We present explicit forms of general breather (GB), Akhmediev breather (AB), Ma soliton (MS) and rogue wave (RW) solutions of the two component nonlinear Schrödinger (NLS) equation, namely Manakov equation. We derive these solutions through two different routes. In the forward route we first construct a suitable periodic envelope soliton solution to this model from which we derive GB, AB, MS and RW solutions. We then consider the RW solution as the starting point and derive AB, MS and GB in the reverse direction. The second approach has not been illustrated so far for the two component NLS equation. Our results show that the above rational solutions of the Manakov system can be derived from the standard scalar nonlinear Schrödinger equation with a modified nonlinearity parameter. Through this two way approach we establish a broader understanding of these rational solutions which will be of interest in a variety of situations.
We construct dark-dark soliton, general breather (GB), Akhmediev breather (AB), Ma soliton (MS), and rogue wave (RW) solutions of a coupled generalized nonlinear Schrödinger (CGNLS) equation. While dark-dark solitons are captured in the defocusing regime of the 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 shrinks. 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 the RW solution as the starting point, derive AB, MS, and GB in the reverse direction, and show that the solutions obtained in both directions match each other. In the course of the reverse analysis we also demonstrate how to capture the RW solutions directly from AB and MS.
We construct a generalized Darboux transformation (GDT) of a general coupled nonlinear Schrödinger (GCNLS) system. Using GDT method we derive a recursive formula and present determinant representations for N-th order rogue wave solution of this system. Using these representations we derive first, second and third order rogue wave solutions with certain free parameters. By varying these free parameters we demonstrate the formation of triplet, triangle and hexagonal patterns of rogue waves.
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