Relative Poincar\'e--Hopf bifurcation and galloping instability of detonation waves.

Kevin Zumbrun, Indiana University (Joint work with Benjamin Texier, Univ. Paris 7)

Transition to longitudinal instability of detonation waves is typified by ``galloping'' or ``pulsating'' behavior, appearing as time-periodic perturbations of the wave. We discuss from several points of view the rigorous characterization of this phemonenon, within the context of the reactive Navier-Stokes equations, as a relative Poincar\'e-Hopf bifurcation with respect to the underlying group invariance of translation. The main mathematical issue from the point of view of abstract bifurcation theory is the absence of a spectral gap between bifurcating modes and essential spectrum of the linearized operator about the wave.