Spectral stability of undercompressive shock profile solutions of a modified KdV-Burgers equation
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Date
2007-10-13
Authors
Dodd, Jeff
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University-San Marcos, Department of Mathematics
Abstract
It is shown that certain undercompressive shock profile solutions of the modified Korteweg-de Vries-Burgers equation
∂tu + ∂x(u3) = ∂3xu + α∂2xu, α ≥ 0
are spectrally stable when α 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 α is sufficiently small and positive.
Description
Keywords
Travelling waves, Undercompressive shocks, Spectral stability, Evans function
Citation
Dodd, J. (2007). Spectral stability of undercompressive shock profile solutions of a modified KdV-Burgers equation. <i>Electronic Journal of Differential Equations, 2007</i>(135), pp. 1-13.
Rights
Attribution 4.0 International