Multi-Symplectic Method for the Two-Component Camassa–Holm (2CH) System

Abstract

In this paper, the multi-symplectic formulations of the two-component Camassa–Holm system are presented. Both the multi-symplectic structure and two local conservation laws of the generalized two-component Camassa–Holm model are proposed for its first-order canonical form. Then, combining the Fourier pseudo-spectral method in the spatial domain with the midpoint method in the time dimension, the multi-symplectic Fourier pseudo-spectral scheme is constructed for the first-order canonical form. Meanwhile, the discrete scheme of the residuals of the multi-symplectic structure and two local conservation laws are also provided. By using the multi-symplectic Fourier pseudo-spectral scheme, the evolution of one- and two-soliton solutions for the generalized two-component Camassa–Holm model is regained. The structure-preserving properties and the reliability of the numerical scheme are illustrated by the tiny numerical residuals (less than 3.5 × 10⁻⁸) of the conservation laws as well as the tiny numerical variations (less than 1 × 10⁻⁹) of the amplitudes and the propagating velocities of the solitons.