An integrable semi-discretization of the Camassa-Holm equation and its determinant solution

TL;DR

This paper introduces an integrable semi-discretization of the Camassa-Holm equation using bilinear forms and determinant solutions, enabling accurate numerical simulations of complex wave interactions.

Contribution

It presents a novel semi-discrete integrable model of the Camassa-Holm equation with explicit determinant formulas for multi-soliton solutions.

Findings

Accurate numerical results for soliton and cuspon interactions

Determinant formulas for N-soliton solutions

Effective simulation of non-exact initial conditions

Abstract

An integrable semi-discretization of the Camassa-Holm equation is presented. The keys of its construction are bilinear forms and determinant structure of solutions of the CH equation. Determinant formulas of $N$-soliton solutions of the continuous and semi-discrete Camassa-Holm equations are presented. Based on determinant formulas, we can generate multi-soliton, multi-cuspon and multi-soliton-cuspon solutions. Numerical computations using the integrable semi-discrete Camassa-Holm equation are performed. It is shown that the integrable semi-discrete Camassa-Holm equation gives very accurate numerical results even in the cases of cuspon-cuspon and soliton-cuspon interactions. The numerical computation for an initial value condition, which is not an exact solution, is also presented.