On two new types of modified short pulse equation

Abstract

A complex modified short pulse (mSP) equation and a nonlocal mSP equation are presented. The integrability and the covariant property of both new equations are shown by constructing their corresponding Lax pair and Darboux transformation, respectively. For the complex mSP equation, some multiple smooth soliton, cuspon soliton, loop soliton, breather and rogue wave solutions are constructed by N-fold Darboux transformation. For the nonlocal mSP equation, some unstable soliton and stable soliton solutions are obtained. The dynamical characteristics of the obtained solutions are shown through some figures.