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Dissipative solutions for the modified two-component Camassa-Holm system. (English) Zbl 1310.35080
Summary: Camassa-Holm model is capable of characterizing the dynamic behavior of shallow water wave, thus has been extensively studied. This paper is concerned with shallow water wave behavior after wave breaking. To better reflect the whole process, the modified two-component Camassa-Holm system is considered. The continuation of solutions of such system after wave braking is investigated. By introducing a skillfully defined characteristic, together with a set of newly defined variables, the original system is converted into a Lagrangian equivalent system, from which global dissipative solutions are obtained. The results obtained herein are deemed useful in understanding the dynamic behavior of shallow water wave during and after wave breaking.

##### MSC:
 35G55 Initial value problems for systems of nonlinear higher-order PDEs 35L65 Hyperbolic conservation laws 35L40 First-order hyperbolic systems 35Q35 PDEs in connection with fluid mechanics
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