Spectral stability of undercompressive shock profile solutions of a modified KdV-Burgers equation
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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.
CitationDodd, J. (2007). Spectral stability of undercompressive shock profile solutions of a modified KdV-Burgers equation. Electronic Journal of Differential Equations, 2007(135), pp. 1-13.
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