=0}^N{b}_j{Q}^j\left(\zeta \right),$$

(10)

where N is calculated in the same way as for the EDAM and $b_N\ne 0$. The function $Q\left(\zeta \right)$ satisfies

$$Q^{\prime }\left(\zeta \right)=d_0+d_{1}Q\left(\zeta \right)+d_2{Q}^2\left(\zeta \right),$$

(11)

where $d_0$, $d_1$, and $d_2$ are unknown constants. Twenty-five nontrivial solutions for Equation (11) corresponding to distinct values of $d_0$, $d_1$, and $d_2$ are reported in [27].

4.

Substituting Equation (10) into Equation (7), using Equation (11) and equating the coefficients of $Q^j\left(\zeta \right)$ to zero, we obtain a system of nonlinear algebraic equations that may be solved using Maple to obtain the unknown constants $b_j,d_0,d_1,$ and $d_2$. Substituting the unknown constants back into the solution form in Equation (10), we find solutions to the nonlinear PDE (5).

2.2. The Application of the Methods

In this section, the application of the EDAM and the GREM for extracting exact traveling wave solutions of the generalized modified UNLSE

$$i{u}_t+u_{xx}+g{\left|u\right|}^{2}u+b{u}_{xt}=0,$$

(12)

will be illustrated. First, we construct an equivalent ordinary differential equation for Equation (12) using the following wave variables:

$$u(x,t)=U\left(\xi \right)e^{i\theta },\xi ={\lambda }_{1}x-{\lambda }_{2}t,\mathrm{and}\theta =dx+\nu t,$$

(13)

where ${\lambda }_1$, ${\lambda }_2$, d, and $\nu$ are constants. Applying Equation (13) to Equation (12), we obtain

$$-i{\lambda }_2{U}^{\prime }-\nu U+{\lambda }_1^2{U}^{\prime \prime }+2id{\lambda }_1{U}^{\prime }-d^{2}U+g{U}^3-b{\lambda }_1{\lambda }_2{U}^{\prime \prime }+ib\nu {\lambda }_1{U}^{\prime }-ibd{\lambda }_2{U}^{\prime }-bd\nu U=0.$$

(14)

We continue by splitting the imaginary and real components of Equation (14). Its real part is expressed as

$$U^{″}-{\displaystyle \frac{(bd\nu +\nu +d^2)}{({\lambda }_1^2-b{\lambda }_1{\lambda }_2)}}U+{\displaystyle \frac{g}{({\lambda }_1^2-b{\lambda }_1{\lambda }_2)}}{U}^3=0,$$

(15)

and the imaginary part is written as

$$(-{\lambda }_2+2d{\lambda }_1+b\nu {\lambda }_1-bd{\lambda }_2)U^{\prime }=0.$$

(16)

From Equation (16), we get

$$-{\lambda }_2+2d{\lambda }_1+b\nu {\lambda }_1-bd{\lambda }_2=0,$$

which implies

$${\lambda }_2={\displaystyle \frac{{\lambda }_1(b\nu +2d)}{1+bd}}.$$

(17)

2.2.1. The Application of the EDAM

This part will use the highlighted EDAM to establish numerous solutions for the modified generalized UNLSE in Equation (12). Balancing $U^{″}$ and $U^3$ in Equation (15) using the homogeneous balancing method, we get $N+2=3N$, which gives $N=1$. Thus, the EDAM solution of Equation (15) takes the form

$$U\left(\xi \right)=b_0+b_{1}B\left(\xi \right).$$

(18)

Substituting Equation (18) into Equation (15) and setting the coefficients of $B^0\left(\xi \right)$, $B^1\left(\xi \right)$, $B^2\left(\xi \right)$, and $B^3\left(\xi \right)$ to zero, we then have the following nonlinear equations:

$$\begin{array}{cc}\hfill B^0\left(\xi \right):& {\displaystyle \frac{{ln}^2\left(A\right)b{\lambda }_2{\lambda }_1{b}_1{m}_1{m}_2-{ln}^2\left(A\right){\lambda }_1^2{b}_1{m}_1{m}_2+bd\nu b_0-g{b}_0^3+d^2{b}_0+\nu b_0}{{\lambda }_1(b{\lambda }_2-{\lambda }_1)}}=0,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill B^1\left(\xi \right):& {\displaystyle \frac{2{\left(ln\left(A\right)\right)}^{2}b{b}_1{\lambda }_1{\lambda }_2{m}_1{m}_3+{\left(ln\left(A\right)\right)}^{2}b{b}_1{\lambda }_1{\lambda }_2{m}_2^2-2{\left(ln\left(A\right)\right)}^2{b}_1{\lambda }_1^2{m}_1{m}_3}{{\lambda }_1\left(b{\lambda }_2-{\lambda }_1\right)}}\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & +{\displaystyle \frac{-{\left(ln\left(A\right)\right)}^2{b}_1{\lambda }_1^2{m}_2^2+bd\nu b_1-3g{b}_0^2{b}_1+d^2{b}_1+\nu b_1}{{\lambda }_1\left(b{\lambda }_2-{\lambda }_1\right)}}=0,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill B^2\left(\xi \right):& {\displaystyle \frac{3{\left(ln\left(A\right)\right)}^{2}b{b}_1{\lambda }_1{\lambda }_2{m}_2{m}_3-3{\left(ln\left(A\right)\right)}^2{b}_1{\lambda }_1^2{m}_2{m}_3-3g{b}_0{b}_1^2}{{\lambda }_1\left(b{\lambda }_2-{\lambda }_1\right)}}=0,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill B^3\left(\xi \right):& {\displaystyle \frac{2{\left(ln\left(A\right)\right)}^{2}b{b}_1{\lambda }_1{\lambda }_2{m}_3^2-2{\left(ln\left(A\right)\right)}^2{b}_1{\lambda }_1^2{m}_3^2-g{b}_1^3}{{\lambda }_1\left(b{\lambda }_2-{\lambda }_1\right)}}=0.\hfill \end{array}$$

(23)

Solving the resulting system using Maple, we obtain

$$\begin{array}{cc}\hfill & \nu ={\displaystyle \frac{2{ln}^2\left(A\right)b{\lambda }_2{\lambda }_1{b}_0{m}_3^2-2{ln}^2\left(A\right)b{\lambda }_2{\lambda }_1{m}_1{m}_3{b}_1^2-2{ln}^2\left(A\right)b{\lambda }_1^2{b}_0^2{m}_3^2+2{ln}^2\left(A\right){\lambda }_1^2{m}_1{m}_3{b}_1^2-d^2{b}_1^2}{b_1^2(bd+1)}},\hfill \\ & g={\displaystyle \frac{2{\lambda }_1{ln}^2\left(A\right)m_3^2(b{\lambda }_2-{\lambda }_1)}{b_1^2}},b_0=b_0,b_1=b_1,m_1=m_1,m_2={\displaystyle \frac{2b_0{m}_3}{b_1}},\mathrm{and}{m}_3=m_3.\hfill \end{array}$$

(24)

We substitute Equation (24) into Equation (18) by incorporating various solutions of Equation (9) to derive different solutions of Equation (15) based on the following cases.

