A note on the stability for Kawahara-KdV type equations

TL;DR

This paper proves the nonlinear stability of solitary wave solutions for Kawahara-KdV and modified Kawahara-KdV equations using Albert's theoretical framework, contributing to the understanding of wave stability in higher-order dispersive equations.

Contribution

It establishes the nonlinear stability of solitary traveling-wave solutions for specific Kawahara-KdV type equations, applying and extending Albert's stability theory.

Findings

Solitary waves are nonlinearly stable for the studied equations.

The stability analysis confirms the robustness of these waves under perturbations.

The approach can be applied to similar higher-order dispersive equations.

Abstract

In this paper we establish the nonlinear stability of solitary traveling-wave solutions for the Kawahara-KdV equation $$u_t+uu_x+u_{xxx}-\gamma_1 u_{xxxxx}=0,$$ and the modified Kawahara-KdV equation $$u_t+3u^2u_x+u_{xxx}-\gamma_2 u_{xxxxx}=0,$$ where $\gamma_i\in\mathbb{R}$ is a positive number when $i=1,2$. The main approach used to determine the stability of solitary traveling-waves will be the theory developed by Albert