Lixin Tian

New peaked solitary wave solutions of the generalized Camassa–Holm equation

Abstract In this paper, we consider generalized Camassa–Holm equations and the generalized weakly dissipative Camassa–Holm equations and derive some new exact peaked solitary wave solutions. For m=3, where m is representative of the strength of the nonlinearity, we give two types new exact traveling wave solutions of the generalized weakly dissipative Camassa–Holm equations.

New compacton solutions and solitary wave solutions of fully nonlinear generalized Camassa–Holm equations

Abstract In this paper, we introduce the fully nonlinear generalized Camassa–Holm equation C ( m , n , p ) and by using four direct ansatzs, we obtain abundant solutions: compactons (solutions with the absence of infinite wings), solitary patterns solutions having infinite slopes or cups, solitary waves and singular periodic wave solutions and obtain kink compacton solutions and nonsymmetry compacton solutions. We also study other forms of fully nonlinear generalized Camassa–Holm equation, and their compacton solutions are governed by linear equations.