Study of a Model Equation in Detonation Theory

Abstract

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 $ u_{t}+\tfrac{1}{2}\left(u^{2}-uu\left(0^{-},t\right)\right)_{x}=f\left(x,u\left(0^{-},t\right)\right),\;x\le0,\; 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.