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.