Exact solutions, Lagrangians and first integrals for generalized Camassa–Holm equation

Abstract

In this paper, Lagrangians and the first integrals of a new class of Liénard-type equations have been investigated. This class can be obtained using the traveling wave reduction of the generalized Camassa–Holm equation. This task is achieved using the direct definitions of the Lagrangians and first integrals. Infinite standard and non-standard Lagrangians are obtained for this class of equations. Some autonomous and nonautonomous first integrals for this class of equations have been obtained. The obtained Lagrangians and first integrals are new. Also, using the obtained first integrals, we have obtained some new exact solutions for the generalized Camassa–Holm equation. The obtained solutions are doubly periodic, dark, and bright soliton solutions which are very important in interpreting physical phenomena modeled using the generalized Camassa–Holm equation.