• Case I: For $m_1{m}_3-{\displaystyle \frac{b_0^2{m}_3^2}{b_1^2}}>0$ and $m_3\ne 0$, we have

  $$U_1\left(\xi \right)=\left(-{\displaystyle \frac{b_0}{b_1}}-{\displaystyle \frac{\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}{tan}_A\left(\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)}{m_3}}\right)b_1+b_0,$$

  $$U_2\left(\xi \right)=\left(-{\displaystyle \frac{b_0}{b_1}}+{\displaystyle \frac{\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}{cot}_A\left(\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)}{m_3}}\right)b_1+b_0,$$

  $$U_3\left(\xi \right)=\left(-{\displaystyle \frac{b_0}{b_1}}+{\displaystyle \frac{\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\left({tan}_A\left(2\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\pm \sqrt{pq}{sec}_A\left(2\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\right)}{m_3}}\right)b_1+b_0,$$

  $$U_4\left(\xi \right)=\left(-{\displaystyle \frac{b_0}{b_1}}-{\displaystyle \frac{\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\left({cot}_A\left(2\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\pm \sqrt{pq}{csc}_A\left(2\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\right)}{m_3}}\right)b_1+b_0,$$

  $$U_5\left(\xi \right)=\left(-{\displaystyle \frac{b_0}{b_1}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\left({tan}_A\left(\frac{1}{2}\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)-{cot}_A\left(\frac{1}{2}\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\right)}{m_3}}\right)b_1+b_0.$$
• Case II: For $m_1{m}_3-{\displaystyle \frac{b_0^2{m}_3^2}{b_1^2}}<0$ and $m_3\ne 0$, we have

  $$U_6\left(\xi \right)=\left(-{\displaystyle \frac{b_0}{b_1}}-{\displaystyle \frac{\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}{tanh}_A\left(\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)}{m_3}}\right)b_1+b_0,$$

  $$U_7\left(\xi \right)=\left(-{\displaystyle \frac{b_0}{b_1}}-{\displaystyle \frac{\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}{coth}_A\left(\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)}{m_3}}\right)b_1+b_0,$$

  $$U_8\left(\xi \right)=\left(-{\displaystyle \frac{b_0}{b_1}}-{\displaystyle \frac{\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\left({tanh}_A\left(2\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\pm i\sqrt{pq}{sech}_A\left(2\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\right)}{m_3}}\right)b_1+b_0,$$

  $$U_9\left(\xi \right)=\left(-{\displaystyle \frac{b_0}{b_1}}-{\displaystyle \frac{\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\left({coth}_A\left(2\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\pm \sqrt{pq}{csch}_A\left(2\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\right)}{m_3}}\right)b_1+b_0,$$

  $$U_{10}\left(\xi \right)=\left(-{\displaystyle \frac{b_0}{b_1}}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\left({tanh}_A\left(\frac{1}{2}\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)+{coth}_A\left(\frac{1}{2}\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\right)}{m_3}}\right)b_1+b_0.$$
• Case III: If $m_1{m}_3>0$ and $m_2=0$ in Equation (9), the algebraic system gives the following values:

  $$\begin{array}{cc}\hfill & b_0=0,b_1=b_1,g={\displaystyle \frac{2{ln}^2\left(A\right)m_3^2(b{\lambda }_1-{\lambda }_2)}{b_1^2}},\hfill \\ & m_1={\displaystyle \frac{1}{2}}{\displaystyle \frac{bd\nu +d^2+\nu }{{\lambda }_2{ln}^2\left(A\right)m_3(b{\lambda }_1-{\lambda }_2)}},m_2=0,\mathrm{and}{m}_3=m_3.\hfill \end{array}$$

  (25)

Substituting Equation (25) into Equation (18), we have the following solutions for $\overline{\Omega }:={\displaystyle \frac{bd\nu +d^2+\nu }{{\lambda }_2{ln}^2\left(A\right)m_3^2(b{\lambda }_1-{\lambda }_2)}}<0:$

$$U_{11}\left(\xi \right)={\displaystyle \frac{1}{2}}\sqrt{-2\overline{\Omega }}{tan}_A\left({\displaystyle \frac{1}{2}}\sqrt{-2\overline{\Omega }}\xi \right)b_1,$$

$$U_{12}\left(\xi \right)=-{\displaystyle \frac{1}{2}}\sqrt{-2\overline{\Omega }}{cot}_A\left({\displaystyle \frac{1}{2}}\sqrt{-2\overline{\Omega }}\xi \right)b_1,$$

$$U_{13}\left(\xi \right)={\displaystyle \frac{1}{2}}\sqrt{-2\overline{\Omega }}\left({tan}_A\left(\sqrt{-2\overline{\Omega }}\xi \right)\pm \sqrt{pq}{sec}_A\left(\sqrt{-2\overline{\Omega }}\xi \right)\right)b_1,$$

$$U_{14}\left(\xi \right)=-{\displaystyle \frac{1}{2}}\sqrt{-2\overline{\Omega }}\left({cot}_A\left(\sqrt{-2\overline{\Omega }}\xi \right)\pm \sqrt{pq}{csc}_A\left(\sqrt{-2\overline{\Omega }}\xi \right)\right)b_1,$$

$$U_{15}\left(\xi \right)={\displaystyle \frac{1}{4}}\sqrt{-2\overline{\Omega }}\left({tan}_A\left({\displaystyle \frac{1}{4}}\sqrt{-2\overline{\Omega }}\xi \right)-{cot}_A\left({\displaystyle \frac{1}{4}}\sqrt{-2\overline{\Omega }}\xi \right)\right)b_1.$$

• Case IV: We have the following solutions for $\overline{\Omega }>0:$

  $$U_{16}\left(\xi \right)=-{\displaystyle \frac{1}{2}}\sqrt{2\overline{\Omega }}{tanh}_A\left({\displaystyle \frac{1}{2}}\sqrt{2\overline{\Omega }}\xi \right)b_1,$$

  $$U_{17}\left(\xi \right)=-{\displaystyle \frac{1}{2}}\sqrt{2\overline{\Omega }}{coth}_A\left({\displaystyle \frac{1}{2}}\sqrt{2\overline{\Omega }}\xi \right)b_1,$$

  $$U_{18}\left(\xi \right)=-{\displaystyle \frac{1}{2}}\sqrt{2\overline{\Omega }}\left({tanh}_A\left(\sqrt{2\overline{\Omega }}\xi \right)\pm i\sqrt{pq}{\sec \mathrm{h}}_A\left(\sqrt{2\overline{\Omega }}\xi \right)\right)b_1,$$

  $$U_{19}\left(\xi \right)=-{\displaystyle \frac{1}{2}}\sqrt{2\overline{\Omega }}\left({coth}_A\left(\sqrt{2\overline{\Omega }}\xi \right)\pm \sqrt{pq}{\csc \mathrm{h}}_A\left(\sqrt{2\overline{\Omega }}\xi \right)\right)b_1,$$

  $$U_{20}\left(\xi \right)=-{\displaystyle \frac{1}{4}}\sqrt{2\overline{\Omega }}\left({tanh}_A\left({\displaystyle \frac{1}{4}}\sqrt{2\overline{\Omega }}\xi \right)+{coth}_A\left({\displaystyle \frac{1}{4}}\sqrt{2\overline{\Omega }}\xi \right)\right)b_1.$$
• Case V: If $m_2=0$ and $m_1=m_3$ in Equation (9), the algebraic system gives the following solutions:

  $$\begin{array}{cc}\hfill & g=-{\displaystyle \frac{bd\nu +d^2+\nu }{b_1^2}},b_0=0,b_1=b_1,\mathrm{and}{m}_1=\pm {\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_2-2{\lambda }_2^2}}}{ln\left(A\right)}}.\hfill \end{array}$$

