Abstract
Whitham modulation equations1 are investigated very actively last time. This is caused by theoretical aspects of general theory of hydrodynamic type systems2, as well as by the importance of physical applications, most interesting of which are nondissipative shock waves (NSW). It is well known that wave breaking of the simple Riemann wave3 in dispersive hydrodynamics leads to the appearance of continuously expanding region filled with undamped small-scale nonlinear oscillations. This is NSW. First the shock wave problem in dispersive hydrodynamics - Gurevich-Pitaevsky (GP) problem - was considered in Ref. 4, where analytic solution of the problem of an initial discontinuity decay in KdV hydrodynamics was obtained and numerical analysis of the simple wave breaking was done. Then this topic was developed in Refs. 5–10. The important property of modulation sets for KdV, NLS and SG-equations which was effectively used in these works is the possibility to represent them in characteristic (Riemann) form.11–13 partially this fact allowed to introduce important concept of quasi-simple wave and receive some exact NSW-solutions.8,9
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El’, G.A., Gurevich, A.V., Krylov, A.L. (1994). Breaking Problem in Dispersive Hydrodynamics. In: Ercolani, N.M., Gabitov, I.R., Levermore, C.D., Serre, D. (eds) Singular Limits of Dispersive Waves. NATO ASI Series, vol 320. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2474-8_7
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DOI: https://doi.org/10.1007/978-1-4615-2474-8_7
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