Evolution of small-amplitude intermediate shocks in a dissipative and dispersive system

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Abstract
Our study of the relationship between shock structure and evolutionarity is extended to include the effects of dispersion as well as dissipation. We use the derivative nonlinear Schrödinger-Burgers equation (DNLSB), which reduces to the Cohen-Kulsrud-Burgers equation (CKB) when finite ion inertia dispersion can be neglected. As in our previous CKB analysis, the fast shock solution is again unique, and the intermediate shock structure solutions are non-unique. With dispersion, the steady intermediate shock structure solutions continue to be labelled by the integral through the shock of the non-co-planar component of the magnetic field, whose value now depends upon the ratio of the dispersion and dissipation lengths. This integral helps to determine the solution of the Riemann problem. With dispersion, this integral is also non-zero for fast shocks. Thus, even for fast shocks, the solution of the Riemann problem depends upon shock structure.

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