Explicit solutions of the generalized Kudryashov’s equation with truncated M-fractional derivative

Abstract

The main purpose of this article is to study the generalized Kudryashov’s equation with truncated M-fractional derivative, which is commonly used to describe the propagation of wide pulses in nonlinear optical fibers. By employing the complete discriminant system of fourth-order polynomials, various types of explicit solutions are systematically classified, which include periodic solutions, the trigonometric functions, the double-period solutions, and the elliptic function solutions. Additionally, a series of 2D, 3D, and contour plots are generated to visually depict the spatial distribution and evolution of various solutions. This not only advances the development of nonlinear equations in theory but also provides valuable guidance in practical applications.

Introduction

Nonlinear science is a discipline that studies nonlinear phenomena^1–11. Many nonlinear sciences are modeled through nonlinear fractional partial differential equations (NLFPDE)^12–14. However, constructing explicit solutions for NLFPDE have been a hot topic of research by many experts and scholars in recent years. Researchers worldwide are committed to continually proposing new methods and improving existing ones, gradually offering effective approaches for constructing exact solutions to NLFPDE. With the further development of symbolic computing software such as Matlab, Maple, and Mathematica¹⁵, the construction of analytical solutions for NLFPDE have been promoted, which is a powerful computational tools for the study of NLFPDE. In recent years, many effective methods have been proposed to construct explicit solutions for this type of equation. These methods mainly include: the modified extended tanh-function method¹⁶, the He-Laplace algorithm¹⁷, the trial solution method¹⁸, the sine-Gordon expansion method¹⁹, the improved $(G^{\prime }/G)$ method²⁰, the $(G^{\prime }/G)$-expansion method²¹, the complete discrimination system for polynomial method²², the Exp-function method²³.

The generalized Kudryashov’s equation with truncated M-fractional derivative is described as follows²⁴

$$\begin{array}{c}\hfill \begin{array}{c}i{D}_{M,t}^{\mu ,\beta }u+a{D}_{M,x}^{2\mu ,\beta }u+(\frac{w_1}{{|u|}^{4n}}+\frac{w_2}{{|u|}^{3n}}+\frac{w_3}{{|u|}^{2n}}+\frac{w_4}{{|u|}^n}+w_5{|u|}^n+w_6{|u|}^{2n}+w_7{|u|}^{3n}+w_8{|u|}^{4n})u\\ =i[\lambda D_{M,x}^{\mu ,\beta }{(u|u|}^{2m})+\theta u{D}_{M,x}^{\mu ,\beta }{|u|}^{2m}+{\delta |u|}^{2m}{D}_{M,x}^{\mu ,\beta }u],0<\mu <1,\beta >0,\end{array}\end{array}$$ 1.1

where $u=u(t,x)$ is a complex-valued wave profile. $D_{M,t}^{\mu ,\beta }$ represents the truncated M-fractional derivative. a stands for the coefficient of chromatic dispersion. $w_i$ $(1\le i\le 8)$ are the coefficient of self-phase modulation. $\lambda$ and $\theta$ represents the coefficient of self-steepening for short pules and the higher-order dispersion, respectively. m and n ($0<n<\frac{1}{2}$) stand for the maximum intensity and the power law non-linearity. The Kudryashov’s equation of integer order represented in²⁵. The main purpose of this article is to study the generalized Kudryashov’s equation with truncated M-fractional derivative. By using the complete discrimination system for polynomial method, a series of explicit solutions to Eq. (1.1) will be obtained.

Definition 1.1

^26,27 Let be $f(t):[0,\infty )\to (-\infty ,+\infty )$. Then, the truncated M-fractional derivative is given by

$$\begin{array}{c}\hfill D_{M,t}^{\mu ,\beta }f=\underset{\tau \to \infty }{lim}\frac{f(t{E}_{\beta }(\tau t^{1-\tau }))}{\tau },0<\mu <1,\beta >0,\end{array}$$ 1.2

where $E_{\beta }(·)$ stands for the truncated Mittag-Leffler function. $E_{\beta }(z)={\sum }_{j=0}^i\frac{z^j}{\mathrm{\Gamma }({\beta }_j+1)}$, $\beta >0$ and $z\in C$. The properties of truncated M-fractional derivatives can be referred to in reference²⁸.

This paper is organized as follows: In "Mathematical analysis", Eq. (1.1) is transformed into a nonlinear ordinary differential equation. In "Explicit solutions of equation ", the explicit solutions of Eq. (1.1) are obtained. In "Method description", a series of 2D, 3D, and contour plots are plotted. Finally, a brief summary is provided in "Explicit solutions of equation".

