Soliton solutions of a novel nonlocal Hirota system and a nonlocal complex modified Korteweg–de Vries equation

Abstract

This paper proposes a novel nonlocal Hirota system by adding constraints to the coupled Hirota equation, which can be transformed into a nonlocal complex modified Korteweg–de Vries (mKdV) equation with a Galilean transformation. Since the nonlocal Hirota equation and nonlocal complex mKdV equation can be converted into each other, we only need to solve one of them. Starting from the eigenfunctions of the Lax pair and the adjoint Lax pair of the nonlocal complex mKdV equation, the N-fold Darboux transformation (DT) of this equation is constructed. Subsequently, bright soliton solutions, double-hump soliton solutions, breather-like solutions under zero background, and multi-breather solutions under nonzero constant background are obtained for the nonlocal complex mKdV equation. Besides that, the interaction behaviors of the multi-soliton waves are also studied. There are some interesting phenomenons. The breather-II soliton wave, that can be seen as a combination of bright wave and dark wave, will appear. It does not appear alone but in the multi-breather solution.