Dark solitons, Breathers and Rogue Wave Solutions of the Coupled Generalized Nonlinear Schr\"odinger Equations

Abstract

We construct dark-dark solitons, general breather (GB), Akhmediev breather (AB), Ma soliton (MS) and rogue wave (RW) solutions of a coupled generalized nonlinear Schr\"odinger (CGNLS) equation. While the dark-dark solitons are captured in the defocusing regime of CGNLS system, the other solutions, namely GB, AB, MS and RW, are identified in the focusing regime. We also analyze the structures of GB, AB, MS and RW profiles with respect to the four-wave mixing parameter. We show that when we increase the value of the real part of the four wave mixing parameter, the number of peaks in the breather profile increases and the width of each peak gets shrunk. Interestingly the direction of this profile also changes due to this change. As far as the RW profile is concerned the width of the peak becomes very thin when we increase the value of this parameter. Further, we consider RW solution as the starting point and derive AB, MS and GB in the reverse direction and show that the solutions obtained in both the directions match with each other. In the course of the reverse analysis we also demonstrate how to capture the RW solutions directly from AB and MS.