  (26)

Substituting Equation (26) into Equation (18), we have the following solutions for ${\displaystyle \frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}<0:$

$$U_{21}\left(\xi \right)={tan}_A\left({\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)b_1,$$

$$U_{22}\left(\xi \right)=-{cot}_A\left({\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)b_1,$$

$$U_{23}\left(\xi \right)=\left({tan}_A\left(2{\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\pm \sqrt{pq}{sec}_A\left(2{\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\right)b_1,$$

$$U_{24}\left(\xi \right)=\left(-{cot}_A\left(2{\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\pm \sqrt{pq}{csc}_A\left(2{\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\right)b_1,$$

$$U_{25}\left(\xi \right)=\left({\displaystyle \frac{1}{2}}{tan}_A\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)-{\displaystyle \frac{1}{2}}{cot}_A\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\right)b_1.$$

• Case VI: If $m_2=0$ and $m_1=-m_3$ in Equation (9), the algebraic system provides the following solutions:

  $$\begin{array}{cc}\hfill & m_1=\pm {\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_2-2{\lambda }_2^2}}}{ln\left(A\right)}},g={\displaystyle \frac{bd\nu +d^2+\nu }{b_1^2}},b_0=0,\mathrm{and}{b}_1=b_1.\hfill \end{array}$$

  (27)

Substituting Equation (26) into Equation (18), we have the following solutions for ${\displaystyle \frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}>0:$

$$U_{26}\left(\xi \right)=-{tanh}_A\left(-{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)b_1,$$

$$U_{27}\left(\xi \right)=-{coth}_A\left(-{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)b_1,$$

$$U_{28}\left(\xi \right)=\left(-{tanh}_A\left(-2{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\pm i\sqrt{pq}{sech}_A\left(-2{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\right)b_1,$$

$$U_{29}\left(\xi \right)=\left(-{coth}_A\left(-2{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\pm \sqrt{pq}{csch}_A\left(-2{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\right)b_1,$$

$$U_{30}\left(\xi \right)=\left(-{\displaystyle \frac{1}{2}}{tanh}_A\left(-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)-{\displaystyle \frac{1}{2}}{coth}_A\left(-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\right)b_1.$$

From Equation (13) and the obtained solutions $U_1\left(\xi \right),U_2\left(\xi \right),\dots ,U_{30}\left(\xi \right)$ as shown above, we get the following solutions to the generalized modified UNLSE (12) using the EDAM based on the above cases:

$$u_1(x,t)=\left(\left(-{\displaystyle \frac{b_0}{b_1}}-{\displaystyle \frac{\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}{tan}_A\left(\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)}{m_3}}\right)b_1+b_0\right)e^{i\theta },$$

$$u_2(x,t)=\left(\left(-{\displaystyle \frac{b_0}{b_1}}+{\displaystyle \frac{\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}{cot}_A\left(\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)}{m_3}}\right)b_1+b_0\right)e^{i\theta },$$

$$u_3(x,t)=\left(\left(-{\displaystyle \frac{b_0}{b_1}}+{\displaystyle \frac{\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\left({tan}_A\left(2\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\pm \sqrt{pq}{sec}_A\left(2\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\right)}{m_3}}\right)b_1+b_0\right)e^{i\theta },$$

$$u_4(x,t)=\left(\left(-{\displaystyle \frac{b_0}{b_1}}-{\displaystyle \frac{\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\left({cot}_A\left(2\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\pm \sqrt{pq}{csc}_A\left(2\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\right)}{m_3}}\right)b_1+b_0\right)e^{i\theta },$$

$$u_5(x,t)=\left(\left(-{\displaystyle \frac{b_0}{b_1}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\left({tan}_A\left(\frac{1}{2}\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)-{cot}_A\left(\frac{1}{2}\sqrt{m_1{m}_3-\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\right)}{m_3}}\right)b_1+b_0\right)e^{i\theta },$$

$$u_6(x,t)=\left(\left(-{\displaystyle \frac{b_0}{b_1}}-{\displaystyle \frac{\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}{tanh}_A\left(\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)}{m_3}}\right)b_1+b_0\right)e^{i\theta },$$

$$u_7(x,t)=\left(\left(-{\displaystyle \frac{b_0}{b_1}}-{\displaystyle \frac{\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}{coth}_A\left(\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)}{m_3}}\right)b_1+b_0\right)e^{i\theta },$$

$$u_8(x,t)=\left(\left(-{\displaystyle \frac{b_0}{b_1}}-{\displaystyle \frac{\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\left({tanh}_A\left(2\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\pm i\sqrt{pq}{sech}_A\left(2\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\right)}{m_3}}\right)b_1+b_0\right)e^{i\theta },$$

$$u_9(x,t)=\left(\left(-{\displaystyle \frac{b_0}{b_1}}-{\displaystyle \frac{\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\left({coth}_A\left(2\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\pm \sqrt{pq}{csch}_A\left(2\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\right)}{m_3}}\right)b_1+b_0\right)e^{i\theta },$$

$$u_{10}(x,t)=\left(\left(-{\displaystyle \frac{b_0}{b_1}}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\left({tanh}_A\left(\frac{1}{2}\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)+{coth}_A\left(\frac{1}{2}\sqrt{-m_1{m}_3+\frac{b_0^2{m}_3^2}{b_1^2}}\xi \right)\right)}{m_3}}\right)b_1+b_0\right)e^{i\theta },$$

$$u_{11}(x,t)=\left({\displaystyle \frac{1}{2}}\sqrt{-2\overline{\Omega }}{tan}_A\left({\displaystyle \frac{1}{2}}\sqrt{-2\overline{\Omega }}\xi \right)b_1\right)e^{i\theta },$$

$$u_{12}(x,t)=\left(-{\displaystyle \frac{1}{2}}\sqrt{-2\overline{\Omega }}{cot}_A\left({\displaystyle \frac{1}{2}}\sqrt{-2\overline{\Omega }}\xi \right)b_1\right)e^{i\theta },$$

$$u_{13}(x,t)=\left({\displaystyle \frac{1}{2}}\sqrt{-2\overline{\Omega }}\left({tan}_A\left(\sqrt{-2\overline{\Omega }}\xi \right)\pm \sqrt{pq}{sec}_A\left(\sqrt{-2\overline{\Omega }}\xi \right)\right)b_1\right)e^{i\theta },$$

$$u_{14}(x,t)=\left(-{\displaystyle \frac{1}{2}}\sqrt{-2\overline{\Omega }}\left({cot}_A\left(\sqrt{-2\overline{\Omega }}\xi \right)\pm \sqrt{pq}{csc}_A\left(\sqrt{-2\overline{\Omega }}\xi \right)\right)b_1\right)e^{i\theta },$$

$$u_{15}(x,t)=\left({\displaystyle \frac{1}{4}}\sqrt{-2\overline{\Omega }}\left({tan}_A\left({\displaystyle \frac{1}{4}}\sqrt{-2\overline{\Omega }}\xi \right)-{cot}_A\left({\displaystyle \frac{1}{4}}\sqrt{-2\overline{\Omega }}\xi \right)\right)b_1\right)e^{i\theta },$$