Mathematical analysis

Firstly, we consider the wave transformation

$$\begin{array}{c}\hfill u(t,x)=U(\xi )e^{i\vartheta (t,x)},\xi =\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu },\vartheta (t,x)=\frac{(-k{x}^{\mu }+w{t}^{\mu }+{\theta }_0)\mathrm{\Gamma }(\beta +1)}{\mu },\end{array}$$ 2.1

where k, w and ${\theta }_0$ represent the frequency, the wave number and the phase of the solion, respectively.

Plugging Eq. (2.1) into Eq. (1.1), we have the real part

$$\begin{array}{c}\hfill \begin{array}{c}\hfill a{c}^{2}U_{}^{_{}^{\prime \prime }}-(w-ak)U+w_1{U}^{1-4n}+w_2{U}^{1-3n}+w_3{U}^{1-2n}+w_4{U}^{1-n}+w_5{U}^{1+n}+w_6{U}^{1+2n}\\ \hfill +w_7{U}^{1+3n}+w_8{U}^{1+4n}-k(\lambda -\delta )U^{1+2m}=0,\end{array}\end{array}$$ 2.2

and imaginary part

$$\begin{array}{c}\hfill 2ak+v+(2\lambda m+\delta +\lambda +2\theta m)=0.\end{array}$$ 2.3

From Eq. (2.3), we have

$$\begin{array}{c}\hfill 2\lambda m+\delta +\lambda +2\theta m=0\text{and}v=-2ak.\end{array}$$ 2.4

Let be $U=Y^{\frac{1}{2n}}$ and $m=2n$. Then, we can obtain

$$\begin{array}{c}\hfill a{c}^2[(1-2n)(Y_{}^{_{}^{\prime }})_{}^2+2nYY_{}^{_{}^{\prime \prime }}]-4n^2(a{k}^2+\omega )Y^2+4n^2{\omega }_6{Y}^3+4n^2{\omega }_{3}Y+4n^2{\omega }_1+4n^2[{\omega }_8-k(\delta +\lambda )]Y^4=0.\end{array}$$ 2.5

According to the principle of rank homogeneous balance, we assume that the approximate solution of Eq. (2.5) as

$$\begin{array}{c}\hfill Y_{}^{_{}^{\prime \prime }}=a_0+a_{1}Y+a_2{Y}^2+a_3{Y}^3.\end{array}$$ 2.6

Explicit solutions of equation (1.1)

The complete discriminant system method of polynomials is a very important method for constructing traveling wave solutions of nonlinear partial differential equations, which was first proposed by Professor Liu Chengshi²⁹. In recent years, many experts and scholars have applied this method to the solution of traveling wave solutions for nonlinear partial differential equations, fractional order partial differential equations, and stochastic nonlinear partial differential equations.

Method description

Firstly, we present the following general fractional order partial differential equation

$$\begin{array}{c}\hfill P(u,u^2,\cdots ,D_{M,t}^{\mu ,\beta },D_{M,x}^{\mu ,\beta },D_{M,t}^{2\mu ,\beta },D_{M,x}^{2\mu ,\beta },\cdots )=0.\end{array}$$ 3.1

Applying the traveling wave transformation (2.1) to Eq. (3.1) yields

$$\begin{array}{c}\hfill P(U,U^2,\cdots ,U_{}^{_{}^{\prime }},U_{}^{_{}^{\prime \prime }},\cdots )=0.\end{array}$$ 3.2

If Eq. (3.2) can be transformed into the following ordinary differential equation,

$$\begin{array}{c}\hfill (U_{}^{_{}^{\prime }})_{}^2=G(U,d_1,d_2,\cdots ,d_m).\end{array}$$ 3.3

where $d_1,\cdots ,d_m$ are parameters. Then, we have

$$\begin{array}{c}\hfill \xi -{\xi }_0=\pm \int \frac{\mathit{du}}{\sqrt{G(U,d_1,d_2,\cdots ,d_m)}}.\end{array}$$ 3.4

Explicit solutions of equation (1.1)

Integrating both sides of Eq. (2.6) simultaneously once yields

$$\begin{array}{c}\hfill (Y_{}^{_{}^{\prime }})_{}^2=\frac{1}{2}{a}_3{Y}^4+\frac{2}{3}{a}_2{Y}^3+a_1{Y}^2+2a_{0}Y+d,\end{array}$$ 3.5

where d is the integral constant.

Inserting Eqs. (2.6) and (3.5) into Eq. (2.5), we can obtain a polynomial equation

$$\begin{array}{c}\hfill r_4{Y}^4+r_3{Y}^3+r_2{Y}^2+r_{1}Y+r_0=0,\end{array}$$ 3.6

where $r_4=a_{3}a{c}^2(\frac{1}{2}+n)+4n^2[{\omega }_8-k(\delta +\mathrm{\Delta })]$, $r_3=\frac{2}{3}{a}_{2}a{c}^2(1+n)+4n^2{\omega }_6$, $r_2=a_{1}a{c}^2-4n^2(a{k}^2+\omega )$, $r_1=2a_{0}a{c}^2(1-n)+4n^2{\omega }_3$, $r_0=da{c}^2(1-2n)+4n^2{\omega }_1$.

