Space-time analysis and beyond: toward a better understanding of Camassa–Holm equation

Camassa-Holm (CH) equation for modeling shallow water wave will be analyzed through direct scattering analysis. Thanks to this analysis in spectral domain, some physically meaningful wave propagation features that can by no means be obtained from the analysis in the space-time domain are exhibited. To broaden our understanding of the CH equation, in this article the study underlying the scattering analysis is performed in spectral domain and the solution of CH equation is computed by the finite difference method in time domain. We aim to get some missing details in the space-time analysis. The detailed wave transmission and reflection in the CH equation, subject to the chosen initial condition, shall be theoretically enlightened.