Refraction of dispersive shock waves

Details Refraction of dispersive shock waves Refraction of dispersive shock waves

2011-05-10

arxiv.org/abs/1105.1920v1

Nonlinear Sciences

Pattern Formation and Solitons

44 pages, 23 figures

Scientific paper

We study a dispersive counterpart of the classical gas dynamics problem of the interaction of a shock wave with a counter-propagating simple rarefaction wave often referred to as the shock wave refraction. The refraction of a dispersive shock wave (DSW) due to its head-on collision with the centred rarefaction wave (RW) is considered in the frameworks of the one-dimensional defocusing nonlinear Schr\"odinger (NLS) equations with cubic as well as with saturable nonlinearity. For the integrable cubic nonlinearity case we present a full asymptotic description of the DSW refraction by constructing appropriate exact solutions of the Whitham modulation equations in Riemann invariants. For the NLS equation with saturable nonlinearity, whose modulation system does not possess Riemann invariants, we take advantage of the recently developed method for the DSW description in non-integrable dispersive systems to obtain key parameters of the DSW refraction. Our modulation theory analytical results are supported by direct numerical simulations of the corresponding full dispersive initial-value problem.

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