Analysis Modulation Instability and Parametric Effect on Soliton Solutions for M-Fractional Landau–Ginzburg–Higgs (LGH) Equation Through Two Analytic Methods

Abstract

This manuscript studies the M-fractional Landau–Ginzburg–Higgs (M-fLGH) equation in comprehending superconductivity and drift cyclotron waves in radially inhomogeneous plasmas, especially for coherent ion cyclotron wave propagation, aiming to explore the soliton solutions, the parameter’s effect, and modulation instability. Here, we propose a novel approach, namely a newly improved Kudryashov’s method that integrates the combination of the unified method with the generalized Kudryashov’s method. By employing the modified F-expansion and the newly improved Kudryashov’s method, we investigate the soliton wave solutions for the M-fLGH model. The solutions are in trigonometric, rational, exponential, and hyperbolic forms. We present the effect of system parameters and fractional parameters. For special values of free parameters, we derive some novel phenomena such as kink wave, anti-kink wave, periodic lump wave with soliton, interaction of kink and periodic lump wave, interaction of anti-kink and periodic wave, periodic wave, solitonic wave, multi-lump wave in periodic form, and so on. The modulation instability criterion assesses the conditions that dictate the stability or instability of soliton solutions, highlighting the interplay between fractional order and system parameters. This study advances the theoretical understanding of fractional LGH models and provides valuable insights into practical applications in plasma physics, optical communication, and fluid dynamics.

1. Introduction

Nonlinear evolution equations (NLEEs) play a fundamental role in describing complex dynamical phenomena in various branches of science and engineering [1,2,3,4,5,6,7,8,9,10]. In fields like fluid dynamics, plasma physics, optical fibers, and quantum mechanics, these equations show how waves move, interact, and transfer energy in nonlinear ways. In contrast to linear equations, which often lead to simplified approximations, nonlinear equations capture the complex dynamics of systems where nonlinearity is essential for understanding the real world. A significant characteristic of NLEEs is their association with soliton solutions. Solitons are self-sustaining, confined waveforms that maintain their shape as they travel at a steady speed. They emerge from a fragile equilibrium between dispersion and nonlinearity in the fundamental medium. Looking into soliton solutions is important for both theory and real life, like in optical communication systems, Bose–Einstein condensates, and modeling water waves.

Mathematically, solitons are often derived using sophisticated analytical techniques such as the Bilinear approach [11], direct algebraic procedure [12], the modified rational sine–cosine [13], the (1/G)-expansion [14], improved F-expansion [15], improved Kudryashov [16], novel Kudryashov, Exp_a-function [17], tanh [18], $\left({\displaystyle \frac{G^{\prime }}{G^{\prime }+G+A}}\right)$-expansion [19], bifurcation theory [20], modified exp-function [21], Hirota bilinear [22], enhanced modified simple equation [23], new extended direct algebraic [24], unified [25,26], and extended tanh expansion [27] methods, and so on [28,29,30]. These techniques help us find exact and close solutions to complicated nonlinear evolution equations. This helps us learn more about nonlinear processes and guides the creation of advanced technologies for wave-based systems. We are always learning more about NLEEs and their soliton solutions. This helps us understand nonlinear wave dynamics better and gives us strong ways to solve problems in science and engineering.

The Landau–Ginzburg–Higgs (LGH) equation is important for understanding many physical things, like how phases change in condensed matter systems and how gauge field theories work at very high energies. It is possible to look at spontaneous symmetry breakdown and the dynamics of scalar and gauge fields using this equation. Its applications encompass superconductivity, quantum field theory, and cosmology, establishing it as a fundamental element in both theoretical and applied physics.

Studying the LGH equation enables researchers to reveal essential insights into nonlinear dynamics, the emergence of topological defects, and energy conservation mechanisms. Also, improvements in mathematical and analytical methods, like bifurcation analysis and soliton theory, have made it easier to look into new solutions. The results of these studies help make new materials better, help us understand quantum technologies better, and answer important questions in modern physics. Mathematically, the Landau–Ginzburg–Higgs (LGH) equation can be written as follows:

$${\mathrm{D}}_{\mathrm{M},\mathrm{t}}^{2\mathrm{P},\mathrm{n}}H-H_{xx}-{\alpha }^{2}H+{\mu }^2{H}^3=0.$$

(1)

Here, $t$ and $x$ represent the temporal and spatial coordinates, respectively, while the real constants $a$ and $b$ correspond to the ion cyclotron wave and the electrostatic potential. The LGH equation, as expressed in Equation (1), was initially formulated to describe the drift cyclotron dynamics associated with coherent ion cyclotron waves in a plasma exhibiting geometric chaos. Many researchers have investigated diverse solutions and applied different analytic methods, including Power Index [31], improved Kudryashov (IKM), Kudryashov’s R (KRM), Sardar’s sub equation [32], Jacobi elliptic functions [33], inverse scattering transformation [34], generalized Kudryashov [35], the ($G’/G,1/G$), modified $\left(G’/G2\right)$, and new auxiliary equation [36], generalized projective Riccati [37], extended tanh [38], Multiple Auxiliary Equation [39], He’s semi-inverse [40], the homotopy perturbation [41], the new generalized [42], the NMSE [43], the IBSEF [44] methods, and so on [45,46,47]. In this work, we investigate the chaotic nature and soliton solution of the M-fractional LGH model. To study the chaotic nature, we apply the bifurcation theory with trigonometric perturbation function. Additionally, we apply the newly improved Kudryashov’s and modified f-expansion techniques to find the soliton solutions analytically. Under some conditions, the solutions are expressed as trigonometric, hyperbolic, exponential, and rational function forms.

2. Fractional Derivative

A fractional derivative extends the conventional notion of a derivative to non-integer (fractional) orders. Ordinary integer-order derivatives show how fast a function changes at a certain point. Fractional derivatives [48,49,50], on the other hand, make it easier to separate a function into any real or complex order. This expansion helps when modeling systems that have memory effects, non-local behavior, and long-range interactions that cannot be properly shown by standard integer-order calculus. Methodologies such as Riemann–Liouville, Caputo, or Grünwald–Letnikov [51,52] typically specify fractional derivatives. The M-fractional derivative [53,54,55] is a more general version of the fractional derivative. It includes more factors or changes, such as a weighted memory effect or different boundary conditions. This change enhances the flexibility in modeling intricate physical, technical, and biological systems, necessitating a more sophisticated depiction of anomalous phenomena, such as viscoelastic materials or turbulent flows. The M-fractional derivative maintains the capacity to address non-local interactions while incorporating an extra dimension of adaptation to accommodate the particular traits of the system in question. The updated version of the M-fractional derivative makes it easier to deal with a wider range of real-world phenomena. It does this by providing a reliable tool for situations where regular fractional derivatives might not work well, allowing for the more accurate and complex modeling of systems that have fractional-order dynamics.

Preliminary of M-Fractional Operator

Consider a mapping $\Im :{ℝ}^{+}\to ℝ$ the Truncated M-fraction derivative of $\Im$ with order $P$ shown as follow:

$$D_{M,t}^{P, n}\left\{\Im \left(t\right)\right\}=\underset{r\to 0}{\mathit{lim}}{\displaystyle \frac{\Im \left(t{E}_n\left(r{t}^{1-P}\right)\right)-\Xi \left(t\right)}{r}}; 0<P<1, n>0.$$

Here, ${\mathrm{E}}_{\mathrm{n}}\left(.\right)$ is a truncated Mittag–Leffler function of one parameter that is defined as [53,54,55]:

$$E_n\left(z\right)={{\displaystyle \sum }}_{j=0}^i{\displaystyle \frac{z^j}{\mathit{\Gamma }\left(nj+1\right)}}.$$

Characteristics:

Suppose $\mathrm{n}>0,{ \mathrm{C}}_1,{\mathrm{C}}_2\in \Re , 0<\mathrm{P}<1$ and $\mathcal{H},\mathcal{M}$ is the $\mathrm{P}$-differentiable at a point $\mathrm{t}>0$; then,

i.

