Breather and Rogue Wave Solutions on the Different Periodic Backgrounds in the Focusing Nonlinear Schrödinger Equation

Abstract

In this paper, we construct breather and rogue wave solutions on the different periodic backgrounds in the focusing nonlinear Schrödinger equation by using the Darboux transformation. First, we present solutions of the Lax pair related to the periodic seed solutions with trivial and nontrivial phases. In this process, different from the previous approaches of employing the nonlinearization of the Lax pair or the traveling wave transformation, we mainly combine the proper assumption with the method of separation of variables. This strategy is more direct and simpler and can be extended to other nonlinear integrable equations. Second, we construct the Kuznetsov–Ma breather and the spatiotemporally periodic breather on the periodic background. Their asymptotic expressions are obtained, which can be used to show that the related nonlinear waves appear on the periodic background. The corresponding dynamical properties and evolution states are illustrated graphically. Finally, at branch points of breathers, the rogue waves on the periodic background are derived and their characteristics are analyzed. For breather and rogue wave solutions, we both investigate the relationship between parameters and solutions' structures and the limits when the elliptic modulus approach to 0 and 1. All the results in this paper might be helpful for us to understand the dynamics of breathers and rogue waves on the periodic background.