THE INTERACTION OF SHOCKS WITH DISPERSIVE WAVES .1. WEAK-COUPLING LIMIT

We introduce and analyze a model for the interaction of shocks with a dispersive wave envelope. The model mimicks the Zakharov system from w eak plasma turbulence theory but replaces the linear wave equation in that system by a nonlinear wave equation allowing the formation of sho cks. This paper considers a weak coupling in which the nonlinear wave evolves independently but appears as the potential in the time-depende nt Schrodinger equation governing the dispersive wave. We first solve the Riemann problem for the system by constructing solutions to the Sc hrodinger equation that are steady in a frame of reference moving with the shock. Then we add a viscous diffusion term to the shock equation and by explicitly constructing asymptotic expansions in the (small) d iffusion coefficient, we show that these solutions are zero diffusion limits of the regularized problem. The expansions are unusual in that it is necessary to keep track of exponentially small terms to obtain a lgebraically small terms. The expansions are compared to numerical sol utions. We then construct a family of time-dependent solutions in the case that the initial data for the nonlinear wave equation evolves to a shock as t --> t < infinity. We prove that the shock formation driv es a finite time blow-up in the phase gradient of the dispersive wave. While the shock develops algebraically in time, the phase gradient bl ows up logarithmically in time. We construct several explicit time-dep endent solutions to the system, including ones that: (a) evolve to the steady states previously constructed, (b) evolve to steady states wit h phase discontinuities (which we call phase kinked steady states), (c ) do not evolve to steady states.