Electron. J. Diff. Eqns., Vol. 2007(2007), No. 135, pp. 1-13.

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

Jeff Dodd

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.

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Jeff Dodd
Department of Mathematical, Computing, and Information Sciences
Jacksonville State University, Jacksonville, AL 36265, USA
email: jdodd@jsu.edu

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