Integrable discretizations of the short pulse equation

TL;DR

This paper develops integrable semi-discrete and full-discrete versions of the short pulse equation, providing explicit solutions and a self-adaptive numerical scheme for accurate simulation.

Contribution

It introduces new integrable discretizations of the short pulse equation using bilinear forms and determinant solutions, along with an adaptive numerical method.

Findings

Derived explicit N-soliton solutions for discrete models

Demonstrated convergence to the continuous short pulse equation

Proposed an effective self-adaptive moving mesh numerical scheme

Abstract

In the present paper, we propose integrable semi-discrete and full-discrete analogues of the short pulse (SP) equation. The key of the construction is the bilinear forms and determinant structure of solutions of the SP equation. We also give the determinant formulas of N-soliton solutions of the semi-discrete and full-discrete analogues of the SP equations, from which the multi-loop and multi-breather solutions can be generated. In the continuous limit, the full-discrete SP equation converges to the semi-discrete SP equation, then to the continuous SP equation. Based on the semi-discrete SP equation, an integrable numerical scheme, i.e., a self-adaptive moving mesh scheme, is proposed and used for the numerical computation of the short pulse equation.