Galloping instability of viscous shock waves

Galloping instability of viscous shock waves

Motivated by physical and numerical observations of time oscillatory “galloping”, “spinning”, and “cellular” instabilities of detonation waves, we study Poincaré–Hopf bifurcation of traveling-wave solutions of viscous conservation laws. The main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum, preventing standard reduction to a finite-dimensional cen...

Stability and Dynamics of Viscous Shock Waves

We examine from a classical dynamical systems point of view stability, dynamics, and bifurcation of viscous shock waves and related solutions of nonlinear pde. The central object of our investigations is the Evans function: its meaning, numerical approximation, and behavior in various asymptotic limits.

Nonlinear Stability of Viscous Shock Waves

The refined inviscid stability condition and cellular instability of viscous shock waves

Combining work of Serre and Zumbrun, Benzoni-Gavage, Serre, and Zumbrun, and Texier and Zumbrun, we propose as a mechanism for the onset of cellular instability of viscous shock and detonation waves in a finite-cross-section duct the violation of the refined planar stability condition of Zumbrun–Serre, a viscous correction of the inviscid planar stability condition of Majda. More precisely, we ...

Alternate Evans Functions and Viscous Shock Waves

The Evans function is known as a helpful tool for locating the spectrum of some variational diierential operators. This is of special interest regarding the stability analysis of traveling waves, such as reaction-diiusion waves, solitary waves, viscous shock waves, etc., and has been used in numerous contexts. The rst aim of this paper is to present an overview of the various ways to deene an E...