Séminaire exceptionnel – 6 déc. 2017 – 11h salle 250

Dispersive shock waves in integrable and non-integrable systems

El Gennady – Loughborough Univ., RU

Dispersive shock waves (DSWs) are dynamic, oscillatory structures forming due to dis- persive regularisation of hydrodynamic singularities in conservative or nearly conservative media. Physical manifestations of DSWs include undular bores on shallow water and in the atmosphere, nonlinear diffraction patterns in laser optics, and blast waves in Bose-Einstein condensates. The mathematical description of DSWs involves a synthesis of methods from hyperbolic quasi-linear systems, asymptotics and soliton theory. In my talk I will describe the modu- lation theory of DSWs which is based on the analysis of the Whitham averaged equations associated with the original dispersive system. This theory was pioneered by Gurevich and Pitaevskii in 1973 by constructing modulation solution for the Riemann problem for the Korteweg – de Vries (KdV) equation, and since then has been developed into a very active research area encompassing a broad spectrum of nonlinear dispersive equations and physical applications. In particular, rich families of solutions have been recently found for systems exhibiting non-convex hyperbolic flux and/or non-convex linear dispersion relation. The talk will emphasise general aspects of the theory but I will also provide concrete examples which will include both integrable and non-integrable models. The talk is based on the recent review articles: • G.A. El and M.A. Hoefer, Dispersive shock waves and modulation theory, Physica D 33 (2016) 11-65. • G.A. El, M.A. Hoefer and M. Shearer, Dispersive and diffusive-dispersive shock waves for non-convex conservation laws, SIAM Review 59 (2017) 3 – 61.