$$u_{16}(x,t)=\left(-{\displaystyle \frac{1}{2}}\sqrt{2\overline{\Omega }}{tanh}_A\left({\displaystyle \frac{1}{2}}\sqrt{2\overline{\Omega }}\xi \right)b_1\right)e^{i\theta },$$

$$u_{17}(x,t)=\left(-{\displaystyle \frac{1}{2}}\sqrt{2\overline{\Omega }}{coth}_A\left({\displaystyle \frac{1}{2}}\sqrt{2\overline{\Omega }}\xi \right)b_1\right)e^{i\theta },$$

$$u_{18}(x,t)=\left(-{\displaystyle \frac{1}{2}}\sqrt{2\overline{\Omega }}\left({tanh}_A\left(\sqrt{2\overline{\Omega }}\xi \right)\pm i\sqrt{pq}{sech}_A\left(\sqrt{2\overline{\Omega }}\xi \right)\right)b_1\right)e^{i\theta },$$

$$u_{19}(x,t)=\left(-{\displaystyle \frac{1}{2}}\sqrt{2\overline{\Omega }}\left({coth}_A\left(\sqrt{2\overline{\Omega }}\xi \right)\pm \sqrt{pq}{csch}_A\left(\sqrt{2\overline{\Omega }}\xi \right)\right)b_1\right)e^{i\theta },$$

$$u_{20}(x,t)=\left(-{\displaystyle \frac{1}{4}}\sqrt{2\overline{\Omega }}\left({tanh}_A\left({\displaystyle \frac{1}{4}}\sqrt{2\overline{\Omega }}\xi \right)+{coth}_A\left({\displaystyle \frac{1}{4}}\sqrt{2\overline{\Omega }}\xi \right)\right)b_1\right)e^{i\theta },$$

$$u_{21}(x,t)=\left({tan}_A\left({\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)b_1\right)e^{i\theta },$$

$$u_{22}(x,t)=\left(-{cot}_A\left({\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)b_1\right)e^{i\theta },$$

$$u_{23}(x,t)=\left(\left({tan}_A\left(2{\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\pm \sqrt{pq}{sec}_A\left(2{\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\right)b_1\right)e^{i\theta },$$

$$u_{24}(x,t)=\left(\left(-{cot}_A\left(2{\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\pm \sqrt{pq}{csc}_A\left(2{\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\right)b_1\right)e^{i\theta },$$

$$u_{25}(x,t)=\left(\left({\displaystyle \frac{1}{2}}tanA\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)-{\displaystyle \frac{1}{2}}{cot}_A\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{-\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\right)b_1\right)e^{i\theta },$$

$$u_{26}(x,t)=\left(-{tanh}_A\left(-{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)b_1\right)e^{i\theta },$$

$$u_{27}(x,t)=\left(-{coth}_A\left(-{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)b_1\right)e^{i\theta },$$

$$u_{28}(x,t)=\left(\left(-{tanh}_A\left(-2{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\pm i\sqrt{pq}{sech}_A\left(-2{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\right)b_1\right)e^{i\theta },$$

$$u_{29}(x,t)=\left(\left(-{coth}_A\left(-2{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\pm \sqrt{pq}{csch}_A\left(-2{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\right)b_1\right)e^{i\theta },$$

$$u_{30}(x,t)=\left(\left(-{\displaystyle \frac{1}{2}}{tanh}_A\left(-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)-{\displaystyle \frac{1}{2}}{coth}_A\left(-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{\frac{bd\nu +d^2+\nu }{2b{\lambda }_1{\lambda }_2-2{\lambda }_2^2}}\xi }{ln\left(A\right)}}\right)\right)b_1\right)e^{i\theta }.$$

Definitions of the generalized trigonometric and hyperbolic functions such as ${tan}_A,$${cot}_A,$${sec}_A,$${csc}_A,$${tanh}_A,$${coth}_A,$${sech}_A,$ and ${csch}_A$ can be found from [39] in which the deformation parameters p and q are greater than zero.

2.2.2. The Application of the GREM

In this section, we will apply the GREM to derive analytical solutions for the generalized modified UNLSE in Equation (12). Using the balance principle, we have $N=1$ for the solution form in Equation (10), which is expressed as

$$U\left(\xi \right)=b_0+b_{1}Q\left(\xi \right).$$

(28)

Now, substituting Equation (28) into Equation (15) and setting all the coefficients of all powers of $Q^j$$(j=0,1,2,3)$ to zero, we obtain the following system of algebraic equations:

$$\begin{array}{cc}\hfill Q^0\left(\xi \right):& {\displaystyle \frac{b{b}_1{d}_0{d}_1{\lambda }_1{\lambda }_2-b_1{d}_0{d}_1{\lambda }_1^2+bd\nu b_0-g{b}_0^3+d^2{b}_0+\nu b_0}{{\lambda }_1\left(b{\lambda }_2-{\lambda }_1\right)}}=0,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill Q\left(\xi \right):& {\displaystyle \frac{2b{b}_1{d}_0{d}_2{\lambda }_1{\lambda }_2+b{b}_1{d}_1^2{\lambda }_1{\lambda }_2-2b_1{d}_0{d}_2{\lambda }_1^2-b_1{d}_1^2{\lambda }_1^2+bd\nu b_1-3g{b}_0^2{b}_1+d^2{b}_1+\nu b_1}{{\lambda }_1\left(b{\lambda }_2-{\lambda }_1\right)}}=0,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill Q^2\left(\xi \right):& {\displaystyle \frac{3b{b}_1{d}_1{d}_2{\lambda }_1{\lambda }_2-3b_1{d}_1{d}_2{\lambda }_1^2-3g{b}_0{b_1}^2}{{\lambda }_1\left(b{\lambda }_2-{\lambda }_1\right)}}=0,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill Q^3\left(\xi \right):& {\displaystyle \frac{2b{b}_1{d}_2^2{\lambda }_1{\lambda }_2-2b_1{d}_2^2{\lambda }_1^2-g{b}_1^3}{{\lambda }_1\left(b{\lambda }_2-{\lambda }_1\right)}}=0.\hfill \end{array}$$

(32)

Using Maple to solve the above system, we obtain

$$g={\displaystyle \frac{2d_2^2{\lambda }_1(b{\lambda }_2-{\lambda }_1)}{b_1^2}},b_0=b_0,b_1=b_1,d_0=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{2{\lambda }_1^2{b}_0^2{d}_2^2-2b{\lambda }_1{\lambda }_2{b}_0^2{d}_2^2+bd{b}_1^2+\nu b_1^2}{{\lambda }_1{b}_1^2{d}_2(b{\lambda }_2-{\lambda }_1)}},d_1={\displaystyle \frac{2d_0{b}_0}{b_1}},\mathrm{and}{d}_2=d_2.$$

Let

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Delta & =-2b{b}_0^2{d}_2^2{\lambda }_1{\lambda }_2+2b_0^2{d}_2^2{\lambda }_1^2+bd\nu b_1^2+d^2{b}_1^2+\nu b_1^2,\hfill \\ \hfill \gamma & ={\lambda }_1{b}_1{d}_2(b{\lambda }_2-{\lambda }_1),\hfill \\ \hfill \psi & ={\displaystyle \frac{2(-2b{\lambda }_1{\lambda }_2{b}_0^2{d}_2^2+2{\lambda }_1^2{b}_0^2{d}_2^2+bd{b}_1^2+d^2{b}_1^2+\nu b_1^2)}{{\lambda }_1{b}_1^2(b{\lambda }_2-{\lambda }_1)}}+{\displaystyle \frac{4d_2^2{b}_0^2}{b_1^2}}.\hfill \end{array}\end{array}$$

Then, the solutions of Equation (15) using the GREM are as follows.

