Nonlinear Stability of the Periodic Traveling Wave Solution for a Class of Coupled KdV Equations

Cong Sun

doi:10.1155/2020/3875038

Abstract

In this paper, by applying the Jacobian ellipse function method, we obtain a group of periodic traveling wave solution of coupled KdV equations. Furthermore, by the implicit function theorem, the relation between some wave velocity and the solution’s period is researched. Lastly, we show that this type of solution is orbitally stable by periodic perturbations of the same wavelength as the underlying wave.

Highlights

In fluid mechanics of the density stratification, the mechanism of propagation of nonlinear long wave is being researched by physicists and mathematicians
The KdV equation is often used to describe the wave of this type when the depth of fluid is much shorter than the length of it
Weak interactions occur in the internal of the nonlinear long waves when wave phase speeds are unequal

Summary

Introduction

In fluid mechanics of the density stratification, the mechanism of propagation of nonlinear long wave is being researched by physicists and mathematicians. I am interested in the existence of a smooth periodic solution and the orbital stability of the solution of a coupled Korteweg-de Vries (equation (1)). By implying the Jacobian ellipse function method, we will show the existence of a class of smooth cnoidal wave solution for system (1).

Results

Conclusion