Stability properties of non-constant stationary solutions of the periodic korteweg-de vries Equation

Abstract

In this paper we study the Korteweg-de Vries partial differential equation on a finite spatial interval. The solution and its second derivative satisfy periodic boundary requirements while the first derivative satisfies a corresponding condition leading to dissipative behavior of solutions with respect to the L2 norm. In attempting to extend earlier local asymptotic behavior of constant solutions one is led to consider a class ofnon-constant stationary, i.e., time independent, solutions which provide an infinite collection of periodic equilibrium states for the system. Here we examine the stability properties of these isolated equilibria. They are shown, modulo one assumption not yet verified, to constitute saddlenode type critical points for the system. As a result it is seen that the constant equilibrium states cannot possess any sort of global asymptotic stability, even in an “almost everywhere” sense.