Integrable discretizations for the short wave model of the Camassa-Holm equation

TL;DR

This paper establishes integrable discretizations of the short wave model of the Camassa-Holm equation, linking it to Toda lattice equations, and provides explicit solutions for these discrete models.

Contribution

It introduces new integrable semi-discrete and full-discrete versions of the SCHE with explicit determinant solutions, expanding the understanding of discrete integrable systems.

Findings

Derived bilinear equations linking SCHE and 2DTL

Constructed explicit N-cuspon solutions in Casorati form

Presented determinant solutions for discrete SCHE models

Abstract

The link between the short wave model of the Camassa-Holm equation (SCHE) and bilinear equations of the two-dimensional Toda lattice (2DTL) is clarified. The parametric form of N-cuspon solution of the SCHE in Casorati determinant is then given. Based on the above finding, integrable semi-discrete and full-discrete analogues of the SCHE are constructed. The determinant solutions of both semi-discrete and fully discrete analogues of the SCHE are also presented.