In order to determine $a_j$ $(j=0,...,3)$ and d, we assume that $r_j=0$ $(j=0,...,4)$, and then we obtain $a_3=\frac{8n^2[k(\delta +\mathrm{\Delta })-{\omega }_8]}{a{c}^2(1+2n)}$, $a_2=-\frac{6n^2{\omega }_6}{a{c}^2(1+n)}$, $a_1=\frac{4n^2(a{k}^2+\omega )}{a{c}^2}$, $a_0=-\frac{2n^2{\omega }_3}{a{c}^2(1-n)}$, $d=-\frac{4n^2{\omega }_1}{a{c}^2(1-2n)}$.

When $a_3>0$, we perform the following transformation

$$\begin{array}{c}\hfill W={(\frac{1}{2}{a}_3)}^{\frac{1}{4}}(Y+\frac{a_2}{3a_3}),{\xi }_1={(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\xi .\end{array}$$ 3.7

Inserting Eq. (3.7) into Eq. (3.5), we can obtain

$$\begin{array}{c}\hfill [W_{{\xi }_1}^{_{}^{\prime }}{]}^2=F(W)=W^4+p{W}^2+qW+r,\end{array}$$ 3.8

where $p=\frac{a_1-\frac{a_2^2}{3a_3}}{\sqrt{\frac{1}{2}{a}_3}}$, $q=(\frac{4a_2^3}{27a_3^2}-\frac{2a_1{a}_2}{3a_3}+2a_0){(\frac{1}{2}{a}_3)}^{-\frac{1}{4}}$, $r=-\frac{a_2^4}{54a_3^3}+\frac{a_1{a}_2^2}{9a_3^2}-\frac{2a_0{a}_2}{3a_3}+d$.

When $a_3<0$, we perform the following transformation

$$\begin{array}{c}\hfill \begin{array}{c}\hfill W={(-\frac{1}{2}{a}_3)}^{\frac{1}{4}}(Y+\frac{a_2}{3a_3}){\xi }_1={(-\frac{1}{2}{a}_3)}^{\frac{1}{4}}\xi .\end{array}\end{array}$$ 3.9

Inserting Eq. (3.9) into Eq. (3.5), we can obtain

$$\begin{array}{c}\hfill [W_{{\xi }_1}^{_{}^{\prime }}{]}^2=-F(W)=-(W^4+p{W}^2+qW+r),\end{array}$$ 3.10

where $p=-\frac{a_1-\frac{a_2^2}{3a_3}}{\sqrt{-\frac{1}{2}{a}_3}}$, $q=(-\frac{4a_2^3}{27a_3^2}+\frac{2a_1{a}_2}{3a_3}-2a_0){(-\frac{1}{2}{a}_3)}^{-\frac{1}{4}}$, $r=\frac{a_2^4}{54a_3^3}-\frac{a_1{a}_2^2}{9a_3^2}-\frac{2a_0{a}_2}{3a_3}-d$.

On the basis of the complete discriminant system for the quartic polynomial $f(W)=W^4+p{W}_2+qW+r$. it is presented as $D_1=4$, $D_2=-p$, $D_3=-2p^3+8pr-9q^2$, $D_4=-p^3{q}^2+4p^{4}r+36p{q}^{2}r-32p^2{r}^2-\frac{27}{4}{q}^4+64r^3$, $E_2=9p^2-32pr$.

From Eqs. (3.8) and (3.10), we can obtain

$$\begin{array}{c}\hfill [W_{{\xi }_1}^{_{}^{\prime }}{]}^2=ϵF(W)=ϵ(W^4+p{W}^2+qW+r),\end{array}$$ 3.11

where $ϵ=\pm 1$.

Therefore, we can rewrite its integral expression as

$$\begin{array}{c}\hfill \pm ({\xi }_1-\xi )=\int \frac{\mathit{dW}}{\sqrt{ϵ(W^4+p{W}^2+qW+r)}}.\end{array}$$ 3.12

Based on the complete discriminant system for the polynomial F(W) as given in Eq. (3.12), the classification of all solutions o Eq. (3.12) can be obtained.

Case 1. $D_2<0,D_3=0,D_4=0$. Here, F(W) has a pair of complex conjugate roots with multiplicity two, i.e. $F(W)={[{(W-l)}^2+s^2]}^2$, where l and s are real numbers, here $s>0$.

When $ϵ=1$, we can obtain from Eq. (3.12)

$$\begin{array}{c}\hfill {\xi }_1-{\xi }_0=\int \frac{\mathit{dW}}{{(W-l)}^2+s^2}=\frac{1}{s}arctan\frac{W-l}{s},\end{array}$$ 3.13

where ${\xi }_0$ is the integral constant.

