The new integrable deformations of short pulse equation and sine-Gordon equation, and their solutions

TL;DR

This paper introduces new integrable deformations of the short pulse and sine-Gordon equations, deriving their hierarchies, Lax representations, and explicit multisoliton, negaton, and positon solutions.

Contribution

It presents novel integrable deformations of the short pulse and sine-Gordon equations, along with their solutions and Lax pairs, expanding the understanding of these equations.

Findings

Derived integrable deformed hierarchies and Lax representations.

Constructed multisoliton, negaton, and positon solutions.

Obtained reduced solutions for the original short pulse equation.

Abstract

We first derive an integrable deformed hierarchy of short pulse equation and their Lax representation. Then we concentrated on the solution of integrable deformed short pulse equation (IDSPE). By proposing a generalized reciprocal transformation, we find a new integrable deformed sine-Gordon equation (IDSGE) and its Lax representation. The multisoliton solutions, negaton solutions and positon solutions for the IDSGE and the N-loop soliton solutions, N-negaton and N-positon solutions for the IDSPE are presented. In the reduced case the new N-positon solutions and N-negaton solutions for short pulse equation are obtained.