Camassa-Holm equation on shallow water wave equation

Abstract

In this paper we can consider the problem of week solutions for the general shallow water wave equation. In the first part of this paper, we deal to the well-known Kdv equation. We obtain the Camassa-Holm equation in particular. Both of them describe unidirectional shallow water waves equation. Moreover, all these equations have a bi-Hamiltonian structure, they are completely integrable, they have infinitely many conserved quantities. From a mathematical point of view the Camassa-Holm equation is well studied. In the second part of this paper, we obtain a global weak solution as a limit of approximation under the assumption Some concepts related to high dimensional spaces are considered. Then the Cauchy problem is considered. It has an admissible weak solution to the Cauchy problem for Existence, uniqueness, and basic energy estimate on this approximate solution sequence are given in some lemmas. Finally, the main theorem and the proof is given