It is shown that certain undercompressive shock profile solutions of the modified Korteweg-de Vries-Burgers equation
are spectrally stable when 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 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.
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| Jeff Dodd |
Department of Mathematical, Computing, and Information Sciences
Jacksonville State University, Jacksonville, AL 36265, USA
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