${\mathrm{D}}_{\mathrm{M},\mathrm{t}}^{\mathrm{P}, \mathrm{n}}\left({\mathrm{C}}_1\mathcal{H}\left(\mathrm{t}\right)+{\mathrm{C}}_2\mathcal{M}\left(\mathrm{t}\right)\right)={\mathrm{C}}_1{\mathrm{D}}_{\mathrm{M},\mathrm{t}}^{\mathrm{P}, \mathrm{n}}\mathcal{H}\left(\mathrm{t}\right)+{\mathrm{C}}_2{\mathrm{D}}_{\mathrm{M},\mathrm{t}}^{\mathrm{P}, \mathrm{n}}\mathcal{M}\left(\mathrm{t}\right)$.

ii.

${\mathrm{D}}_{\mathrm{M},\mathrm{t}}^{\mathrm{P}, \mathrm{n}}\left(\mathcal{H}\left(\mathrm{t}\right)\mathcal{M}\left(\mathrm{t}\right)\right)=\mathcal{H}\left(\mathrm{t}\right){\mathrm{D}}_{\mathrm{M},\mathrm{t}}^{\mathrm{P}, \mathrm{n}}\mathcal{M}\left(\mathrm{t}\right)+\mathcal{M}\left(\mathrm{t}\right){\mathrm{D}}_{\mathrm{M},\mathrm{t}}^{\mathrm{P}, \mathrm{n}}\mathcal{H}\left(\mathrm{t}\right)$.

iii.

${\mathrm{D}}_{\mathrm{M},\mathrm{t}}^{\mathrm{P}, \mathrm{n}}\left({\displaystyle \frac{\mathcal{H}\left(\mathrm{t}\right)}{\mathsf{\Theta }\left(\mathrm{t}\right)}}\right)={\displaystyle \frac{\mathcal{M}\left(\mathrm{t}\right){\mathrm{D}}_{\mathrm{M},\mathrm{t}}^{\mathrm{P}, \mathrm{n}}\mathcal{H}\left(\mathrm{t}\right)-\mathcal{H}\left(\mathrm{t}\right){\mathrm{D}}_{\mathrm{M},\mathrm{t}}^{\mathrm{P}, \mathrm{n}}\mathcal{M}\left(\mathrm{t}\right)}{\mathcal{M}{\left(\mathrm{t}\right)}^2}}$.

iv.

${\mathrm{D}}_{\mathrm{M},\mathrm{t}}^{\mathrm{P}, \mathrm{n}}\left(\mathcal{M}\left(\mathrm{t}\right)\right)=0; \mathcal{M}\left(\mathrm{t}\right)=\mathrm{a}$; here, $\mathrm{a}$ is an arbitrary constant.

v.

$D_{M,t}^{P, n}\mathcal{M}\left(t\right)={\displaystyle \frac{t^{1-P}}{\mathit{\Gamma }\left(n+1\right)}}{\displaystyle \frac{d\mathcal{M}\left(t\right)}{dt}}$.

3. Analytical Method

Let NLPDEs in the following form:

$$\psi \left({\mathrm{D}}_{\mathrm{M},\mathrm{t}}^{\mathrm{P},\mathrm{n}}H,H^2{H}_x, {\mathrm{D}}_{\mathrm{M},\mathrm{t}}^{2\mathrm{k},\mathrm{n}}H, H_{xxx}\right)=0.$$

(2)

Consider a transformation variable:

$$\chi =kx-v{\displaystyle \frac{\mathit{\Gamma }\left(n+1\right)}{P}}{\mathrm{t}}^P; H\left(x,t\right)=H\left(\chi \right).$$

(3)

Using Equation (3) in Equation (4), we obtain the ODE of Equation (2) as follows:

$${\psi }_1\left(v{H}_{\chi },k{H}^2{H}_{\chi }, k^2{H}_{\chi },k^3{H}_{\chi \chi \chi }\right)=0.$$

(4)

3.1. Modified F-Expansion Method

The solution of Equation (4) has the following form [56,57]:

$$H\left(\chi \right)={\lambda }_0+{{\displaystyle \sum }}_{i=1}^N\left[{\lambda }_i{\left(W\left(\chi \right)\right)}^i+{\mu }_i{\left(W\left(\chi \right)\right)}^i\right];{\lambda }_n,{\mu }_n\ne 0.$$

(5)

Here, $W\left(\chi \right)$ satisfied the auxiliary equation,

$${\displaystyle \frac{dW\left(\chi \right)}{d\chi }}=s_0+s_{1}W\left(\chi \right)+s_{2}W{\left(\chi \right)}^2.$$

(6)

Insert the solution in Equation (5) with the help of Equation (6) into Equation (4); then, we obtain a polynomial of $W\left(\chi \right)$. If we set each coefficient of this polynomial, a system of algebraic equations is attained. Now we obtain the values of ${\lambda }_0,{\lambda }_i,{\mu }_i,v,k$ by solving the obtained system. If we insert the solution sets in the trial solution Equation (5), the required solutions are obtained.

The solutions of the auxiliary differential equations are as follows:

• If $s_1=-s_2=1, s_0=0; W\left(\chi \right)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}\tan \mathrm{h}\left({\displaystyle \frac{\chi }{2}}\right)$.
• If $s_1=-s_2=1, s_0=0; W\left(\chi \right)={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}\cot \mathrm{h}\left({\displaystyle \frac{\chi }{2}}\right)$.
• If $s_0=-s_2={\displaystyle \frac{1}{2}}, s_1=0; W\left(\chi \right)=\cot \mathrm{h}\left(\chi \right)\pm cosceh\left(\chi \right)$.
• If $s_0=1, s_1=0, s_2=-1; W\left(\chi \right)=\tan \mathrm{h}\left(\chi \right), \cot \mathrm{h}\left(\chi \right)$.
• If $s_0={\displaystyle \frac{1}{2}}, s_1=0, s_2={\displaystyle \frac{1}{2}}; W\left(\chi \right)=\sec \left(\chi \right)+tan\left(\chi \right)$.
• If $s_0=-{\displaystyle \frac{1}{2}}, s_1=0, s_2=-{\displaystyle \frac{1}{2}}; W\left(\chi \right)=\sec \left(\chi \right)-tan\left(\chi \right)$.
• If $s_0=s_1=0; W\left(\chi \right)={\displaystyle \frac{1}{ s_2\chi +\mathsf{\delta }}}$.
• If $s_1=s_2=0; W\left(\chi \right)={\mathrm{s}}_0\chi$.
• If $s_0=0, s_1>0, s_2<0; W\left(\chi \right)={\displaystyle \frac{s_{1}exp\left(s_1\left(v+\delta \right)\right)}{1-s_{2}exp\left({\mathrm{s}}_1\left(\chi +\delta \right)\right)}}$.
• If $s_0=0, s_1〈0, s_2〉0; W\left(\chi \right)=-{\displaystyle \frac{s_{1}exp\left(s_1\left(v+\delta \right)\right)}{1+s_{2}exp\left({\mathrm{s}}_1\left(\chi +\delta \right)\right)}}$.
• If $s_0=0;W\left(\chi \right)={\displaystyle \frac{-s_2\delta }{{\mathrm{s}}_2\left(\delta +cosh\left(s_2\left(\chi +\delta \right)\right)-sinh\left(s_2\left(\chi +\delta \right)\right)\right)}}$.
• If $s_2=0; W\left(\chi \right)={\displaystyle \frac{-s_1+e^{s_2\chi }}{s_2}}$.