Therefore, the solutions to Eq. (1.1) are given by

$$\begin{array}{c}\hfill u_1(t,x)=[{(\frac{1}{2}{a}_3)}^{-\frac{1}{4}}(stan(s({(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0))+l)-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)}.\end{array}$$

Case 2. $D_2=0,D_3=0,D_4=0$. Here, F(W) has a quadruple real root at zero, i.e. $F(W)=W^4$. Therefore, it follows that from Eq. (3.12) when $a_3>0$

$$\begin{array}{c}\hfill {\xi }_1-{\xi }_0=\int \frac{\mathit{dW}}{W^2}=-W^{-1}.\end{array}$$ 3.14

Therefore, we can derive the solutions to Eq. (1.1)

$$\begin{array}{c}\hfill u_2(t,x)=[-{(\frac{1}{2}{a}_3)}^{-\frac{1}{4}}{({(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0)}^{-1}-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)},\end{array}$$

which represents rational solutions.

Case 3. $D_2>0,D_3=0,D_4=0,E_2>0$. Obviously, F(W) has two distinct real roots with multiplicity two, i.e. $F(W)={(W-\alpha )}^2{(W-\beta )}^2$, where $\alpha$ and $\beta$ are real numbers, here $\alpha >\beta$.

If $ϵ=1$, we can obtain from Eq. (3.12) when $W>\alpha$ or W $<\beta$.

$$\begin{array}{c}\hfill \pm ({\xi }_1-{\xi }_0)=\int \frac{\mathit{dW}}{(W-\alpha )(W-\beta )}=\frac{1}{\alpha -\beta }ln|\frac{W-\alpha }{W-\beta }|.\end{array}$$ 3.15

The explicit solutions of (1.1) can be obtained from Eq. (3.12).

$$\begin{array}{c}\hfill u_3(t,x)=[{(\frac{1}{2}{a}_3)}^{-\frac{1}{4}}[\frac{\beta -\alpha }{2}(coth\frac{(\alpha -\beta )({(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0)}{2}-1)+\beta ]-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)}.\end{array}$$

When $\alpha >W>\beta$, the explicit solutions of (1.1) can be obtained from Eq. (3.12)

$$\begin{array}{c}\hfill u_4(t,x)=[{(\frac{1}{2}{a}_3)}^{-\frac{1}{4}}(\frac{\beta -\alpha }{2}(tanh\frac{(\alpha -\beta )({(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0)}{2}-1)+\beta )-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)}.\end{array}$$

Case 4. $D_2>0,D_3>0,D_4=0$. Clearly, F(W) has a double real root and two distinct single real roots, i.e. $F(W)=(W-\beta )(W-\gamma ){(W+\frac{\beta +\gamma }{2})}^2$, where $\beta$ and $\gamma$ are real numbers, here $\beta >\gamma ,\gamma \ne -3\beta ,\gamma \ne -\frac{\beta }{3}$.

Denote $\theta =(3\beta +\gamma )(\beta +3\gamma )$, ${\theta }_1=\frac{1}{2}{(\frac{1}{2}{a}_3)}^{-\frac{1}{4}}(\beta +\gamma )+\frac{a_2}{3a_3}$, ${\theta }_2=\frac{1}{2}{(-\frac{1}{2}{a}_3)}^{-\frac{1}{4}}(\beta +\gamma )+\frac{a_2}{3a_3}$.

When $a_3>0$, $W>\beta$ or $W<\gamma$, if $\gamma \le 0\text{and}\gamma <\beta <-\frac{\gamma }{3}\text{or}\beta >-3\gamma$ or $\beta >\gamma >0$, the explicit solution of (1.1) can be obtained

$$\begin{array}{c}\hfill u_5(t,x)=[\frac{{(\frac{1}{2}{a}_3)}^{-\frac{1}{4}}\theta }{2(\beta -\gamma )cosh[\frac{\sqrt{\theta }}{2{(\frac{1}{2}{a}_3)}^{-\frac{1}{4}}}(\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0)]+4(\beta +\gamma )}-{\theta }_1]e^{i\vartheta (t,x)}.\end{array}$$

If $-3\beta <\gamma <-\frac{\beta }{3}<0$, the solution of Eq. (1.1) is

$$\begin{array}{c}\hfill u_6(t,x)=[\frac{{(\frac{1}{2}{a}_3)}^{-\frac{1}{4}}\theta }{4(\beta +\gamma )\pm 2(\beta -\gamma )sin[\frac{\sqrt{-\theta }}{2{(\frac{1}{2}{a}_3)}^{-\frac{1}{4}}}(\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0)]}-{\theta }_1]e^{i\vartheta (t,x)},\end{array}$$

when $a_3<0$, $\beta >W>\gamma$.

