^{1}

^{2}

^{3}

^{1}

^{2}

^{1}

^{2}

^{3}

In this paper, the bifurcation, phase portraits, traveling wave solutions, and stability analysis of the fractional generalized Hirota–Satsuma coupled KdV equations are investigated by utilizing the bifurcation theory. Firstly, the fractional generalized Hirota–Satsuma coupled KdV equations are transformed into two-dimensional Hamiltonian system by traveling wave transformation and the bifurcation theory. Then, the traveling wave solutions of the fractional generalized Hirota–Satsuma coupled KdV equations corresponding to phase orbits are easily obtained by applying the method of planar dynamical systems; these solutions include not only the bell solitary wave solutions, kink solitary wave solutions, anti-kink solitary wave solutions, and periodic wave solutions but also Jacobian elliptic function solutions. Finally, the stability criteria of the generalized Hirota–Satsuma coupled KdV equations are given.

In this paper, we consider the fractional generalized Hirota–Satsuma coupled KdV (FGHSCKdV) equations as follows [

In addition, when

Equation (

Fractional partial differential equation (FPDE), usually used to simulate phenomena in natural science, plays a very important role in the field of applied sciences (such as fluid mechanics, nonlinear optics, and so on). The most commonly used definitions of fractional derivatives include Caputo derivative, Riemann–Liouville derivative, conformable derivative, etc. But, for FGHSCKdV equations, the research progress is very slow due to the complexity of fractional derivative. The main reason is that most of the fractional derivatives do not obey the chain rule. For example, in Ref. [

This paper is organized as follows. In Section

We first do the following traveling wave transformation:

Substituting (

In order to eliminate the coupling relationship of the above equations, we assume that

Substituting (

Multiplying equation (

On the one hand, we obtain the following from (

On the other hand, integrating (

Therefore, by substituting (

We find from (

Suppose that

Here, we conclude that system (

Note that

Phase portraits of system (

When

where

Substituting (

That is,

When

When selecting the appropriate parameters, the bell solitary wave solution

For

Then, we obtain the periodic solutions

That is,

The bell solitary wave solution

When

where

Substituting (

When

When selecting the appropriate parameters, the kink solitary wave solution

The kink solitary wave solution

In the above discussion, we have obtained the solution

Consider the perturbed solution of system (

Substituting (

Linearizing equation (

Suppose that (

Substituting (

From (

In this paper, we have drawn the bifurcations of phase portraits of FGHSCKdV equations, and we have constructed the traveling wave solutions of FGHSCKdV equations which include implicit analytical solutions, hyperbolic function solutions, and Jacobian elliptic function solutions. Through a fractional traveling wave transformation, the FGHSCKdV equations can be simplified into coupled nonlinear ordinary differential equations. In order to eliminate the coupling relationship, we assume that the solutions

No data were used to support this study.

The authors declare that they have no conflicts of interest.

This study was supported by Institutions of Higher Education of Sichuan Province under grant no. MSSB-2021-13.