06 Resolution of a shock in hyperbolic systems modified by weak dispersion

We present a way to deal with dispersion-dominated " shock-type " transition in the absence of completely integrable structure for the systems that one may characterize as strictly hyperbolic regularized by a small amount of dispersion. The analysis is performed by assuming that, the dispersive shock transition between two different constant states can be modelled by an expansion fan solution of the associated modulation (Whitham) system for the short-wavelength nonlinear oscillations in the transition region (the so-called Gurevich – Pitaevskii problem). We consider as single-wave so bi-directional systems. The main mathematical assumption is that of hyperbolicity of the Whitham system for the solutions of our interest. By using general properties of the Whitham averaging for a certain class of nonlinear dispersive systems and specific features of the Cauchy data prescription on characteristics we derive a set of transition conditions for the dispersive shock, actually bypassing full integration of the modulation equations. Along with model KdV and mKdV examples, we consider a non-integrable system describing fully nonlinear ion-acoustic waves in collisionless plasma. In all cases our transition conditions are in complete agreement with previous analytical and numerical results. 1 Modern theory of dispersive shocks is based on the analysis of the Whitham averaged equations describing modulations of nonlinear short-wavelength oscillations in the transition region between two smooth regimes. If the wave dynamics is governed by one of the completely integrable equations, exact solutions in terms of the Riemann invariants are also available for the corresponding Whitham system providing full asymptotic description of such a transition. For nonintegrable systems describing many physically important cases of nonlinear dispersive wave propagation such modulation solutions are not readily (if at all) available. In this paper we show that, by using some general properties of the Whitham equations connected with their " averaged " origin one is able to obtain a set of transition conditions representing the " dispersive " analog of the traditional shock conditions of classical dissipative gas dynamics. The developed method does not make use of the Riemann invariants for the modulation equations and can be applied to nonintegrable conservative systems. In particular, the obtained conditions allow for determination of the lead solitary wave amplitude in terms of the jumps for hydrodynamic variables across the dispersive shock.