If $0<-\frac{\gamma }{3}<\beta <-3\gamma$, the solution of Eq. (1.1) is

$$\begin{array}{c}\hfill u_7(t,x)=[\frac{-{(-\frac{1}{2}{a}_3)}^{-\frac{1}{4}}\theta }{2(\beta -\gamma )cosh[\frac{\sqrt{-\theta }}{2{(-\frac{1}{2}{a}_3)}^{-\frac{1}{4}}}(\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0)]-4(\beta +\gamma )}-{\theta }_2]e^{i\vartheta (t,x)}.\end{array}$$

If $\gamma \le 0\text{and}\gamma <\beta <-\frac{\gamma }{3}\text{or}\beta >-3\gamma$ or $\beta >\gamma >0$, the solution of Eq. (1.1) is

$$\begin{array}{c}\hfill u_8(t,x)=[\frac{-{(-\frac{1}{2}{a}_3)}^{-\frac{1}{4}}\theta }{\pm 2(\beta -\gamma )sin[\frac{\sqrt{\theta }}{2{(-\frac{1}{2}{a}_3)}^{-\frac{1}{4}}}(\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0)]-4(\beta +\gamma )}-{\theta }_2]e^{i\vartheta (t,x)}.\end{array}$$

Case 5.$D_2>0,D_3>0,D_4>0$. Obviously, F(W) has four real roots, i.e. $F(W)=(W-{\beta }_1)(W-{\beta }_2)(W-{\beta }_3)(W-{\beta }_4)$, where ${\beta }_1,{\beta }_2,{\beta }_3,{\beta }_4$ are real numbers. ${\beta }_1>{\beta }_2>{\beta }_3>{\beta }_4$ and ${\beta }_1+{\beta }_2+{\beta }_3+{\beta }_4=0$.

Denote $m_1=\sqrt{\frac{({\beta }_2-{\beta }_3)({\beta }_1-{\beta }_4)}{({\beta }_1-{\beta }_3)({\beta }_2-{\beta }_4)}},m_2=\sqrt{\frac{({\beta }_1-{\beta }_2)({\beta }_3-{\beta }_4)}{({\beta }_1-{\beta }_3)({\beta }_2-{\beta }_4)}},\theta =\frac{1}{2}\sqrt{({\beta }_1-{\beta }_3)({\beta }_2-{\beta }_4)}$. When $a_3>0$, if $W<{\beta }_4$ or $W>{\beta }_1$, we make the following transformation:

$$\begin{array}{c}\hfill W=\frac{{\beta }_2({\beta }_1-{\beta }_4){sin}^2\phi -{\beta }_1({\beta }_2-{\beta }_4)}{({\beta }_1-{\beta }_4){sin}^2\phi -({\beta }_2-{\beta }_4)}.\end{array}$$

If ${\beta }_3<W<{\beta }_2$, we make the following transformation:

$$\begin{array}{c}\hfill W=\frac{{\beta }_4({\beta }_2-{\beta }_3){sin}^2\phi -{\beta }_3({\beta }_2-{\beta }_4)}{({\beta }_2-{\beta }_3){sin}^2\phi -({\beta }_2-{\beta }_4)}.\end{array}$$ 3.16

From Eq. (3.12), we obtain

$$\begin{array}{c}\hfill {\xi }_1-{\xi }_0=\int \frac{\mathit{dW}}{\sqrt{(W-{\beta }_1)(W-{\beta }_2)(W-{\beta }_3)(W-{\beta }_4)}}=\frac{1}{\theta }\int \frac{d\phi }{\sqrt{1-m_1^2{sin}^2\phi }}.\end{array}$$ 3.17

By using Eq. (3.17) and the definition of the Jacobian elliptic sine function, we can obtain

$$\begin{array}{c}\hfill \mathbf{sn}[\frac{1}{2}\sqrt{({\beta }_1-{\beta }_3)({\beta }_2-{\beta }_4)}({\xi }_1-{\xi }_0),m_1]=\mathbf{sn}(\theta {\xi }_1-{\xi }_0,m_1)=sin\phi .\end{array}$$

If $W<{\beta }_4$, the solution of Eq. (1.1)

$$\begin{array}{c}\hfill u_9(t,x)=[\frac{({\beta }_3-{\beta }_1){\beta }_4+{\beta }_3({\beta }_1-{\beta }_4){\mathbf{sn}}^2(\theta {(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0,m_1)}{{(\frac{1}{2}{a}_3)}^{\frac{1}{4}}({\beta }_3-{\beta }_1+({\beta }_1-{\beta }_4){\mathbf{sn}}^2(\theta {(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0,m_1))}-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)}.\end{array}$$

