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Spectral stability of undercompressive shock profile solutions of a modified KdV-Burgers equation. (English) Zbl 1137.76023
Summary: It is shown that certain undercompressive shock profile solutions of 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 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.

MSC:
76E17 Interfacial stability and instability in hydrodynamic stability
76L05 Shock waves and blast waves in fluid mechanics
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