Orbital stability of periodic traveling wave solutions to the coupled compound KdV and MKdV equations with two components

Xiaoxiao Zheng; Hui Wu

doi:10.3934/mfc.2020002

Article Contents

2020, Volume 3, Issue 1: 11-24. Doi: 10.3934/mfc.2020002

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Orbital stability of periodic traveling wave solutions to the coupled compound KdV and MKdV equations with two components

Xiaoxiao Zheng^, and
Hui Wu

School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, China

^* Corresponding author: Xiaoxiao Zheng
^* Corresponding author: Xiaoxiao Zheng

Received: November 30, 2019

Published: February 2020

The first author is supported by the Natural Science Foundation of Shandong Province (No. ZR2018BA016), the second author is supported by the National Natural Science Foundation of China (No. 11801306)

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Abstract

In this article, the authors consider the orbital stability of periodic traveling wave solutions for the coupled compound KdV and MKdV equations with two components

$ \begin{equation*} \left\{ \begin{aligned} &u_{t}+vv_{x}+\beta u^{2}u_{x}+u_{xxx}-uu_{x} = 0, \ \ \beta>0, \\ &v_{t}+(uv)_{x}+2vv_{x} = 0, \end{aligned} \right. \end{equation*} $

Firstly, we show that there exist a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period $ L $ for the coupled compound KdV and MKdV equations. Then, combining the orbital stability theory presented by Grillakis et al., and detailed spectral analysis given by using Lamé equation and Floquet theory, we show that the dnoidal type periodic wave solution with period $ L $ is orbitally stable. As the modulus of the Jacobian elliptic function $ k\rightarrow 1 $, we obtain the orbital stability results of solitary wave solution with zero asymptotic value for the coupled compound KdV and MKdV equations from our work. In addition, we also obtain the stability results for the coupled compound KdV and MKdV equations with the degenerate condition $ v = 0 $, called the compound KdV and MKdV equation.

Keywords:

coupled compound KdV and MKdV equations,
periodic traveling waves,
orbital stability.

Mathematics Subject Classification: 35Q53, 37K45, 39A23.

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References

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