• CASE I: For $d_1^2-4d_0{d}_2>0$ and $d_1{d}_2\ne 0$ (or, $d_0{d}_2\ne 0$), we have the hyperbolic solutions as follows:

  $$\begin{array}{cc}\hfill U_1\left(\xi \right)& =\left(-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{\psi }tanh\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)}{d_2}}-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_2\left(\xi \right)& =\left(-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{\psi }coth\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)}{d_2}}-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_3^{\pm }\left(\xi \right)& =\left(-{\displaystyle \frac{1}{2d_2}}\left(\sqrt{\psi }\left(tanh\left(\sqrt{\psi }(\xi +C)\right)\pm isech\left(\sqrt{\psi }(\xi +C)\right)\right)\right)-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_4\left(\xi \right)& =\left(-{\displaystyle \frac{1}{2d_2}}\left(\sqrt{\psi }\left(coth\left(\sqrt{\psi }(\xi +C)\right)+csch\left(\sqrt{\psi }(\xi +C)\right)\right)\right)-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_5\left(\xi \right)& =\left(-{\displaystyle \frac{1}{4d_2}}\left(\sqrt{\psi }\left(tanh\left({\displaystyle \frac{1}{4}}\sqrt{\psi }(\xi +C)\right)+coth\left({\displaystyle \frac{1}{4}}\sqrt{\psi }(\xi +C)\right)\right)\right)-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_6\left(\xi \right)& ={\displaystyle \frac{\left(\Delta cosh\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\sqrt{\psi }sinh\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)-\frac{2d_2{b}_{0}cosh\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)}{b_1}\right)\right)}}+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_7\left(\xi \right)& ={\displaystyle \frac{\left(\Delta sinh\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\sqrt{\psi }cosh\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)-\frac{2d_2{b}_{0}sinh\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)}{b_1}\right)\right)}}+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_8\left(\xi \right)& ={\displaystyle \frac{\left(\Delta cosh\left(\sqrt{\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\sqrt{\psi }sinh\left(\sqrt{\psi }(\xi +C)\right)-\frac{2d_2{b}_{0}cosh\left(\sqrt{\psi }(\xi +C)\right)}{b_1}+i\sqrt{\psi }\right)\right)}}+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_9\left(\xi \right)& ={\displaystyle \frac{\left(\Delta sinh\left(\sqrt{\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\sqrt{\psi }cosh\left(\sqrt{\psi }(\xi +C)\right)-\frac{2d_2{b}_{0}sinh\left(\sqrt{\psi }(\xi +C)\right)}{b_1}+\sqrt{\psi }\right)\right)}}+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_{10}\left(\xi \right)& ={\displaystyle \frac{\left(2\Delta sinh\left(\frac{1}{4}\sqrt{\psi }(\xi +C)\right)cosh\left(\frac{1}{4}\sqrt{\psi }(\xi +C)\right)\right)}{\left(\gamma \left(2\sqrt{\psi }cosh{\left(\frac{1}{4}\sqrt{\psi }(\xi +C)\right)}^2-\frac{1}{b_1}\left(4d_2{b}_{0}sinh\left(\frac{1}{4}\sqrt{\psi }(\xi +C)\right)cosh\left(\frac{1}{4}\sqrt{\psi }(\xi +C)\right)\right)-\sqrt{\psi }\right)\right)}}+b_0,\hfill \end{array}$$

  where C is an arbitrary constant.
• CASE II: For $d_1^2-4d_0{d}_2<0$ and $d_1{d}_2\ne 0$ (or, $d_0{d}_2\ne 0$), we have the trignometric solutions as follows:

  $$\begin{array}{cc}\hfill U_{11}\left(\xi \right)& =\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{-\psi }tan\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)}{d_2}}-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_{12}\left(\xi \right)& =\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{-\psi }cot\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)}{d_2}}-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_{13}^{\pm }\left(\xi \right)& =\left({\displaystyle \frac{1}{2d_2}}\left(\sqrt{-\psi }\left(tan\left(\sqrt{-\psi }(\xi +C)\right)\pm sec\left(\sqrt{-\psi }(\xi +C)\right)\right)\right)-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_{14}^{\pm }\left(\xi \right)& =\left(-{\displaystyle \frac{1}{2d_2}}\left(\sqrt{-\psi }\left(cot\left(\sqrt{-\psi }(\xi +C)\right)\pm csc\left(\sqrt{-\psi }(\xi +C)\right)\right)\right)-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_{15}\left(\xi \right)& =\left({\displaystyle \frac{1}{4d_2}}\left(\sqrt{-\psi }\left(tan\left({\displaystyle \frac{1}{4}}\sqrt{-\psi }(\xi +C)\right)-cot\left({\displaystyle \frac{1}{4}}\sqrt{-\psi }(\xi +C)\right)\right)\right)-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_{16}\left(\xi \right)& =-{\displaystyle \frac{\left(\Delta cos\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\sqrt{-\psi }sin\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)-\frac{2d_2{b}_{0}cos\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)}{b_1}\right)\right)}}+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_{17}\left(\xi \right)& ={\displaystyle \frac{\left(\Delta sin\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\sqrt{-\psi }cos\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)-\frac{2d_2{b}_{0}sin\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)}{b_1}\right)\right)}}+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_{18}\left(\xi \right)& =-{\displaystyle \frac{\left(\Delta cos\left(\sqrt{-\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\frac{2d_2{b}_{0}cos\left(\sqrt{-\psi }(\xi +C)\right)}{b_1}+\sqrt{-\psi }sin\left(\sqrt{-\psi }(\xi +C)\right)+\sqrt{-\psi }\right)\right)}}+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_{19}\left(\xi \right)& =-{\displaystyle \frac{\left(\Delta sin\left(\sqrt{-\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\frac{2d_2{b}_{0}sin\left(\sqrt{-\psi }(\xi +C)\right)}{b_1}+\sqrt{-\psi }cos\left(\sqrt{-\psi }(\xi +C)\right)+\sqrt{-\psi }\right)\right)}}+b_0,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill U_{20}\left(\xi \right)& ={\displaystyle \frac{\left(2\Delta sin\left(\frac{1}{4}\sqrt{-\psi }(\xi +C)\right)cos\left(\frac{1}{4}\sqrt{-\psi }(\xi +C)\right)\right)}{\left(\gamma \left(2\sqrt{-\psi }cos{\left(\frac{1}{4}\sqrt{-\psi }(\xi +C)\right)}^2-\frac{1}{b_1}\left(4d_2{b}_{0}sin\left(\frac{1}{4}\sqrt{-\psi }(\xi +C)\right)cos\left(\frac{1}{4}\sqrt{-\psi }(\xi +C)\right)\right)-\sqrt{-\psi }\right)\right)}}+b_0,\hfill \end{array}$$

  where C is an arbitrary constant.