If $W>{\beta }_1$, the solution of Eq. (1.1)

$$\begin{array}{c}\hfill u_{10}(t,x)=[\frac{({\beta }_4-{\beta }_2){\beta }_1+{\beta }_2({\beta }_1-{\beta }_4){\mathbf{sn}}^2(\theta {(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\xi -{\xi }_0,m_1)}{{(\frac{1}{2}{a}_3)}^{\frac{1}{4}}({\beta }_4-{\beta }_2+({\beta }_1-{\beta }_4){\mathbf{sn}}^2(\theta {(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\xi -{\xi }_0,m_1))}-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)}.\end{array}$$

If ${\beta }_3<W<{\beta }_2$ the solution is

$$\begin{array}{c}\hfill u_{11}(t,x)=[\frac{({\beta }_4-{\beta }_2){\beta }_3+{\beta }_4({\beta }_2-{\beta }_3){\mathbf{sn}}^2(\theta {(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0,m_1)}{{(\frac{1}{2}{a}_3)}^{\frac{1}{4}}({\beta }_4-{\beta }_2+({\beta }_2-{\beta }_3){\mathbf{sn}}^2(\theta {(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0,m_1))}-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)}.\end{array}$$

When $a_3<0$, if ${\beta }_1>W>{\beta }_2$, we perform the following transformation:

$$\begin{array}{c}\hfill W=\frac{{\beta }_3({\beta }_1-{\beta }_2){sin}^2\phi -{\beta }_2({\beta }_1-{\beta }_3)}{({\beta }_1-{\beta }_2){sin}^2\phi -({\beta }_1-{\beta }_3)}.\end{array}$$ 3.18

If ${\beta }_4<W<{\beta }_3$, we perform the following transformation:

$$\begin{array}{c}\hfill W=\frac{{\beta }_1({\beta }_3-{\beta }_4){sin}^2\phi -{\beta }_4({\beta }_3-{\beta }_1)}{({\beta }_3-{\beta }_4){sin}^2\phi -({\beta }_3-{\beta }_1)}.\end{array}$$

Then, the solution of Eq. (3.12)

$$\begin{array}{c}\hfill {\xi }_1-{\xi }_0=\int \frac{\mathit{dW}}{\sqrt{(W-{\beta }_1)(W-{\beta }_2)(W-{\beta }_3)(W-{\beta }_4)}}=\frac{1}{\theta }\int \frac{d\phi }{\sqrt{1-m_2^2{sin}^2\phi }}.\end{array}$$ 3.19

By using Eq. (3.19) and the definition of the Jacobian elliptic sine function, we can obtain:

$$\begin{array}{c}\hfill \mathbf{sn}[\frac{1}{2}\sqrt{({\beta }_1-{\beta }_3)({\beta }_2-{\beta }_4)}({\xi }_1-{\xi }_0),m_2]=\mathbf{sn}(\theta {\xi }_1-{\xi }_0,m_2)=sin\phi .\end{array}$$

If ${\beta }_2<W<{\beta }_1$, the solution of Eq. (1.1)

$$\begin{array}{c}\hfill u_{12}(t,x)=[\frac{({\beta }_4-{\beta }_2){\beta }_1-{\beta }_4({\beta }_1-{\beta }_2){\mathbf{sn}}^2(\theta {(-\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0,m_2)}{{(-\frac{1}{2}{a}_3)}^{\frac{1}{4}}({\beta }_4-{\beta }_2+({\beta }_2-{\beta }_1){\mathbf{sn}}^2(\theta {(-\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0,m_2))}-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)}.\end{array}$$

If ${\beta }_4<W<{\beta }_3$, the solution of Eq. (1.1)

$$\begin{array}{c}\hfill u_{13}(t,x)=[\frac{({\beta }_1-{\beta }_3){\beta }_4+{\beta }_1({\beta }_3-{\beta }_4){\mathbf{sn}}^2(\theta {(-\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0,m_2)}{{(-\frac{1}{2}{a}_3)}^{\frac{1}{4}}({\beta }_1-{\beta }_3+({\beta }_3-{\beta }_4){\mathbf{sn}}^2(\theta {(-\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0,m_2))}-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)}.\end{array}$$

Case 6.$D_4<0,D_2{D}_3\ge 0$. Evidently, F(W) has two distinct real roots and a pair of complex conjugate roots, i.e. $F(W)=(W-{\beta }_1)((W-{\beta }_2)[(W+\frac{{\beta }_1+{\beta }_2}{2})+{\beta }_3^2]$, where ${\beta }_1,{\beta }_2,{\beta }_3$ are real numbers. ${\beta }_1>{\beta }_2$ and ${\beta }_3\ne 0$.