3.2. Newly Improved Kudryashov’s Method

Here, we propose a novel approach (newly improved Kudryashov’s method) that integrates the unified method [58,59] with Kudryashov’s method [60]. The solution of Equation (4) is considered as the following form [60]:

$$H\left(\chi \right)={\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^M\left[q_i{\left(W\left(\chi \right)\right)}^i\right]}{{{\displaystyle \sum }}_{i=1}^N\left[p_i{\left(W\left(\chi \right)\right)}^i\right]}};q_M,p_N\ne 0.$$

(7)

Here, $W\left(\chi \right)$ satisfies the auxiliary equation [58,59],

$${\displaystyle \frac{dW\left(\chi \right)}{d\chi }}=\epsilon +s_{2}W{\left(\chi \right)}^2.$$

(8)

Insert the solution in Equation (7) with the help of Equation (8) into Equation (4); then, we obtain a polynomial of $W\left(\chi \right)$. If we set each coefficient of this polynomial, a system of algebraic equations is attained. Now, we obtain the values of ${\lambda }_0,{\lambda }_i,{\mu }_i,v,k$ by solving the obtained system. If we insert the solution sets in the trial solution Equation (8), the required solutions are obtained.

The solutions of the auxiliary differential equations are as follows:

For $\epsilon <0,$

$$F\left( \phi \right)={\displaystyle \frac{\sqrt{-\left({\mathsf{\Theta }}^2+{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{-\epsilon }\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sinh}\left(2\sqrt{-\epsilon }\left(\chi +l\right)\right)}},$$

$$F\left( \phi \right)={\displaystyle \frac{-\sqrt{-\left(m^2+{\xi }^2\right)\epsilon }-m\sqrt{-\epsilon }\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}},$$

$$F\left( \phi \right)=\sqrt{-\epsilon }+{\displaystyle \frac{-2\xi \sqrt{-\epsilon }}{-\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)+\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)+k}},$$

$$F\left( \phi \right)=-\sqrt{-\epsilon }+{\displaystyle \frac{2\xi \sqrt{-\epsilon }}{\xi +\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)-\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}},$$

For $\epsilon <0,$

$$F\left( \phi \right)={\displaystyle \frac{\sqrt{\left({\mathsf{\Theta }}^2-{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{\epsilon }\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}},$$

$$F\left( \phi \right)={\displaystyle \frac{-\sqrt{\left({\mathsf{\Theta }}^2-{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{\epsilon }\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}{\mathsf{\Theta }\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)+\xi }},$$

$$F\left( \phi \right)=i\sqrt{\epsilon }+{\displaystyle \frac{-2\xi i\sqrt{\epsilon }}{\xi +\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)-i\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}},$$

$$F\left( \phi \right)=-i\sqrt{\epsilon }+{\displaystyle \frac{2\xi i\sqrt{\epsilon }}{\xi +\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +Q\right)\right)+i\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +Q\right)\right)}},$$

For $\epsilon =0$, $F\left( \phi \right)={\displaystyle \frac{1}{ \chi +l}}$.

4. Soliton Solutions of M-Fractional Landau–Ginzburg–Higgs Equation

At this point, we investigate the soliton solution from the M-fractional Landau–Ginzburg–Higgs model by using MFE and NIK methods. The LGH equation is a significant nonlinear evolution equation (NLEE) that characterizes the interior dynamics of intricate physical systems. This equation is crucial for comprehending superconductivity and drift cyclotron waves in radially inhomogeneous plasmas, especially for coherent ion cyclotron wave propagation. We utilize the MFE and NIK methods to locate appropriate solitary wave solutions.

At first, we insert a traveling wave variable $H\left(x,t\right)=H\left(\chi \right); \chi =kx-v{\displaystyle \frac{\mathit{\Gamma }\left(n+1\right)}{P}}{t}^P$ in Equation (1); then, we obtain the following ODE:

$$\left(v^2-k^2\right)H^{″}-{\alpha }^{2}H+{\mu }^2{H}^3=0.$$

(9)

4.1. Application of the Modified F-Expansion Method

After balancing between $H^{″}$ and $H^3$, $N=1$. Then, the solution of Equation (9) is

$$H\left(x,t\right)={\beta }_0+{\beta }_{1}W\left(\chi \right)+{\displaystyle \frac{{\delta }_1}{W\left(\chi \right)}}.$$

(10)

Now, the solution in Equation (10) is used in Equation (9) with the help of Equation (6). After some calculations according to the MFE method, we obtain the following solution sets:

$$k=\sqrt{{\displaystyle \frac{-4v^2{s}_0{s}_2+v^2{s}_1^2+2{\alpha }^2}{s_1^2-4s_0{s}_1}}}, {\beta }_0={\displaystyle \frac{\alpha s_1\sqrt{s_1^2-4s_0{s}_1}}{\left(s_1^2-4s_0{s}_1\right)\mu }}, {\beta }_1=0,{\delta }_1={\displaystyle \frac{2\alpha s_0}{\mu \sqrt{s_1^2-4s_0{s}_1}}};$$

$$v=\sqrt{{\displaystyle \frac{-4k^2{s}_0{s}_2+k^2{s}_1^2-2{\alpha }^2}{s_1^2-4s_0{s}_1}}}, {\beta }_0={\displaystyle \frac{\alpha s_1\sqrt{s_1^2-4s_0{s}_1}}{\left(s_1^2-4s_0{s}_1\right)\mu }}, {\beta }_1={\displaystyle \frac{2\alpha s_0}{\mu \sqrt{s_1^2-4s_0{s}_1}}},{\delta }_1=0.$$

$$\mathrm{Case}\text{-}01: k=\sqrt{{\displaystyle \frac{-4v^2{s}_0{s}_2+v^2{s}_1^2+2{\alpha }^2}{s_1^2-4s_0{s}_1}}}, {\beta }_0={\displaystyle \frac{\alpha s_1\sqrt{s_1^2-4s_0{s}_1}}{\left(s_1^2-4s_0{s}_1\right)\mu }}, {\beta }_1=0,{\delta }_1={\displaystyle \frac{2\alpha s_0}{\mu \sqrt{s_1^2-4s_0{s}_1}}}.$$

$$H\left(x,t\right)={\displaystyle \frac{\alpha s_1\sqrt{s_1^2-4s_0{s}_1}}{\left(s_1^2-4s_0{s}_1\right)\mu }}+{\displaystyle \frac{\frac{2\alpha s_0}{\mu \sqrt{s_1^2-4s_0{s}_1}}}{\cot \mathrm{h}\left(\chi \right)\pm cosceh\left(\chi \right)}};s_0=-s_2={\displaystyle \frac{1}{2}}, s_1=0.$$

(11)

$$H\left(x,t\right)={\displaystyle \frac{\alpha s_1\sqrt{s_1^2-4s_0{s}_1}}{\left(s_1^2-4s_0{s}_1\right)\mu }}+{\displaystyle \frac{\frac{2\alpha s_0}{\mu \sqrt{s_1^2-4s_0{s}_1}}}{\tan \mathrm{h}\left(\chi \right)}};s_0=1, s_1=0, s_2=-1.$$

