Spectral stability of undercompressive shock profile solutions of a modified KdV-Burgers equation

Abstract

It is shown that certain undercompressive shock profile solutions of the modified Korteweg-de Vries-Burgers equation $$ partial_t u + partial_x(u^3) = partial_x^3 u + alpha partial_x^2 u, quad alpha geq 0 $$ are spectrally stable when $alpha$ is sufficiently small, in the sense that their linearized perturbation equations admit no eigenvalues having positive real part except a simple eigenvalue of zero (due to the translation invariance of the linearized perturbation equations). This spectral stability makes it possible to apply a theory of Howard and Zumbrun to immediately deduce the asymptotic orbital stability of these undercompressive shock profiles when $alpha$ is sufficiently small and positive.

Keywords

• Travelling waves
• undercompressive shocks
• spectral stability
• Evans function