Denote

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p_1& =\sqrt{\frac{{(3{\beta }_1+{\beta }_2)}^2}{4}+{\beta }_3^2},p_2=\sqrt{\frac{{({\beta }_1+3{\beta }_2)}^2}{4}+{\beta }_3^2},\hfill \\ \hfill m_1& =\frac{1}{2}\sqrt{\frac{{(p_1+p_2)}^2-{({\beta }_1-{\beta }_2)}^2}{p_1{p}_2}},m_2=\frac{1}{2}\sqrt{\frac{-{(p_1-p_2)}^2+{({\beta }_1-{\beta }_2)}^2}{p_1{p}_2}},\hfill \end{array}\end{array}$$

when $a_3>0$.

If $W>{\beta }_1$ or $W<{\beta }_2$, the solution of Eq. (1.1)

$$\begin{array}{c}\hfill u_{14}(t,x)=[\frac{-{\beta }_2{p}_1+{\beta }_1{p}_2+({\beta }_2{p}_1+{\beta }_1{p}_2)\mathbf{cn}(\sqrt{p_1{p}_2}{(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0,m_1)}{{(\frac{1}{2}{a}_3)}^{\frac{1}{4}}[-p_1+p_2+(p_1+p_2)\mathbf{cn}(\sqrt{p_1{p}_2}{(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0,m_1)]}-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)}.\end{array}$$

If ${\beta }_1<W<{\beta }_2$, the solution of Eq. (1.1)

$$\begin{array}{c}\hfill u_{15}(t,x)=[\frac{{\beta }_2{p}_1+{\beta }_1{p}_2+({\beta }_2{p}_1-{\beta }_1{p}_2)\mathbf{cn}(\sqrt{p_1{p}_2}{(-\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0,m_2)}{{(-\frac{1}{2}{a}_3)}^{\frac{1}{4}}[p_1+p_2+(p_1-p_2)\mathbf{cn}(\sqrt{p_1{p}_2}{(-\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0,m_2)]}-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)}.\end{array}$$

Case 7.$D_2>0,D_3=D_4=E_2=0$. Apparently, F(W) has one triple real root and one single real root, i.e. $p<0$, $q=\pm \sqrt{-\frac{8}{27}{p}^3}$, $r=\frac{2p^3+9q^2}{8p}$, then $F(W)={(W-\beta )}^3((W+3\beta )$, where ${\beta }^2=-\frac{p}{6}$.

When $a_3>0$, if $W>max(\beta ,-3\beta )$ or $W<min(\beta ,-3\beta )$, the solution of Eq. (1.1)

$$\begin{array}{c}\hfill u_{16}(t,x)=[{(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\beta \frac{3+4{\beta }^2{({(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0)}^2}{-1+4{\beta }^2{({(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0)}^2}-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)}.\end{array}$$

When $a_3<0$, if $min(\beta ,-3\beta )<W<max(\beta ,-3\beta )$, the solution of Eq. (1.1)

$$\begin{array}{c}\hfill u_{17}(t,x)=[{(-\frac{1}{2}{a}_3)}^{\frac{1}{4}}\beta \frac{-3+4{\beta }^2{({(-\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0)}^2}{1+4{\beta }^2{({(-\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0)}^2}-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)}.\end{array}$$

Case 8. $D_4=0,D_2{D}_3<0$. Obviously, F(W) has one double real root and a pair of complex conjugate roots, i.e. F(W) = ${(W-\alpha )}^2[{(W-l)}^2+s^2]$.

When $ϵ=1$, Eq. (3.12) can be rewritten as

$$\begin{array}{c}\hfill \pm ({\xi }_1-{\xi }_0)=\int \frac{\mathit{dW}}{(W-\alpha )\sqrt{{(W-l)}^2+s^2}}=\frac{1}{\sqrt{{(\alpha -l)}^2+s^2}}ln|\frac{\gamma W+\delta -\sqrt{{(W-l)}^2+s^2}}{W-\alpha }|,\end{array}$$ 3.20

where $\gamma =\frac{\alpha -2l}{\sqrt{{(\alpha -l)}^2+s^2}},\delta =\sqrt{{(\alpha -l)}^2+s^2}-\frac{\alpha (\alpha -2l)}{\sqrt{{(\alpha -l)}^2+s^2}}.$

Correspondingly, the solution of Eq. (1.1)

$$\begin{array}{c}\hfill u_{18}(t,x)=[{(\frac{1}{2}{a}_3)}^{-\frac{1}{4}}\frac{[e^{\pm \sqrt{{(\alpha -l)}^2+s^2}({(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0)}-\gamma ]+\sqrt{{(\alpha -l)}^2+s^2}(2-\gamma )}{{(e^{\pm \sqrt{{(\alpha -l)}^2+s^2}[{(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0]}-\gamma )}^2-1}-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)}.\end{array}$$

This represents a solitary wave solution.