(12)

$$H\left(x,t\right)={\displaystyle \frac{\alpha s_1\sqrt{s_1^2-4s_0{s}_1}}{\left(s_1^2-4s_0{s}_1\right)\mu }}+{\displaystyle \frac{\frac{2\alpha s_0}{\mu \sqrt{s_1^2-4s_0{s}_1}}}{\cot \mathrm{h}\left(\chi \right)}};s_0=1, s_1=0, s_2=-1.$$

(13)

$$H\left(x,t\right)={\displaystyle \frac{\alpha s_1\sqrt{s_1^2-4s_0{s}_1}}{\left(s_1^2-4s_0{s}_1\right)\mu }}+{\displaystyle \frac{\frac{2\alpha s_0}{\mu \sqrt{s_1^2-4s_0{s}_1}}}{\sec \left(\chi \right)+tan\left(\chi \right)}};s_0={\displaystyle \frac{1}{2}}, s_1=0, s_2={\displaystyle \frac{1}{2}}.$$

(14)

$$H\left(x,t\right)={\displaystyle \frac{\alpha s_1\sqrt{s_1^2-4s_0{s}_1}}{\left(s_1^2-4s_0{s}_1\right)\mu }}+{\displaystyle \frac{\frac{2\alpha s_0}{\mu \sqrt{s_1^2-4s_0{s}_1}}}{\sec \left(\chi \right)-tan\left(\chi \right)}};s_0=-{\displaystyle \frac{1}{2}}, s_1=0, s_2=-{\displaystyle \frac{1}{2}}.$$

(15)

Here, $\chi =\sqrt{{\displaystyle \frac{-4v^2{s}_0{s}_2+v^2{s}_1^2+2{\alpha }^2}{s_1^2-4s_0{s}_1}}}x-v{\displaystyle \frac{\Gamma \left(n+1\right)}{P}}{t}^P$.

$$\mathrm{Case}\text{-}02: v=\sqrt{{\displaystyle \frac{-4k^2{s}_0{s}_2+k^2{s}_1^2-2{\alpha }^2}{s_1^2-4s_0{s}_1}}}, {\beta }_0={\displaystyle \frac{\alpha s_1\sqrt{s_1^2-4s_0{s}_1}}{\left(s_1^2-4s_0{s}_1\right)\mu }}, {\beta }_1={\displaystyle \frac{2\alpha s_0}{\mu \sqrt{s_1^2-4s_0{s}_1}}},{\delta }_1=0.$$

$$H\left(x,t\right)={\displaystyle \frac{\alpha s_1\sqrt{s_1^2-4s_0{s}_1}}{\left(s_1^2-4s_0{s}_1\right)\mu }}+{\displaystyle \frac{2\alpha s_0\left(\cot \mathrm{h}\left(\chi \right)\pm cosceh\left(\chi \right)\right)}{\mu \sqrt{s_1^2-4s_0{s}_1}}};s_0=-s_2={\displaystyle \frac{1}{2}}, s_1=0.$$

(16)

$$H\left(x,t\right)={\displaystyle \frac{\alpha s_1\sqrt{s_1^2-4s_0{s}_1}}{\left(s_1^2-4s_0{s}_1\right)\mu }}+{\displaystyle \frac{2\alpha s_0\left(\tan \mathrm{h}\left(\chi \right)\right)}{\mu \sqrt{s_1^2-4s_0{s}_1}}};s_0=1, s_1=0, s_2=-1.$$

(17)

$$H\left(x,t\right)={\displaystyle \frac{\alpha s_1\sqrt{s_1^2-4s_0{s}_1}}{\left(s_1^2-4s_0{s}_1\right)\mu }}+{\displaystyle \frac{2\alpha s_0\left(\cot \mathrm{h}\left(\chi \right)\right)}{\mu \sqrt{s_1^2-4s_0{s}_1}}};s_0=1, s_1=0, s_2=-1.$$

(18)

$$H\left(x,t\right)={\displaystyle \frac{\alpha s_1\sqrt{s_1^2-4s_0{s}_1}}{\left(s_1^2-4s_0{s}_1\right)\mu }}+{\displaystyle \frac{2\alpha s_0\left(\sec \left(\chi \right)+tan\left(\chi \right)\right)}{\mu \sqrt{s_1^2-4s_0{s}_1}}};s_0={\displaystyle \frac{1}{2}}, s_1=0, s_2={\displaystyle \frac{1}{2}}.$$

(19)

$$H\left(x,t\right)={\displaystyle \frac{\alpha s_1\sqrt{s_1^2-4s_0{s}_1}}{\left(s_1^2-4s_0{s}_1\right)\mu }}+{\displaystyle \frac{2\alpha s_0\left(\sec \left(\chi \right)-tan\left(\chi \right)\right)}{\mu \sqrt{s_1^2-4s_0{s}_1}}};s_0=-{\displaystyle \frac{1}{2}}, s_1=0, s_2=-{\displaystyle \frac{1}{2}}.$$

(20)

Here, $\chi =kx-\sqrt{{\displaystyle \frac{-4k^2{s}_0{s}_2+k^2{s}_1^2-2{\alpha }^2}{s_1^2-4s_0{s}_1}}}{\displaystyle \frac{\Gamma \left(n+1\right)}{P}}{t}^P$.

4.2. Application of the Newly Improved Kudryashov’s Method

After balancing between $H^{″}$ and $H^3$, $M=N+1$. For $N=1$, we obtain $M=2$. Then, the solution of Equation (9) is

$$H\left(x,t\right)={\displaystyle \frac{q_0+q_{1}W\left(\chi \right)+q_{2}W{\left(\chi \right)}^2}{p_0+p_{1}W\left(\chi \right)}}.$$

(21)

Now, solution Equation (21) is used in Equation (9) with the help of Equation (8). After some calculations according to the NIK method, we obtain the following solution sets:

$$v=\sqrt{-{\displaystyle \frac{-2v^2\epsilon +{\alpha }^2}{2\epsilon }}}, p_1={\displaystyle \frac{q_2\mu }{\alpha }}\sqrt{-\epsilon },q_0=0,q_1=-{\displaystyle \frac{\alpha p_0\sqrt{-\epsilon }}{\mu \epsilon }};$$

$$k=\sqrt{-{\displaystyle \frac{-4k^2\epsilon +{\alpha }^2}{4\epsilon }}}, p_0=0,p_1={\displaystyle \frac{q_2\mu }{\alpha }}\sqrt{2\epsilon },q_0=q_2\epsilon , q_1=0.$$

$$\mathrm{Case}\text{-}01: v=\sqrt{-{\displaystyle \frac{-2v^2\epsilon +{\alpha }^2}{2\epsilon }}}, p_1={\displaystyle \frac{q_2\mu }{\alpha }}\sqrt{-\epsilon },q_0=0,q_1=-{\displaystyle \frac{\alpha p_0\sqrt{-\epsilon }}{\mu \epsilon }}.$$

For $\epsilon <0,$

$$H\left(x,t\right)={\displaystyle \frac{-\frac{\alpha p_0\sqrt{-\epsilon }}{\mu \epsilon }\left(\frac{\sqrt{-\left({\mathsf{\Theta }}^2+{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{-\epsilon }\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sinh}\left(2\sqrt{-\epsilon }\left(\chi +l\right)\right)}\right)+q_2{\left(\frac{\sqrt{-\left({\mathsf{\Theta }}^2+{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{-\epsilon }\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sinh}\left(2\sqrt{-\epsilon }\left(\chi +l\right)\right)}\right)}^2}{p_0+\frac{q_2\mu }{\alpha }\sqrt{-\epsilon }\left(\frac{\sqrt{-\left({\mathsf{\Theta }}^2+{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{-\epsilon }\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sinh}\left(2\sqrt{-\epsilon }\left(\chi +l\right)\right)}\right)}}.$$

