Question

brief history on Camassa-Holm equation for fun!

Answer 1

The Camassa-Holm equation, a completely integrable evolution equation, contains rich geometric structures. For the existence of the bi-Hamiltonian structure and the so-called peaked wave solutions, considerable interest has been aroused in the last several decades. Focusing on local geometric properties of the peaked wave solutions for the Camassa-Holm equation, we propose the multisymplectic method to simulate the propagation of the peaked wave in this paper. Based on the multisymplectic theory, we present a multisymplectic formulation of the Camassa-Holm equation and the multisymplectic conservation law. Then, we apply the Euler box scheme to construct the structure-preserving scheme of the multisymplectic form. Numerical results show the merits of the multisymplectic scheme constructed, especially the local conservative properties on the wave form in the propagation process.

The Camassa–Holm (CH) equation u_t −u_xxt +2κu_x = 2u_xu_xx −3uu_x +uu_xxx, is an extensively studied nonlinear equation. It first appeared as an abstract biHamiltonian partial differential equation in an article of Fuchssteiner and Fokas but did not receive much attention until Camassa and Holm derived it as a nonlinear wave equation that models unidirectional wave propagation on shallow water and discovered its rich mathematical structure. In this context u(x,t) represents the fluid velocity in the x direction at time t and the real constant κ is related to the critical shallow water wave speed. Apart from this, the CH equation was also found in as a model for nonlinear waves in cylindrical hyper elastic rods.

Since its discovery, the literature on the CH equation has been growing exponentially (at the moment, the paper by Camassa and Holm has more than 2400 citations in Google Scholar, 1698 in Scopus and 1016 in MathSciNet) and it is impossible to give a comprehensive overview here. In fact our focus lies on understanding the CH equation and its so-called conservative solutions via the inverse scattering transform (IST) approach. It was noticed by Camassa and Holm that the CH equation is completely integrable in the sense that it enjoys a Lax pair structure and hence may be treated with the help of the IST approach in principal. The corresponding isospectral problem is a Sturm–Liouville problem of the form −y ′′ + 1/4 y = zyω(·,t), where ω = u − u_xx + κ is known as the momentum.