Stability of dnoidal waves to Hirota-Satsuma system

Abstract

This work is concerned with nonlinear stability properties of periodic travelling wave solutions of the Hirota-Satsuma system $$ \begin{cases} u_t+ u_{xxx}+6u_xu=2bvv_x\\ v_t+ v_{xxx}+ 3uv_x=0 \end{cases} $$ posed in $\mathbb R$ with $b>0$. We prove that this system is globally well posed in $L^2_{per}([0,L])\times H^1_{per}([0,L])$ by using Bourgain's space framework. Also shown is the existence of at least two nontrivial smooth curves of periodic travelling wave solutions depending on the classical Jacobian elliptic functions. We find dnoidal and cnoidal waves solutions. Then we prove the nonlinear stability of the dnoidal waves solutions in the energy space $L^2_{per}([0,L])\times H^1_{per}([0,L])$. The Floquet theory is used to obtain a detailed spectral analysis of the Jacobian form of Lam\'e's equation which is required in our stability theory.