(22)

$$H\left(x,t\right)={\displaystyle \frac{-\frac{\alpha p_0\sqrt{-\epsilon }}{\mu \epsilon }\left(\frac{-\sqrt{-\left(m^2+{\xi }^2\right)\epsilon }-m\sqrt{-\epsilon }\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}\right)+q_2{\left(\frac{-\sqrt{-\left(m^2+{\xi }^2\right)\epsilon }-m\sqrt{-\epsilon }\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}\right)}^2}{p_0+\frac{q_2\mu }{\alpha }\sqrt{-\epsilon }\left(\frac{-\sqrt{-\left(m^2+{\xi }^2\right)\epsilon }-m\sqrt{-\epsilon }\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}\right)}}.$$

(23)

$$H\left(x,t\right)={\displaystyle \frac{-\frac{\alpha p_0\sqrt{-\epsilon }}{\mu \epsilon }\left(\sqrt{-\epsilon }+\frac{-2\xi \sqrt{-\epsilon }}{-\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)+\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)+k}\right)+q_2{\left(\sqrt{-\epsilon }+\frac{-2\xi \sqrt{-\epsilon }}{-\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)+\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)+k}\right)}^2}{p_0+\frac{q_2\mu }{\alpha }\sqrt{-\epsilon }\left(\sqrt{-\epsilon }+\frac{-2\xi \sqrt{-\epsilon }}{-\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)+\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)+k}\right)}}.$$

(24)

$$H\left(x,t\right)={\displaystyle \frac{-\frac{\alpha p_0\sqrt{-\epsilon }}{\mu \epsilon }\left(-\sqrt{-\epsilon }+\frac{2\xi \sqrt{-\epsilon }}{\xi +\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)-\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}\right)+q_2{\left(-\sqrt{-\epsilon }+\frac{2\xi \sqrt{-\epsilon }}{\xi +\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)-\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}\right)}^2}{p_0+\frac{q_2\mu }{\alpha }\sqrt{-\epsilon }\left(-\sqrt{-\epsilon }+\frac{2\xi \sqrt{-\epsilon }}{\xi +\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)-\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}\right)}}.$$

(25)

$$H\left(x,t\right)={\displaystyle \frac{-\frac{\alpha p_0\sqrt{-\epsilon }}{\mu \epsilon }\left(\frac{\sqrt{\left({\mathsf{\Theta }}^2-{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{\epsilon }\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}\right)+q_2{\left(\frac{\sqrt{\left({\mathsf{\Theta }}^2-{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{\epsilon }\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}\right)}^2}{p_0+\frac{q_2\mu }{\alpha }\sqrt{-\epsilon }\left(\frac{\sqrt{\left({\mathsf{\Theta }}^2-{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{\epsilon }\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}\right)}}.$$

(26)

$$H\left(x,t\right)={\displaystyle \frac{-\frac{\alpha p_0\sqrt{-\epsilon }}{\mu \epsilon }\left(\frac{-\sqrt{\left({\mathsf{\Theta }}^2-{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{\epsilon }\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}{\mathsf{\Theta }\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)+\xi }\right)+q_2{\left(\frac{-\sqrt{\left({\mathsf{\Theta }}^2-{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{\epsilon }\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}{\mathsf{\Theta }\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)+\xi }\right)}^2}{p_0+\frac{q_2\mu }{\alpha }\sqrt{-\epsilon }\left(\frac{-\sqrt{\left({\mathsf{\Theta }}^2-{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{\epsilon }\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}{\mathsf{\Theta }\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)+\xi }\right)}}.$$

(27)

$$H\left(x,t\right)={\displaystyle \frac{-\frac{\alpha p_0\sqrt{-\epsilon }}{\mu \epsilon }\left(i\sqrt{\epsilon }+\frac{-2\xi i\sqrt{\epsilon }}{\xi +\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)-i\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}\right)+q_2{\left(i\sqrt{\epsilon }+\frac{-2\xi i\sqrt{\epsilon }}{\xi +\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)-i\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}\right)}^2}{p_0+\frac{q_2\mu }{\alpha }\sqrt{-\epsilon }\left(i\sqrt{\epsilon }+\frac{-2\xi i\sqrt{\epsilon }}{\xi +\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)-i\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}\right)}}.$$

(28)

$$H\left(x,t\right)={\displaystyle \frac{-\frac{\alpha p_0\sqrt{-\epsilon }}{\mu \epsilon }\left(-i\sqrt{\epsilon }+\frac{2\xi i\sqrt{\epsilon }}{\xi +\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +Q\right)\right)+i\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +Q\right)\right)}\right)+q_2{\left(i\sqrt{\epsilon }-\frac{2\xi i\sqrt{\epsilon }}{\xi +\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +Q\right)\right)+i\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +Q\right)\right)}\right)}^2}{p_0+\frac{q_2\mu }{\alpha }\sqrt{-\epsilon }\left(-i\sqrt{\epsilon }+\frac{2\xi i\sqrt{\epsilon }}{\xi +\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +Q\right)\right)+i\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +Q\right)\right)}\right)}}.$$

(29)

Here, $\chi =kx-\sqrt{-{\displaystyle \frac{-2v^2\epsilon +{\alpha }^2}{2\epsilon }}}{\displaystyle \frac{\Gamma \left(n+1\right)}{P}}{\mathrm{t}}^P$.

$$\mathrm{Case}\text{-}02: k=\sqrt{-{\displaystyle \frac{-4k^2\epsilon +{\alpha }^2}{4\epsilon }}}, p_0=0,p_1={\displaystyle \frac{q_2\mu }{\alpha }}\sqrt{2\epsilon },q_0=q_2\epsilon , q_1=0.$$

For $\epsilon <0,$

$$H\left(x,t\right)={\displaystyle \frac{q_2\epsilon +q_2{\left(\frac{\sqrt{-\left({\mathsf{\Theta }}^2+{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{-\epsilon }\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sinh}\left(2\sqrt{-\epsilon }\left(\chi +l\right)\right)}\right)}^2}{\frac{q_2\mu }{\alpha }\sqrt{2\epsilon }\left(\frac{\sqrt{-\left({\mathsf{\Theta }}^2+{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{-\epsilon }\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sinh}\left(2\sqrt{-\epsilon }\left(\chi +l\right)\right)}\right)}}.$$

(30)

$$H\left(x,t\right)={\displaystyle \frac{q_2\epsilon +q_2{\left(\frac{-\sqrt{-\left(m^2+{\xi }^2\right)\epsilon }-m\sqrt{-\epsilon }\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}\right)}^2}{\frac{q_2\mu }{\alpha }\sqrt{2\epsilon }\left(\frac{-\sqrt{-\left(m^2+{\xi }^2\right)\epsilon }-m\sqrt{-\epsilon }\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}\right)}}.$$

(31)

$$H\left(x,t\right)={\displaystyle \frac{q_2\epsilon +q_2{\left(\sqrt{-\epsilon }+\frac{-2\xi \sqrt{-\epsilon }}{-\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)+\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)+k}\right)}^2}{\frac{q_2\mu }{\alpha }\sqrt{2\epsilon }\left(\sqrt{-\epsilon }+\frac{-2\xi \sqrt{-\epsilon }}{-\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)+\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)+k}\right)}}.$$

