Search

In this paper, we developed, for the first time, the exact expressions of several periodic travelling wave solutions and a solitary wave solution for a shallow water wave model of moderate amplitude. Then, we present the existence theorem of the global weak solutions. Finally, we prove the stability of solution in $$L^{1}(R)$$ space for the Cauchy problem of the equation. [ABSTRACT FROM AUTHOR]

In this paper, we firstly establish the existence theorem of the global weak solutions of the Cauchy problem for the shallow water wave model of moderate amplitude, then following the idea in Xin and Zhang’s work (see Xin and Zhang, 2002), we prove the uniqueness of global weak solutions using the localization analysis in the transport equation theory. Finally, several travelling wave solutions are derived. [ABSTRACT FROM AUTHOR]

Abstract: A shallow water equation including the Camassa–Holm and Degasperis–Procesi equations as special cases is investigated. Provided that initial value , and does not change sign, the existence and uniqueness of global weak solution for the equation are established. [Copyright &y& Elsevier]

Abstract: The global weak solutions for the Degasperis–Procesi equation with weakly dissipative term are investigated. Provided that , , and , the existence and uniqueness of global weak solutions for the equation are shown to be true in the space . [Copyright &y& Elsevier]