Cusped and smooth solitons for the generalized Camassa-Holm equation on the nonzero constant pedestal (Q1724095)

Summary: We investigate the traveling solitary wave solutions of the generalized Camassa-Holm equation \(u_t - u_{x x t} + 3 u^2 u_x = 2 u_x u_{x x} + u u_{x x x}\) on the nonzero constant pedestal \(\lim_{\xi \rightarrow \pm \infty} u \left(\xi\right) = A\). 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.