Shock bifurcation and emergence of diffusive solitons in a nonlinear wave equation with relaxation

Abstract

A hyperbolic generalization of Burgers' equation, which includes relaxation, is examined using analytical and numerical tools. By means of singular surface theory, the evolution of initial discontinuities (i.e. shocks) is fully classified. In addition, the parameter space is explored and the bifurcation experienced by the shock amplitude is identified. Then, by means of numerical simulations based on a Godunov-type scheme, we confirm the theoretical findings and explore the solution structure of a signaling-type initial-boundary-value problem with discontinuous boundary data. In particular, we show that diffusive solitons (or Taylor shocks) can emerge in the solution, behind the wavefront. We also show that, for certain parameter values, a shock wave becomes an acceleration wave in infinite time, an unexpected result that is the exact opposite of the well-known phenomenon of finite-time acceleration wave blow-up. Finally, the ‘red light turning green’ problem is re-examined.