The solutions of Equation (12) using the GREM and Equation (13) are as follows:

$$\begin{array}{cc}\hfill u_1(x,t)& =\left(\left(-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{\psi }tanh\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)}{d_2}}-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_2(x,t)& =\left(\left(-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{\psi }coth\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)}{d_2}}-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_3^{\pm }(x,t)& =\left(\left(-{\displaystyle \frac{1}{2d_2}}\left(\sqrt{\psi }\left(tanh\left(\sqrt{\psi }(\xi +C)\right)\pm isech\left(\sqrt{\psi }(\xi +C)\right)\right)\right)-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_4(x,t)& =\left(\left(-{\displaystyle \frac{1}{2d_2}}\left(\sqrt{\psi }\left(coth\left(\sqrt{\psi }(\xi +C)\right)+csch\left(\sqrt{\psi }(\xi +C)\right)\right)\right)-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_5(x,t)& =\left(\left(-{\displaystyle \frac{1}{4d_2}}\left(\sqrt{\psi }\left(tanh\left({\displaystyle \frac{1}{4}}\sqrt{\psi }(\xi +C)\right)+coth\left({\displaystyle \frac{1}{4}}\sqrt{\psi }(\xi +C)\right)\right)\right)-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_6(x,t)& =\left(-{\displaystyle \frac{\left(\Delta cosh\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\sqrt{\psi }sinh\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)-\frac{2d_2{b}_{0}cosh\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)}{b_1}\right)\right)}}+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_7(x,t)& =\left({\displaystyle \frac{\left(\Delta sinh\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\sqrt{\psi }cosh\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)-\frac{2d_2{b}_{0}sinh\left(\frac{1}{2}\sqrt{\psi }(\xi +C)\right)}{b_1}\right)\right)}}+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_8(x,t)& =\left({\displaystyle \frac{\left(\Delta cosh\left(\sqrt{\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\sqrt{\psi }sinh\left(\sqrt{\psi }(\xi +C)\right)-\frac{2d_2{b}_{0}cosh\left(\sqrt{\psi }(\xi +C)\right)}{b_1}+i\sqrt{\psi }\right)\right)}}+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_9(x,t)& =\left({\displaystyle \frac{\left(\Delta sinh\left(\sqrt{\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\sqrt{\psi }cosh\left(\sqrt{\psi }(\xi +C)\right)-\frac{2d_2{b}_{0}sinh\left(\sqrt{\psi }(\xi +C)\right)}{b_1}+\sqrt{\psi }\right)\right)}}+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_{10}(x,t)& =\left({\displaystyle \frac{\left(2\Delta sinh\left(\frac{1}{4}\sqrt{\psi }(\xi +C)\right)cosh\left(\frac{1}{4}\sqrt{\psi }(\xi +C)\right)\right)}{\left(\gamma \left(2\sqrt{\psi }cosh{\left(\frac{1}{4}\sqrt{\psi }(\xi +C)\right)}^2-\frac{1}{b_1}\left(4d_2{b}_{0}sinh\left(\frac{1}{4}\sqrt{\psi }(\xi +C)\right)cosh\left(\frac{1}{4}\sqrt{\psi }(\xi +C)\right)\right)-\sqrt{\psi }\right)\right)}}+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_{11}(x,t)& =\left(\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{-\psi }tan\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)}{d_2}}-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_{12}(x,t)& =\left(\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{-\psi }cot\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)}{d_2}}-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_{13}^{\pm }(x,t)& =\left(\left({\displaystyle \frac{1}{2d_2}}\left(\sqrt{-\psi }\left(tan\left(\sqrt{-\psi }(\xi +C)\right)\pm sec\left(\sqrt{-\psi }(\xi +C)\right)\right)\right)-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_{14}^{\pm }(x,t)& =\left(\left(-{\displaystyle \frac{1}{2d_2}}\left(\sqrt{-\psi }\left(cot\left(\sqrt{-\psi }(\xi +C)\right)\pm csc\left(\sqrt{-\psi }(\xi +C)\right)\right)\right)-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_{15}(x,t)& =\left(\left({\displaystyle \frac{1}{4d_2}}\left(\sqrt{-\psi }\left(tan\left({\displaystyle \frac{1}{4}}\sqrt{-\psi }(\xi +C)\right)-cot\left({\displaystyle \frac{1}{4}}\sqrt{-\psi }(\xi +C)\right)\right)\right)-{\displaystyle \frac{b_0}{b_1}}\right)b_1+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_{16}(x,t)& =\left(-{\displaystyle \frac{\left(\Delta cos\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\sqrt{-\psi }sin\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)-\frac{2d_2{b}_{0}cos\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)}{b_1}\right)\right)}}+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_{17}(x,t)& =\left({\displaystyle \frac{\left(\Delta sin\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\sqrt{-\psi }cos\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)-\frac{2d_2{b}_{0}sin\left(\frac{1}{2}\sqrt{-\psi }(\xi +C)\right)}{b_1}\right)\right)}}+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_{18}(x,t)& =\left(-{\displaystyle \frac{\left(\Delta cos\left(\sqrt{-\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\frac{2d_2{b}_{0}cos\left(\sqrt{-\psi }(\xi +C)\right)}{b_1}+\sqrt{-\psi }sin\left(\sqrt{-\psi }(\xi +C)\right)+\sqrt{-\psi }\right)\right)}}+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_{19}(x,t)& =\left(-{\displaystyle \frac{\left(\Delta sin\left(\sqrt{-\psi }(\xi +C)\right)\right)}{\left(\gamma \left(\frac{2d_2{b}_{0}sin\left(\sqrt{-\psi }(\xi +C)\right)}{b_1}+\sqrt{-\psi }cos\left(\sqrt{-\psi }(\xi +C)\right)+\sqrt{-\psi }\right)\right)}}+b_0\right)e^{i\theta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_{20}(x,t)& =\left({\displaystyle \frac{\left(2\Delta sin\left(\frac{1}{4}\sqrt{-\psi }(\xi +C)\right)cos\left(\frac{1}{4}\sqrt{-\psi }(\xi +C)\right)\right)}{\left(\gamma \left(2\sqrt{-\psi }cos{\left(\frac{1}{4}\sqrt{-\psi }(\xi +C)\right)}^2-\frac{1}{b_1}\left(4d_2{b}_{0}sin\left(\frac{1}{4}\sqrt{-\psi }(\xi +C)\right)cos\left(\frac{1}{4}\sqrt{-\psi }(\xi +C)\right)\right)-\sqrt{-\psi }\right)\right)}}+b_0\right)e^{i\theta }.\hfill \end{array}$$

3. Results and Discussion

This part will look at the derived solutions discovered by applying the EDAM and the GREM to the generalized modified UNLSE (12). The 3D and 2D graphical results will be illustrated in this section. Next, the Galilean transformation will transform the problem into a dynamical system. Using planar dynamical system theory, we will examine the system’s equilibrium points and the accompanying Hamiltonian polynomial in detail. We will investigate the consequences of introducing external influences to the system and its chaotic behaviors. Finally, we will study the modulation instability analysis and show 2D plots of the accompanying gain spectrum dynamics for Equation (12).

3.1. Graphical Results

Now, we begin with visual demonstrations of the wave dynamics using suitable values for the obtained solutions. We present a variety of instances in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 pointing out the dynamical behaviors of the real and imaginary parts of certain selected wave solutions.

