Cusped and Smooth Solitons for the Generalized Camassa-Holm Equation on the Nonzero Constant Pedestal

Abstract

We investigate the traveling solitary wave solutions of the generalized Camassa-Holm equation $u_t\hspace{0.17em}-\hspace{0.17em}{u}_{xxt}\hspace{0.17em}+\hspace{0.17em}3u^2{u}_x=2u_x{u}_{xx}\hspace{0.17em}+\hspace{0.17em}u{u}_{xxx}$ on the nonzero constant pedestal ${\lim }_{\xi \to \pm \infty }u\left(\xi \right)=A$. Our procedure shows that the generalized Camassa-Holm equation with nonzero constant boundary has cusped and smooth soliton solutions. Mathematical analysis and numerical simulations are provided for these traveling soliton solutions of the generalized Camassa-Holm equation. Some exact explicit solutions are obtained. We show some graphs to explain our these solutions.