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

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