Stability of cnoidal waves

Abstract

This paper is concerned with the stability of periodic travelling-wave solutions of the Korteweg-de Vries equation $$ u_t+uu_x+u_{xxx}=0. $$ Here, $u$ is a real-valued function of the two variables $x,t\in{\mathbb R}$ and subscripts connote partial differentiation. These special solutions were termed cnoidal waves by Korteweg and de Vries. They also appear in earlier work of Boussinesq. It is shown that these solutions are stable to small, periodic perturbations in the context of the initial-value problem. The approach is that of the modern theory of stability of solitary waves, but adapted to the periodic context. The theory has prospects for the study of periodic travelling-wave solutions of other partial differential equations.