(32)

$$H\left(x,t\right)={\displaystyle \frac{q_2\epsilon +q_2{\left(-\sqrt{-\epsilon }+\frac{2\xi \sqrt{-\epsilon }}{\xi +\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)-\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}\right)}^2}{\frac{q_2\mu }{\alpha }\sqrt{2\epsilon }\left(-\sqrt{-\epsilon }+\frac{2\xi \sqrt{-\epsilon }}{\xi +\mathit{cosh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)-\mathit{sinh}\left(2\sqrt{-\epsilon }\left( \chi +l\right)\right)}\right)}}.$$

(33)

For $\epsilon >0,$

$$H\left(x,t\right)={\displaystyle \frac{q_2\epsilon +q_2{\left(\frac{\sqrt{\left({\mathsf{\Theta }}^2-{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{\epsilon }\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}\right)}^2}{\frac{q_2\mu }{\alpha }\sqrt{2\epsilon }\left(\frac{\sqrt{\left({\mathsf{\Theta }}^2-{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{\epsilon }\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}{\xi +\mathsf{\Theta }\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}\right)}}.$$

(34)

$$H\left(x,t\right)={\displaystyle \frac{q_2\epsilon +q_2{\left(\frac{-\sqrt{\left({\mathsf{\Theta }}^2-{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{\epsilon }\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}{\mathsf{\Theta }\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)+\xi }\right)}^2}{\frac{q_2\mu }{\alpha }\sqrt{2\epsilon }\left(\frac{-\sqrt{\left({\mathsf{\Theta }}^2-{\xi }^2\right)\epsilon }-\mathsf{\Theta }\sqrt{\epsilon }\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}{\mathsf{\Theta }\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)+\xi }\right)}}.$$

(35)

$$H\left(x,t\right)={\displaystyle \frac{q_2\epsilon +q_2{\left(i\sqrt{\epsilon }+\frac{-2\xi i\sqrt{\epsilon }}{\xi +\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)-i\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}\right)}^2}{\frac{q_2\mu }{\alpha }\sqrt{2\epsilon }\left(i\sqrt{\epsilon }+\frac{-2\xi i\sqrt{\epsilon }}{\xi +\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)-i\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +l\right)\right)}\right)}}.$$

(36)

$$H\left(x,t\right)={\displaystyle \frac{q_2\epsilon +q_2{\left(-i\sqrt{\epsilon }+\frac{2\xi i\sqrt{\epsilon }}{\xi +\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +Q\right)\right)+i\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +Q\right)\right)}\right)}^2}{\frac{q_2\mu }{\alpha }\sqrt{2\epsilon }\left(-i\sqrt{\epsilon }+\frac{2\xi i\sqrt{\epsilon }}{\xi +\mathit{cos}\left(2\sqrt{\epsilon }\left( \chi +Q\right)\right)+i\mathit{sin}\left(2\sqrt{\epsilon }\left( \chi +Q\right)\right)}\right)}}.$$

(37)

Here, $\chi =\sqrt{-{\displaystyle \frac{-4k^2\epsilon +{\alpha }^2}{4\epsilon }}}x-v{\displaystyle \frac{\Gamma \left(n+1\right)}{P}}{\mathrm{t}}^P$.

5. Figure Analysis

The soliton solution of the M-fractional LGH equation is essential for comprehending nonlinear wave dynamics in diverse physical systems. This equation is essential for understanding superconductivity, phase transitions, and plasma wave interactions. Solitary waves, which preserve their form during propagation, offer insights into steady energy transfer mechanisms inside these systems. The LGH equation in superconductivity elucidates the behavior of order parameters and the production of vortices, which are crucial for comprehending magnetic flux quantization. In plasma physics, it facilitates the analysis of drift cyclotron waves and coherent ion cyclotron structures in radially inhomogeneous media. Identifying precise solitary wave solutions enables researchers to investigate stability criteria, energy conservation, and the interactions of nonlinear waves. These answers also make it easier to look at integrable models and mathematical physics more ultimately. This helps make progress in optical communications, condensed matter physics, and high-energy field theories. Toward this point, we study the time M-fractional LGH equation through MFE and NIK methods. By using these methods, the obtained solutions of the M-fLGH equation are presented as rational, exponential, hyperbolic, and trigonometric function forms. The behaviors of the obtained solutions are kink-shape soliton, anti-kink-shape soliton, periodic lump wave, linked-lump wave soliton, bright periodic soliton, interaction between anti-kink and periodic lump soliton, and multi-lump soliton solutions. The obtained behavior has a significant role in describing different properties of the LGH equation. Kink solitons are a change between two different equilibrium states. They are often seen in plasma physics, especially in ion cyclotron waves. Kink solitons are a type of nonlinear disturbance that can be transmitted steadily in magnetized plasmas. They allow energy to be transferred without spreading out. They are essential in characterizing nonlinear drift waves and extensive plasma structures, which affect the distribution of electrostatic potential in restricted plasma systems. Anti-kink solitons exhibit behavior analogous to kink solids; however, they do so with an inverted profile, signifying a reverse transition between two states. In ion cyclotron waves, anti-kink solitons can show opposite nonlinear wavefronts or reverse energy flows in a magnetized plasma. Their existence means that there is a lot of concentrated energy, which changes the electrostatic potential and makes plasma formations more stable in fusion and space plasma situations. Constrained, non-singular wave formations, known as periodic lump waves, maintain their periodicity while propagating in a nonlinear medium. In ion cyclotron waves, periodic lump waves signify coherent energy structures affected by wave–particle interactions. The fact that they are involved in changes in electrostatic potential suggests that they could be used to control wave turbulence and improve plasma confinement, especially in laboratory and astrophysical plasma environments. Bright periodic solitons refer to confined, oscillatory wave packets that travel without dispersion. In ion cyclotron plasma waves, bright solitons mark places where the electrostatic field changes intensity more than usual. These areas are often connected to mechanisms that heat the plasma more. These structures are very important for understanding plasma instability and can be used to make wave-based plasma heating methods that can be used for controlled fusion studies. An anti-kink soliton and a periodic lump soliton can interact in ion cyclotron waves to make complex systems for moving energy around. These interactions may lead to localized instabilities or energy redistribution in plasma systems. This interaction in electrostatic potential analysis shows nonlinear resonance events that are very important for understanding how to control turbulence, move ions, and connect plasma waves in fusion reactors and space plasmas. By looking at these nonlinear wave patterns, scientists can learn a lot about how waves and particles interact, how energy is distributed, and how magnetized plasmas stay stable. These findings can be used for both astrophysical and laboratory plasma studies. The effect of fractional parameters on the obtained solutions is presented through three-dimensional density diagrams and two-dimensional diagrams.

5.1. Figure Analysis of the Modified F-Expansion Method

In this subsection, we present some phenomena of the solution of LGH obtained numerically and graphically by the MFE method and also present the effect of the fractional parameters and the following parameters: $\alpha$, related to the coherence length, which determines the size of superconducting domains; and $\mu$, associated with the strength of nonlinear self-interactions of the order parameter, affecting vortex formation. We present the effect of M-fractional parameters, and the effect of the nonlinear coefficient $\mu$, (which is associated with the strength of nonlinear self-interactions of the order parameter, affecting vortex formation) in Figure 1, Figure 2 and Figure 3 and Figure 4 and Figure 5; Figure 6 and Figure 7 and Figure 8 and Figure 9 respectively. For $\mu <0$ or $\alpha >0$, solution Equations (11)–(20) provide the anti-kink solution or dark-type solution. But for $\mu >0$ or $\alpha <0$, solution Equations (11)–(20) provide the kink-type solution or bright-type solution.

