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

Abstract

We first derive an integrable deformed hierarchy of a short pulse equation and its Lax representation. Then we concentrate on the solution of the 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 the short pulse equation are obtained.