Abstract
This paper investigates the modulational instability and the interactions among nonlinear localized waves such as breathers, rogue waves and semi-rational solitons of the modified self-steepening nonlinear Schrödinger equation. Firstly, the existence conditions for the modulational instability of the plane wave solution to this system are proposed. Secondly, some physically significant phenomena are investigated based on the generalized (2, N-2)-fold Darboux transformation, such as the breather collision with rogue wave, breather collision with semi-rational soliton, semi-rational soliton collision with rogue wave, and the semi-rational soliton collision in orders N=2 and N=3 cases. Finally, the dynamical behaviors of certain localized wave interaction solutions are discussed by performing numerical simulation so that one can predict whether these solutions are dynamically stable enough to propagate in a short time. It is hoped that the results in the present work can be used to understand related physical phenomena in nonlinear optics and relevant fields.
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