Generalized Darboux transformation and Nth order rogue wave solution of a general coupled nonlinear Schrödinger equations

Abstract

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.

Rational solutions to a KdV-like equation

“…Their rational solutions, both singular and non-singular, or even rogue wave solutions will be a very interesting topic. Particularly, higher-order rogue wave solutions should be connected with generalized Wronskian solutions [19,20] and generalized Darboux transformations [21,22]. …”

On the characterization of breather and rogue wave solutions and modulation instability of a coupled generalized nonlinear Schrödinger equations

“…Using GDT we have derived a recursive formula and presented determinant representations for Nth order RW solution of this system. Using these representations we have displayed second order RW solution with two free parameters and the third order RW solution with four free parameters [18]. We have also analyzed the second and third order RW solutions by varying these free parameters and shown that one can obtain certain interesting structures.…”

“…We have also analyzed the second and third order RW solutions by varying these free parameters and shown that one can obtain certain interesting structures. In the case of second order RW, we have shown that these RWs exhibit a triplet pattern and in the case of third order RW solution we have visualized a triangular and hexagonal structures depending upon the restriction on the free parameters [18]. In continuation of earlier studies, in this work, we aim to investigate the interaction behaviors in nonlinear localized modes such as solitons, breather and rogue wave in the CGNLS system.…”

“…In a follow-up work, we have constructed Nth order RW solution to CGNLS system (1) using generalized Darboux transformation (GDT) method [18]. It is very difficult to construct higher order rogue wave solutions using classical DT method since higher order rogue wave solutions do contain only one critical eigenvalue.…”

Exact traveling wave solutions of an autonomous system via the enhanced (G′/G)-expansion method

“…Thus, the methods for deriving exact solutions for the governing equations have to be developed. As a result, numerous techniques of obtaining traveling wave solutions have been developed over last three decades, such as, the tanh-coth function method, [2,3] the Kudryashov method, [4] the Exp-function method, [5][6][7][8] the modified simple equation method, [9][10][11][12] the (G′/G)-expansion method, [13][14][15][16][17][18][19][20][21][22] the exp(−Φ(ξ))-expansion method, [23,24] the Darboux transformation method, [25] the Hirota method, and [26] the differential transform method. [27][28][29][30] From our point of view, all these methods have some merits and demerits with respect to the problem considered and there is no unified method that can be used to deal with all types of NLEEs.…”