For $\mu >0$ or $\alpha >0$, we obtain the kink-shape solution and for $\mu >0$ or $\alpha >0$, the anti-kink-shape soliton solutions are obtained in Figure 1 and Figure 2, respectively. In Figure 1, we present the fractional parameters on the anti-kink-shape wave of solution Equation (11) for the values $v=1,\mu =-1,\alpha =-1,n=0.5$. In Figure 2, we present the fractional parameters on the kink-shape wave of solution Equation (11) for the values $v=1,\mu =1,\alpha =-1,n=0.5$.

In Figure 3 and Figure 4, we present the effect of $\mu$ on the obtained solution Equation (11) for the values $v=1,P=0.8, \alpha =-1,n=0.5$. For $\mu >0$, we obtain the kink-shaped wave. In Figure 3, the amplitude decreased with an increased value of $\mu$, similarly to in Figure 4.

In Figure 5, solution Equation (14) presents the anti-kinky-periodic wave for the parameters. Here, we present the effect of the fractional parameter $\beta$ for $0.2, 0.4,$ and $0.8$. In Figure 5, we show that the periodicity increases but the amplitude of the kink shape is constant with the values of the fractional parameter $\beta$. For $\mu <0$ or $\alpha >0$, solution Equations (11)–(15) provide the anti-kink solution or dark-type solution. But for $\mu >0$ or $\alpha <0$, solution Equations (11)–(15) provide the kink-type solution or bright-type solution.

In Figure 6, solution Equation (14) presents a periodic wave for the parameters. Here, we present the effect of the fractional parameter $\beta$ for $0.2, 0.4,$ and $0.8$., and we show that the periodicity increases but the amplitude of the periodic wave is decreasing with the values of the fractional parameter $\beta$.

In Figure 7, solution Equation (14) presents a periodic wave for the parameters. Here, we present the effect of the fractional parameter $\beta$ for $0.2, 0.4,$ and $0.8$., and we show that the periodicity increases but the amplitude of the periodic wave also increases with the values of the fractional parameter $\beta$.

In Figure 8, we present the effect of the nonlinear coefficient ($\mu =-0.1,-0.5,-1$) on the solution in Equation (14), which presents an anti-kinky-periodic wave for the parameters. In 3D and 2D diagrams, we show that the periodicity of the periodic wave is increased and the amplitude of the anti-kink-shape wave is also increased with the values of the fractional parameter $\mu$. The effect of $\mu$ on the periodic wave solution of the imaginary and absolute form of Equation (14) is shown in Figure 9 with a 2D diagram. Decreasing the value of $\mu$, the periodicity and amplitude both decreases.

5.2. Figure Analysis of the Newly Improved Kudryashov’s Method

At this subpoint, we numerically and graphically analyze the phenomena associated with the obtained solutions of the time M-fraction LGH equation using the newly improved Kudryashov’s method. Specifically, we examine the effects of the fractional parameter and the parameters $\alpha$ and $\mu$, where $\alpha$ is related to the coherence length, determining the size of superconducting domains, and $\mu$ represents the strength of nonlinear self-interactions of the order parameter, influencing vortex formation. We present the effect of the nonlinear coefficient $\mu$ and the effect of M-fractional parameters on the soliton solutions in Figures 10, 14–16, 20 and 11–13, 17–20 respectively. The analysis reveals that for $\mu <0$ or $\alpha >0$, the solutions in Equations (22)–(37) correspond to anti-kink or dark-type solitons. Conversely, for $\mu >0$ or $\alpha <0$, the solutions exhibit kink-type or bright soliton characteristics.

For $\mu >0$, solution Equation (22) represents the kink-shape solution for the values $\mathsf{\Theta }=1,\mathsf{\xi }=-5, p_0=0.5,q_2=1,\epsilon =-0.5,v=-1,h=0.01,\alpha =1,P=0.8,n=0.5$.

In Figure 10, we present the effect of $\mu$ on the kink-shape solution of Equation (22). The amplitude decreases with an increased value of $\mu$.

In Figure 11, we present the fractional parameters’ ($P$) effect on the kink-shape solution of Equation (22) for the values $\mathsf{\Theta }=1,\mathsf{\xi }=0.5, p_0=0.5,q_2=1,\epsilon =-0.5,v=-1,h=0.01,\alpha =1,\mu =1,n=0.5$. The amplitude is constant but the position changes with the reduced curve form with an increased value of $P$.

For $\mu <0$, solution Equation (22) represents the anti-kink-shape solution for the values $\mathsf{\Theta }=1,\mathsf{\xi }=0.5, p_0=0.5,q_2=1,\epsilon =-0.5,v=-1,h=0.01,\alpha =1,\mu =-1,n=0.5$.

In Figure 12, we present the fractional parameters’ effect on the kink-shape solution of Equation (22) for $\mu =-1<0$. The amplitude is constant but the position changes with the reduced curve form with an increased value of $P$.

In Figure 13, we present the fractional parameters’ ($P$) effect on the soliton solution of Equation (24) for the values $\mathsf{\Theta }=-1,\mathsf{\xi }=0.5, p_0=-0.5,q_2=-1,\epsilon =-1,v=-0.1,h=0.01,\alpha =0.5,\mu =-1,n=0.5$. The amplitude is constant but the position changes with the reduced curve form with an increased value of $P$. In Figure 14, we present the effect of $\mu$ on the soliton solution of Equation (24). The amplitude decreases with an increased value of $\mu$.

Solution Equation (30) represents a periodic wave solution for the values $\mathsf{\Theta }=0.5,\mathsf{\xi }=0.33, q_2=0.5,\epsilon =0.5,k=-1,h=0.5,\alpha =1,\mu =-1,n=0.5$.

In Figure 15, we present the fractional parameters’ ($P$) effect on the periodic wave solution of Equation (30). The amplitude is constant but the periodicity increases with an increased value of $P$. In Figure 16, we present the effect of $\mu$ on the periodic wave solution of Equation (30). The amplitude decreases and the periodicity is constant with an increased value of $\mu$.

Solution Equation (30) represents a periodic multi-lump wave solution for the values $\mathsf{\Theta }=0.5,\mathsf{\xi }=2, q_2=0.5,\epsilon =0.5,k=-1,h=0.5,\mu =1,\alpha =1,n=0.5$. In Figure 17, we present the fractional parameters’ ($P$) effect on the periodic multi-lump wave solution of Equation (30). The amplitude is constant but the periodicity increases with an increased value of $P$. In Figure 18, we present the effect of $\mu$ on the periodic multi-lump wave solution of Equation (30) for the values $\mathsf{\Theta }=0.5,\mathsf{\xi }=2, q_2=0.5,\epsilon =0.5,k=-1,h=0.5,P=0.8,\alpha =1,n=0.5$. The amplitude decreases and the periodicity is constant with an increased value of $\mu$.

Solution Equation (30) represents a periodic lump wave solution for the values $\mathsf{\Theta }=0.5,\mathsf{\xi }=2, q_2=0.5,\epsilon =0.5,k=-1,h=0.5,\mu =1,\alpha =1,n=0.5$. In Figure 19, we present the fractional parameters’ ($P$) effect on the periodic multi-lump wave solution of Equation (30). The amplitude is constant but the depth of the lump decreases with an increased value of $P$. In Figure 20, we present the effect of $\mu$ on the periodic multi-lump wave solution of Equation (30) for the values $\mathsf{\Theta }=0.5,\mathsf{\xi }=2, q_2=0.5,\epsilon =0.5,k=-1,h=0.5,P=0.8,\alpha =1,n=0.5$. The amplitude decreases and the periodicity is constant with an increased value of $\mu$.

A summary of the figurative analysis is given in

Table 1. Summary of various types of soliton solutions and parameter setup.

