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The Interaction of Shocks with Dispersive Waves I. Weak Coupling Limit
Author(s) -
Axel Ralph M.,
Newton Paul K.
Publication year - 1996
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1996962201
Subject(s) - shock wave , nonlinear system , burgers' equation , physics , shock (circulatory) , mathematical analysis , coupling (piping) , limit (mathematics) , nonlinear schrödinger equation , mathematics , envelope (radar) , diffusion , schrödinger equation , classical mechanics , quantum mechanics , mechanics , medicine , mechanical engineering , telecommunications , radar , computer science , engineering
We introduce and analyze a model for the interaction of shocks with a dispersive wave envelope. The model mimicks the Zakharov system from weak plasma turbulence theory but replaces the linear wave equation in that system by a nonlinear wave equation allowing the formation of shocks. This paper considers a weak coupling in which the nonlinear wave evolves independently but appears as the potential in the time‐dependent Schrodinger equation governing the dispersive wave. We first solve the Riemann problem for the system by constructing solutions to the Schrodinger 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) diffusion 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 algebraically small terms. The expansions are compared to numerical solutions. 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 * < ∞. We prove that the shock formation drives a finite time blow‐up in the phase gradient of the dispersive wave. While the shock develops algebraically in time, the phase gradient blows up logarithmically in time. We construct several explicit time‐dependent solutions to the system, including ones that: (a) evolve to the steady states previously constructed, (b) evolve to steady states with phase discontinuities (which we call phase kinked steady states), (c) do not evolve to steady states.