Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions

Abstract

In this paper, we construct a generalized Darboux transformation for the nonlinear Schrödinger equation. The associated N-fold Darboux transformation is given in terms of both a summation formula and determinants. As applications, we obtain compact representations for the Nth-order rogue wave solutions of the focusing nonlinear Schrödinger equation and Hirota equation. In particular, the dynamics of the general third-order rogue wave is discussed and shown to exhibit interesting structures.

Rational W-shaped solitons on a continuous-wave background in the Sasa-Satsuma equation

“…3. The distribution shape is identical with the well-known fundamental RW solution with highest peak value of the simplified NLS [1,[22][23][24]. These characters are different from the fundamental W-shaped soliton in Case 1.…”

“…Notably, we find a new type rational solution on continuous background with some certain conditions on the background's amplitude and frequency c ≥ 2w. We find that the rational solution does not correspond to rogue wave, in contrast to the ones of the simplified NLS [1,[22][23][24]. Its dynamics corresponds to soliton's which has a stable distribution shape with evolution, and the distribution shape like a "W" which has one hump and two valleys on the hump's two sides.…”

High-order rogue wave solutions for the coupled nonlinear Schrödinger equations-II

“…By choosing parameters, we can obtain the different types of RW solution. The three RWs can be superposed together, and construct symmetric structure as the second-order RW with highest peak for scalar NLS [13][14][15][16][17]. However, it is usually very complicated to obtain the symmetric structure since there are much more parameters than the one for scalar NLS.…”

“…We are not sure whether this pattern can emerge in this two-component coupled system. c) Second-order rogue wave solution As the second-order RW in scalar system [13][14][15][16][17], the second-order RW solution here is still obtained by superposition of fundamental RW solutions with the same spectral parameter. Therefore, one can obtain three fundamental RWs with identical pattern in each component for the second-order RW solution, which is similar to the ones in scalar case.…”

“…Secondly, the number of RW in temporal-spatial distribution plane is also different from the ones in scalar systems. We have demonstrated that two or four fundamental RWs can emerge on the distribution plane [6,12], which is absent for scalar systems [13][14][15][16][17].…”