Figure Type	Corresponding Equation	Parameter Values
Anti-kink soliton	Equation (11)	$v=1,\mu =-1,\alpha =-1,n=0.5$.
Kink soliton	Equation (11)	$v=1,\mu =1,\alpha =-1,n=0.5$.
Interaction between anti-kink and periodic lump soliton	Equation (14)	$v=-1,\mu =-1,\alpha =1,n=0.5$.
Periodic linked-lump soliton	Equation (14)	$v=-1,\mu =-1,\alpha =1,n=0.5$.
Interaction between anti-kink and periodic lump soliton	Equation (14)	$v=-1,P=0.8,\alpha =1,n=0.5$.
Interaction between anti-kink and periodic lump soliton.	Equation (14)	$v=-1,P=0.8,\alpha =1,n=0.5$.
Singular soliton	Equation (24)	$\mathsf{\Theta }=-1,\mathsf{\xi }=0.5, p_0=-0.5,q_2=-1,\epsilon =-1,v=-0.1,h=0.01,\alpha =0.5,\mu =-1,n=0.5$.
Bright periodic soliton	Equation (30)	$\mathsf{\Theta }=0.5,\mathsf{\xi }=0.33, q_2=0.5,\epsilon =0.5,k=-1,h=0.5,\alpha =1,\mu =-1,n=0.5$.
Periodic multi-lump soliton	Equation (30)	$\mathsf{\Theta }=0.5,\mathsf{\xi }=2, q_2=0.5,\epsilon =0.5,k=-1,h=0.5,\alpha =1,n=0.5,P=0.8$.

.

6. Modulation Instability

The modulation instability [61] of the Landau–Ginzburg–Higgs (LGH) equation is crucial for comprehending nonlinear wave events, pattern generation, and spontaneous symmetry breaking in diverse physical systems. It aids in forecasting the evolution of minor disturbances, resulting in intricate wave formations such as solitons or breathers. In condensed matter physics, the LGH equation describes superconductivity and superfluidity. Modulation instability, on the other hand, describes phase changes and the appearance of defects like vortices. The instability is important for studying spontaneous symmetry breaking, which is a basic idea in cosmology and high-energy physics, especially when it comes to the Higgs process. It also makes it easier to understand how soliton and breather are made, which is important in fields like quantum field theory and nonlinear optics. The modulation instability in the LGH equation tells us a lot about the stability and dynamics of many different systems, from quantum field theory to condensed matter. This helps us understand more complicated, nonlinear phenomena.

In this section, we analyze the modulation instability of traveling waves in Equation (1). Let us consider the perturbed solution expressed by

$$H=\eta u+\tau .$$

(38)

The constant $\tau$ is known as a steady-state solution for Equation (1).

Insert Equation (38) into Equation (1),

$$\eta u_{tt}-\eta u_{xx}-{\beta }^2\eta u+{\mu }^2{\left(\eta u+\tau \right)}^3=0.$$

(39)

By linearizing, we obtain

$$\eta u_{tt}-\eta u_{xx}-{\beta }^2\eta u+3\eta u{\mu }^2{\tau }^2=0.$$

(40)

Let the solution of Equation (40) be $u\left(x,t\right)=e^{i(kx-v{\displaystyle \frac{\Gamma \left(n+1\right)}{P}}{t}^P)}$

Here, $k$ is frequency and $v$ is the wave speed. Now, putting the solution into Equation (40),

$$-v^2+k^2-{\beta }^2+3{\mu }^2{\tau }^2=0.$$

$$=>v=\sqrt{k^2-{\beta }^2+3{\mu }^2{\tau }^2}.$$

(41)

It is clear from Equation (41) that any combination of the solutions will appear to decay for negative values of $v$, and the dispersion remains stable.

The gain spectrum of modulation instability (MI) for various values of the parameters is manifested in Figure 21 and Figure 22.

7. Comparison and Novelty of This Work

In this section, we compare our findings by modified F-expansion and the newly improved Kudryashov’s methods with the findings of Barman et al. [44] and Iftikhar et al. [46] solutions. The extended tanh method and two-variable method for the Landau–Ginzburg–Higgs equation are used.

7.1. Comparison with the Existing Works

Barman et al. [44] examined diverse wave shapes in the LGH equation, employing the extended Tanh approach, and discerned just hyperbolic solutions. The obtained solutions encompass brilliant and dark solitons, peakon-type, compact, and periodic waveforms in numerical representation. In a similar vein, Iftikhar et al. [46] investigated soliton solutions for the traditional version of Equation (1), utilizing the two-variable method, and derived just two solutions (see Ref. [44]). These answers pertain to two separate categories of singular waveforms in numerical form. Conversely, utilizing the unified approach delineated in this study, we discerned sixteen solutions to Equation (1). The Riccati equation is distinct between the two methodologies.

7.2. Novelty of This Work

In this work, we present some phenomena of the obtained solution of LGH by the modified F-expansion and the newly improved Kudryashov’s methods both numerically and graphically and also present the effect of the fractional parameters as well as the parameters $\alpha$ that are related to the coherence length, which determines the size of superconducting domains, and $\mu$, which is associated with the strength of nonlinear self-interactions of the order parameter, affecting vortex formation.

By implementing the modified F-expansion method, the solutions are presented as rational, hyperbolic, exponential, and trigonometric function forms. The behaviors of the obtained solutions are kink-shape soliton, anti-kink-shape soliton, periodic lump wave, linked-lump wave soliton, bright periodic soliton, the interaction between anti-kink and periodic lump soliton, etc.

Utilizing the newly improved Kudryashov method, the resultant solutions are articulated in the form of rational, hyperbolic, exponential, and trigonometric functions. The resultant wave structures display varied behaviors, encompassing kink-shaped and anti-kink-shaped solitons, bright periodic solitons, singular soliton solutions, periodic lump waves, and periodic multi-lump solitons. These discoveries enhance comprehension of the model’s dynamics and the diverse array of wave events it can accommodate. The prior discussion indicates that certain solutions have been derived for the first time within the framework of this paradigm. This study constitutes the first effort to systematically assess and compare the impact of various fractional parameters on the resultant solutions. By altering these parameters, we have examined their impact on the wave structures, yielding an enhanced understanding of the model’s behavior under fractional-order alterations.

8. Conclusions

This manuscript successfully examined the M-fractional Landau–Ginzburg–Higgs (LGH) equation by using modified F-expansion and the newly improved Kudryashov’s methods. The obtained solutions are significant in describing diverse phenomena in comprehending drift cyclotron waves and superconductivity in radially inhomogeneous plasmas. By implementing the modified F-expansion and the newly improved Kudryashov’s methods, we can obtain precise soliton solutions and examine their stability under a range of parametric conditions. For the special values of free parameters, we derive some novel phenomena such as kink wave, anti-kink wave, periodic lump wave with soliton, interaction of kink and periodic lump wave, interaction of anti-kink and periodic wave, periodic wave, solitonic wave, multi-lump wave in periodic form, and so on. The effects of fractional parameter $\beta$; the parameter α, which is related to the coherence length, which determines the size of superconducting domains; and $\mu$, associated with the strength of nonlinear self-interactions of the order parameter, affecting vortex formation on the obtained solution, are investigated with three- and two-dimensional diagrams. The stability of these solutions is further assessed by the modulation instability criterion, which provides a further in-depth understanding of fractional LGH models with applications in fluid dynamics, optical communication, and plasma physics. This study advances the theoretical understanding of fractional LGH models and provides valuable insights for practical applications in plasma physics, optical communication, and fluid dynamics.