Soliton, breather and rational solutions of a high-order modified derivative nonlinear Schrödinger equation

Nonlinear Schr & ouml;dinger equation and derivative nonlinear Schr & ouml;dinger equation play the pivotal roles in nonlinear optics. In this paper, we mainly investigate a high-order modified derivative nonlinear Schr & ouml;dinger (h-MDNLS) equation through Darboux transformation method, which can be obtained from the generalized modified derivative nonlinear Schr & ouml;dinger (g-MDNLS) hierarchy with constraints. A variety of analytic solutions are constructed. We present exact soliton solutions of the h-MDNLS equation based on the zero seed solution. Furthermore, we analyze the dynamics and asymptotic behavior of the two-soliton solutions. Interestingly, the breather wave is generated by the interaction of two parallel solitary waves. In addition, we also derive the breather solutions, including Kuznetsov-Ma (KM) breather, Akhmediev breather and rational solutions of the h-MDNLS equation with the nonzero seed solution. We must emphasize that breather solutions and rational solutions of the h-MDNLS equation are studied for the first time, which have not been reported in the literature.