Study of a Model Equation in Detonation Theory: Multidimensional Effects

Here we analyze properties of an equation that we previously proposed to model the dynamics of unstable detonation waves [A. R. Kasimov, L. M. Faria, and R. R. Rosales, Model for shock wave chaos, Phys. Rev. Lett., 110 (2013), 104104]. The equation is ut+ 1 2 ( u2 − uu 0−, t x = f ( x, u ( 0−, t )) , x ≤ 0, t > 0. It describes a detonation shock at x = 0 with the reaction zone in x < 0. We investigate the nature of the steady-state solutions of this nonlocal hyperbolic balance law, the linear stability of these solutions, and the nonlinear dynamics. We establish the existence of instability followed by a cascade of period-doubling bifurcations leading to chaos.