Figure 1a, Figure 2a, Figure 3a, Figure 4a, Figure 5a and Figure 6a depict the 3D behavior of the wave function, while Figure 1b, Figure 2b, Figure 3b, Figure 4b, Figure 5b and Figure 6b and Figure 1c, Figure 2c, Figure 3c, Figure 4c, Figure 5c and Figure 6c describe the 3D real and imaginary behaviors of the wave function, respectively. Figure 7 and Figure 8 present 2D contour plots of the selected solutions. The study presents innovative solutions to the generalized modified UNLSE (12), a fundamental equation in physics that describes various phenomena, including energy level, superconductivity, wave–particle duality, phase transitions, particle behaviors, and quantum field theory. By exploring different values of constants in the equation, we have discovered novel solutions for the proposed equation. The solutions have far-reaching applications in various fields of science including the study of wave propagation, heat flow, and other physical phenomena. As a result, there are four types of soliton solutions: dark, bright, periodic, and multiple soliton solutions. It is worth noting that the solutions obtained in this study are novel and have not been previously discovered. The graphical representations support the obtained results and are consistent with the existing literature. Furthermore, the utilized methods have revealed new forms of traveling wave solutions which are not found elsewhere. These findings contribute to the ongoing research in this field and demonstrate the potential of symbolic software in solving complex equations. The applied methodologies yield solutions in the form of the trigonometric, hyperbolic functions, and their generalized functions, which are equivalent to those forms offered by the new extended direct algebraic method (NEDAM) [40]. The advantage of the methods used over other methods is that they provide solution forms covering almost all scientific functions. Also, the proposed solutions in this work generalize the solutions derived using the extended simple equation method [17], the modified Kudryashov method [18], the classical ansatz methods [33], the improved modified Sardar sub-equation method [34], the modified extended mapping method [35], the auxiliary equation method [36], and the extended tangent hyperbolic function method [37]. Thus, the GREM and the EDAM are more efficient than the existing methodologies due to their ability to produce robust traveling wave solutions and wave structures, which are paramount in hyperbolic PDEs and other related phenomena.

3.2. The Bifurcation Analysis

This section will discuss the bifurcation phase portrait (PP) analysis of the generalized modified UNLSE (12) using the theory of planar dynamical systems. We begin by applying the Galilean transformation $\Lambda \left(\xi \right)=U^{\prime }\left(\xi \right)$ to Equation (15) to obtain

$$\left\{\begin{array}{c}{U}^{\prime }\left(\xi \right)=\Lambda \left(\xi \right),\hfill \\ {\Lambda }^{\prime }\left(\xi \right)={\displaystyle \frac{(bd\nu +\nu +d^2)}{({\lambda }_1^2-b{\lambda }_1{\lambda }_2)}}U\left(\xi \right)-{\displaystyle \frac{g}{({\lambda }_1^2-b{\lambda }_1{\lambda }_2)}}{U}^3\left(\xi \right).\hfill \end{array}\right.$$

(33)

The Hamiltonian function corresponding to system (33) is

$$H(U,\Lambda )={\displaystyle \frac{1}{2}}{\Lambda }^2-{\displaystyle \frac{(bd\nu +\nu +d^2)}{2({\lambda }_1^2-b{\lambda }_1{\lambda }_2)}}{U}^2+{\displaystyle \frac{g}{4({\lambda }_1^2-b{\lambda }_1{\lambda }_2)}}{U}^4=w,$$

(34)

where w is the constant of motion. Moreover, the equilibrium points of Equation (33) are computed by solving

$$\left\{\begin{array}{c}\Lambda \left(\xi \right)=0,\hfill \\ {\displaystyle \frac{(bd\nu +\nu +d^2)}{({\lambda }_1^2-b{\lambda }_1{\lambda }_2)}}U\left(\xi \right)-{\displaystyle \frac{g}{({\lambda }_1^2-b{\lambda }_1{\lambda }_2)}}{U}^3\left(\xi \right)=0.\hfill \end{array}\right.$$

(35)

After solving Equation (35), the system admits the equilibrium points, ${\overline{E}}_1=(0,0)$, ${\overline{E}}_2=\left(\sqrt{{\displaystyle \frac{(bd\nu +\nu +d^2)}{-g}}},0\right)$ and ${\overline{E}}_3=\left(-\sqrt{{\displaystyle \frac{(bd\nu +\nu +d^2)}{-g}}},0\right)$. Therefore, the system admits the following Jacobian,

$$\begin{array}{c}\hfill J(U,\Lambda )=\left(\begin{array}{cc}0& 1\\ {\displaystyle \frac{(bd\nu +\nu +d^2)}{({\lambda }_1^2-b{\lambda }_1{\lambda }_2)}}-{\displaystyle \frac{3g}{({\lambda }_1^2-b{\lambda }_1{\lambda }_2)}}{U}^2\left(\xi \right)& 0\end{array}\right),\end{array}$$

(36)

whose determinant is

$$\left|J(U,\Lambda )\right|={\displaystyle \frac{3g}{({\lambda }_1^2-b{\lambda }_1{\lambda }_2)}}{U}^2\left(\xi \right)-{\displaystyle \frac{(bd\nu +\nu +d^2)}{({\lambda }_1^2-b{\lambda }_1{\lambda }_2)}}.$$

(37)

From the theory of the planar dynamical systems [41],

1.

If $\left|J\right(U,\Lambda \left)\right|>0$, then $(U,\Lambda )$ is a center point.

2.

If $\left|J\right(U,\Lambda \left)\right|=0$, then $(U,\Lambda )$ is a degenerate point.

3.

If $\left|J\right(U,\Lambda \left)\right|<0$, then $(U,\Lambda )$ is a saddle point.

Figure 9 indicates the dynamical characteristics of the dynamical system (33) associated with two different cases. Case I is presented in Figure 9a using $b=1.4, d=0.5,$ $\nu =1, {\lambda }_1=0.2, {\lambda }_2=-0.2,$ and $g=1$. Based on the parameter values applied for this case, the equilibrium point ${\overline{E}}_1$ represents a saddle point, while the equilibrium points ${\overline{E}}_2$ and ${\overline{E}}_3$ represent center points. Case II is presented in Figure 9b for $b=1.4, d=0.5, \nu =-1, {\lambda }_1=0.2, {\lambda }_2=-0.2,$ and $g=-1$. With the parameter values mentioned for Case II, the equilibrium point ${\overline{E}}_1$ represents a center point, while the equilibrium points ${\overline{E}}_2$ and ${\overline{E}}_3$ behave as saddle points.

