A model for shock wave chaos

We propose the following simple model equation that describes chaotic shock waves: \[ u_{t}+\frac{1}{2}\left(u^{2}-uu_{s}\right)_{x}=f\left(x,u_{s}\right). \] It is given on the half-line $x<0$ and the shock is located at $x=0$ for any $t\ge0$. Here $u_{s}\left(t\right)$ is the shock state and $f$ is a given source term [1]. The equation is a modification of the Burgers equation that includes non-locality via the presence of the shock-state value of the solution in the equation itself. The model predicts steady-state solutions, their instability through a Hopf bifurcation, and a sequence of period-doubling bifurcations leading to chaos. This dynamics is similar to that observed in the one-dimensional reactive Euler equations that describe detonations. We present nonlinear numerical simulations as well as a complete linear stability theory for the equation.