Existence of Traveling Wave Solutions for the Perturbed Modefied Gardner Equation

Abstract

The modified Gardner equation is widely used to describe the supernonlinear proliferation of ion-acoustic waves and quantum electron-positron ion magneto plasmas. This paper focuses on the investigation of the modified Gardner equation with Kuramoto-Sivashinsky perturbation. The existence results of nonlinear and supernonlinear ion-acoustic solitary and periodic waves are established by employing the geometric singular perturbation theory, invariant manifold theory, and bifurcation theory. The supernonlinear solitary wave is a new class of solitary waves, which was proposed by Dubinov and Kolotkov [12]. In this work, the existence of the novel type of ion-acoustic solitary and periodic waves in the perturbed modified Gardner equation has been proven for the first time. Through rigorous mathematical analysis and numerical simulations, we further substantiate the validity of our proposed methods and model. Overall, this study enhances the understanding of the nonlinear and supernonlinear dynamics of ion-acoustic waves in the modified Gardner equation, while also providing a solid foundation for future investigations and applications in plasma physics and related disciplines.

Appendix A: Discriminant of the Roots of a Cubic Equation

Lemma A.1

[54] For a cubic equation with one unknown

$$\begin{aligned} \text {mx}^3+\text {nx}^2+\text {rx}+l=0,m,n,r,l\in \mathbb {R},m\ne 0,\end{aligned}$$

(A.1)

the discriminant is

$$\begin{aligned} \Delta =N^2-4 \text {MQ}, \end{aligned}$$

where

$$\begin{aligned} M:=n^2-3 \text {mr}, N:=\text {nr}-9 \text {ml}, Q:=r^2-3 \text {nl}, \end{aligned}$$

we have

• (1) if $M=N=0$, the Eq. (A.1) has a triple real root;

• (2) if $\Delta >0$, the Eq. (A.1) has one real root and a pair of conjugate imaginary roots;

• (3) if $\Delta =0$, the Eq. (A.1) has three real roots, one of which is a dual root;

• (4) if $\Delta <0$, the Eq. (A.1) has three distinct real roots.