Case 9.$D_4>0,D_2{D}_3\le 0$. Obviously, F(W) has two pairs of complex conjugate roots, i.e.F(W) = $[{(W-l_1)}^2+s_1^2][{(W-l_2)}^2+s_2^2]$, where $l_1,l_2,s_1,s_2$ are real numbers, here $s_1\ge s_2>0$.

If $ϵ=1$, we make the following transformation:

$$\begin{array}{c}\hfill W=\frac{atan\phi +b}{ctan\phi +d},\end{array}$$

where $a=l_{1}c+s_{1}d,b=l_{1}d-s_{1}c,c=-s_1-\frac{s_2}{m_1},d=l_1-l_2,m_1=E+\sqrt{E^2-1},E=\frac{{(l_1-l_2)}^2+s_1^2+s_2^2}{2s_1{s}_2}$.

From Eq. (3.12), we can obtain:

$$\begin{array}{c}\hfill {\xi }_1-{\xi }_0=\int \frac{\mathit{dW}}{\sqrt{[{(W-l_1)}^2+s_1^2][{(W-l_2)}^2+s_2^2]}}=\frac{c^2+d^2}{s_2\sqrt{(c^2+d^2)+(m_1^2{c}^2+d^2)}}\int \frac{d\phi }{\sqrt{1-m^2{sin}^2\phi }},\end{array}$$ 3.21

where $m^2=\frac{m_1^2-1}{m_1^2}$.

Derived from Eq. (3.21) and the definitions of the Jacobian sine and cosine functions, we obtain:

$$\begin{array}{cc}& \mathbf{sn}[\frac{s_2\sqrt{(c^2+d^2)+(m_1^2{c}^2+d^2)}}{c^2+d^2}({\xi }_1-{\xi }_0),m]=sin\phi .\hfill \\ \hfill & \mathbf{cn}[\frac{s_2\sqrt{(c^2+d^2)+(m_1^2{c}^2+d^2)}}{c^2+d^2}({\xi }_1-{\xi }_0),m]=cos\phi .\hfill \end{array}$$

So, the solution of Eq. (1.1)

$$\begin{array}{c}\hfill u_{19}(t,x)=[\frac{a\mathbf{sn}[\eta ({(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\xi -{\xi }_0),m]+b\mathbf{cn}(\eta ({(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0),m)}{{(\frac{1}{2}{a}_3)}^{\frac{1}{4}}(c\mathbf{sn}(\eta ({(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\frac{c(x^{\mu }-v{t}^{\mu })\mathrm{\Gamma }(\beta +1)}{\mu }-{\xi }_0),m)+d\mathbf{cn}(\eta ({(\frac{1}{2}{a}_3)}^{\frac{1}{4}}\xi -{\xi }_0),m))}-\frac{a_2}{3a_3}]e^{i\vartheta (t,x)}.\end{array}$$

where $\eta =\frac{s_2\sqrt{(c^2+d^2)+(m_1^2{c}^2+d^2)}}{c^2+d^2}$

Graphical illustrations

In this section, we provided specific values for some parameters and plotted three-dimensional, two-dimensional, and contour maps of the obtained solutions of Eq. (1.1) shown as Figs. 1, 2, 3. The solution shown in Fig. 1 is a trigonometric function solution. Figure 2 shows a bell shaped solitary wave. Figure 3 shows a Jacobian function solution, which is a periodic function solution.

Fig. 1.

The explicit solution $|u_1(t,x)|$ of Eq. (1.1) when $a_3=2,a_2=3,a_1=\frac{7}{2},a_0=\frac{5}{4},d=-\frac{3}{8},c=v=1,{\xi }_0=0$.

Fig. 2.

The explicit solution $|u_3(t,x)|$ of Eq. (1.1) when $a_3=2,a_2=3,a_1=-\frac{1}{2},a_0=-\frac{3}{4},d=-\frac{1}{16},c=v=1,{\xi }_0=0$.

Fig. 3.

The explicit solution $|u_9(t,x)|$ of Eq. (1.1) when $a_3=2,a_2=3,a_1=-\frac{7}{2},a_0=\frac{9}{4},d=-\frac{83}{16},c=v=1,{\xi }_0=0$.

Conclusion

In this paper, we successfully employed a complete discriminant system of fourth-order polynomials to solve the traveling wave solutions of the generalized Kudryashov’s equation with truncated M-fractional derivative and classified them. What’s more, we presented 2D and 3D plots of the solutions under different parameter values by using Mathematical software. Compared to previous studies, our work further enriches the traveling wave solutions of the generalized Kudryashov’s equation with truncated M-fractional derivative. In the future, we plan to explore higher-order and more complex fractional partial differential equations to address problems in a broader and more complex range of engineering and physical fields.

Author contributions

All authors contributed equally to this paper.

Data availability 

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.