Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 135, pp. 1-13.
Title: Spectral stability of undercompressive shock profile
solutions of a modified KdV-Burgers equation
Author: Jeff Dodd (Jacksonville State Univ., AL, USA)
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.
Submitted July 17, 2007. Published October 13, 2007.
Math Subject Classifications: 74J30, 74J40, 35Q53, 35P05.
Key Words: Travelling waves; undercompressive shocks;
spectral stability; Evans function.