Integrable semi-discretization of a multi-component short pulse equation

TL;DR

This paper develops an integrable semi-discrete version of a multi-component short pulse equation, extending the continuous model with a new discrete formulation while preserving soliton solutions.

Contribution

It introduces a novel integrable semi-discretization of the multi-component short pulse equation using a Bäcklund transformation and pfaffian solutions.

Findings

Constructed an integrable semi-discrete multi-component short pulse equation.

Proved the existence of N-soliton solutions in pfaffian form.

Established the discrete hodograph transformation for the semi-discrete model.

Abstract

In the present paper, we mainly study the integrable semi-discretization of a multi-component short pulse equation. Firstly, we briefly review the bilinear equations for a multi-component short pulse equation proposed by Matsuno (J. Math. Phys. \textbf{52} 123705) and reaffirm its $N$-soliton solution in terms of pfaffians. Then by using a B\"{a}cklund transformation of the bilinear equations and defining a discrete hodograph (reciprocal) transformation, an integrable semi-discrete multi-component short pulse equation is constructed. Meanwhile, its $N$-soliton solution in terms of pfaffians is also proved.