3.3. The Chaotic Analysis

This section will analyze the chaotic behavior of the system (33) by adding a periodic external force. Now, consider the perturbed dynamical system,

$$\left\{\begin{array}{c}{U}^{\prime }\left(\xi \right)=\Lambda \left(\xi \right),\hfill \\ {\Lambda }^{\prime }\left(\xi \right)={\displaystyle \frac{(bd\nu +\nu +d^2)}{({\lambda }_1^2-b{\lambda }_1{\lambda }_2)}}U\left(\xi \right)-{\displaystyle \frac{g}{({\lambda }_1^2-b{\lambda }_1{\lambda }_2)}}{U}^3\left(\xi \right)+Ecos\left(ft\right),\hfill \end{array}\right.$$

(38)

where E is the amplitude of the external acting force and f is the frequency of the system. In Figure 10, we plot the PP of the perturbed system (38) in 2D and 3D together with the corresponding sensitivity analysis by solving the system via the well-known Runge–Kutta method for $b=1$, $d=1$, $\nu =1$, ${\lambda }_1=2$, ${\lambda }_2=-2$, $g=-1$, $f=10$ while varying the amplitude with the values $E=0.5$ and 1. Figure 10 shows the significance and implications of the small changes in amplitude in the chaotic pattern. In Figure 11, we mainly focus on the effect of varying the frequency of the perturbed system by assigning $b=1$, $d=2$, $\nu =-0.1$, ${\lambda }_1=2$, ${\lambda }_2=-2$, $g=-4.1$, $\nu =1$ while varying the frequency with the values $f=1$ and 5. This figure shows that when we increase the frequency of the perturbed system, the complexity and intricacy of the chaotic pattern in the system drastically reduce. In Figure 12, we vary the values of the parameter $\nu$ with $\nu =-1$ and $-2$ and fix $b=1.4$, $d=0.2$, $f=5$, ${\lambda }_1=0.2$, ${\lambda }_2=-0.2$, $g=-1$, $E=-2$ to derive more complex chaotic patterns for the perturbed system (38).

Furthermore, we will focus our attention on the sensitivity of both the unperturbed and perturbed systems with respect to the initial conditions. Figure 13 presents the sensitivity analysis of the unperturbed system (33) with respect to three initial conditions using the parameters $b=1$, $d=1$, $\nu =1$, ${\lambda }_1=2$, ${\lambda }_2=-2$, and $g=-1$. Figure 14 shows the sensitivity analysis for the perturbed system (38) using the values $b=1$, $d=1$, $\nu =1$, ${\lambda }_1=2$, ${\lambda }_2=-2$, $g=-1$, $E=0.5$, and $f=10$. These two plots indicate that small variations in the initial conditions do not affect the stability of the solution sufficiently.

3.4. The Modulation Instability

This section will examine a modulation instability (MI) of the generalized modified UNLSE in Equation (12). The MI is an intriguing phenomenon that arises from the interaction of nonlinear and dispersive effects. Again, during the long-term development of the continuous wave (CW), structures such as modulated waves and rogue waves may be formed by utilizing the appropriate nonlinear parameters. In an optical fiber, modulated bright solitons are formed due to the interplay of Kerr nonlinearity, group velocity dispersion, and quartic dispersions. We will apply the linear stability analysis to establish the MI of Equation (12) as adopted in [42,43]. First, we assume the steady continuous wave solution for Equation (12) to be

$$u(x,t)=\sqrt{P}exp\left(i\sqrt{gP}x\right),$$

(39)

where P is the normalized power. Moreover, to test for the modulation instability of Equation (12), we assume the perturbed plane wave to be the solution of the generalized modified UNLSE as follows:

$$u(x,t)=\left(\sqrt{P}+\Omega (x,t)\right)exp\left(i\sqrt{gP}x\right),$$

(40)

where $\Omega (x,t)$ is the small perturbation. Now, we substitute Equation (40) into Equation (12) to get the following linearized equation:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Pg\Omega \left(x,t\right)& +Pg{\Omega }^{*}\left(x,t\right)+i\left(b\sqrt{gP}+1\right){\displaystyle \frac{\partial }{\partial t}}\Omega \left(x,t\right)+2i\sqrt{gP}{\displaystyle \frac{\partial }{\partial x}}\Omega \left(x,t\right)+b\left({\displaystyle \frac{{\partial }^2}{\partial x\partial t}}\Omega \left(x,t\right)\right)\hfill \\ & +{\displaystyle \frac{{\partial }^2}{\partial x^2}}\Omega \left(x,t\right)=0,\hfill \end{array}\end{array}$$

(41)

where ${\Omega }^{*}(x,t)$ is the conjugate of $\Omega (x,t)$. Now, we seek the trivial solution for Equation (41) of the form

$$\Omega (x,t)={\beta }_1{\mathrm{e}}^{i\left(kx-wt\right)}+{\beta }_2{\mathrm{e}}^{-i\left(kx-wt\right)},$$

(42)

where ${\beta }_1$, ${\beta }_2$ are constants. Also, k and w stand for the perturbation’s normalized wave number and frequency. Substituting Equation (42) into Equation (41), we then obtain

$$\left(\begin{array}{cc}bkw+gP-k^2+w& Pg\\ Pg& bkw+gP-k^2-w\end{array}\right)\left(\begin{array}{c}{\beta }_1\\ {\beta }_2\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right).$$

(43)

The dispersion relation $w=w\left(k\right)$ defines the connection between temporal oscillations $e^{iwt}$ and spatial oscillations $e^{ikx}$ with wave number k. We get the dispersion relation as

$$w={\displaystyle \frac{\left(-Pbg+b{k}^2+\sqrt{{\left(Pbg\right)}^2-2gP+k^2}\right)k}{b^2{k}^2-1}}.$$

(44)

The dispersion relation shows that the steady-state stability depends on stimulated Raman scattering, self-phase modulation, group velocity dispersion, and wave number. If ${\left(Pbg\right)}^2-2gP+k^2>0$ and $b^2{k}^2\ne 1$, then w is real for all k and the steady state is stable against small perturbations. Also, if ${\left(Pbg\right)}^2-2gP+k^2<0$ and $b^2{k}^2\ne 1$, then w is complex for all k. Consequently, the perturbation grows exponentially and the MI occurs. Therefore, the corresponding MI gain spectrum is

$$h\left(k\right)=2\mathrm{Im}\left(w\right)=2\mathrm{Im}\left({\displaystyle \frac{\left(-Pbg+b{k}^2+\sqrt{{\left(Pbg\right)}^2-2gP+k^2}\right)k}{b^2{k}^2-1}}\right).$$

(45)

Figure 15 contains three distinct MIs by varying the parameters associated with the gain spectrum $h\left(k\right)$. In Figure 15a, the MI’s growth rate increases with increasing value of P and decreases with decreasing value of P. In Figure 15b the MI’s growth rate decreases with decreasing value of b. Finally, Figure 15c shows that the MI’s growth rate increases with increasing value of g with huge differences in amplitude.

4. Conclusions

This paper proposed a generalized modified UNLSE with applications in modulated wavetrain instabilities. The EDAM and GREM were fully utilized and produced novel soliton solutions for the proposed equation. The derived solutions are new and more general in the form of hyperbolic, trigonometric, and elliptic functions. The methods provided dynamical soliton structures, such as dark, bright, and rogue wave profiles. Moreover, the Galilean transformation was applied to transform the proposed model into a corresponding planar dynamical system. We further derived the equilibrium points for the corresponding dynamical system and its Hamiltonian polynomial using the theory of planar dynamical systems. The bifurcation PP analysis of the system was completely analyzed with and without the presence of the external force to study the model’s chaotic behaviors. We provided 2D and 3D plots to support our reported mathematical analysis. Comprehensive sensitivity analyses of both dynamical systems reveal that small variations in the initial conditions do not have much effect on the stability of the solution. The findings presented in this work are novel, fascinating, and theoretically significant for the understanding of the temporal evolution of disturbances in moderately stable or unstable media. Understanding the dynamical behaviors of the equation is essential for predicting and designing new technologies, especially in modulated wavetrain instabilities.