Integrable discretizations and self-adaptive moving mesh method for a coupled short pulse equation

TL;DR

This paper develops integrable semi-discrete and fully discrete versions of a coupled short pulse equation, providing soliton solutions and a self-adaptive moving mesh method that accurately captures the continuous dynamics.

Contribution

It introduces new integrable discretizations of the coupled short pulse equation and applies them as a self-adaptive numerical scheme with verified accuracy.

Findings

Discrete models converge to the continuous CSP equation

N-soliton solutions are explicitly constructed

Numerical simulations match analytical solutions

Abstract

In the present paper, integrable semi-discrete and fully discrete analogues of a coupled short pulse (CSP) equation are constructed. The key of the construction is the bilinear forms and determinant structure of solutions of the CSP equation. We also construct Nsoliton solutions for the semi-discrete and fully discrete analogues of the CSP equations in the form of Casorati determinant. In the continuous limit, we show that the fully discrete CSP equation converges to the semi-discrete CSP equation, then further to the continuous CSP equation. Moreover, the integrable semi-discretization of the CSP equation is used as a selfadaptive moving mesh method for numerical simulations. The numerical results agree with the analytical results very well.