Annals of Mathematical Sciences and Applications

Volume 4 (2019)

Number 2

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

Pages: 367 – 393



Chueh-Hsin Chang (Department of Applied Mathematics, Tunghai University, Taichung, Taiwan)

Ching-Hao Yu (State Key Lab. of Hydraulics and Mountain River Engineering, Sichuan University, Sichuan, China)

Tony Wen-Hann Sheu (Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, Taiwan)


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.

2010 Mathematics Subject Classification

Primary 78A46, 81U40. Secondary 37K10.

The corresponding author, professor Tony Wen-Hann Sheu, would like to thank professor Chang-Shou Lin for his encouragement and support of performing this study.

Received 10 May 2019

Accepted 12 August 2019

Published 2 October 2019