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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.

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