General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation

Abstract

General high-order rogue waves in the nonlinear Schrödinger equation are derived by the bilinear method. These rogue waves are given in terms of determinants whose matrix elements have simple algebraic expressions. It is shown that the general N -th order rogue waves contain N − 1 free irreducible complex parameters. In addition, the specific rogue waves obtained by Akhmediev et al. (Akhmediev et al. 2009 Phys. Rev. E 80, 026601 (doi:10.1103/PhysRevE.80.026601)) correspond to special choices of these free parameters, and they have the highest peak amplitudes among all rogue waves of the same order. If other values of these free parameters are taken, however, these general rogue waves can exhibit other solution dynamics such as arrays of fundamental rogue waves arising at different times and spatial positions and forming interesting patterns.

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

“…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].…”

Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents

“…Note that equation (4.16) is similar to but not the same as our integrable two-dimensional NLS equation owing to the appearance of the term iα 32 A 0ζ ζ ζ . However, fortunately we have another equation (4.14) at our disposal, obtained at a lower order.…”

“…The maximum amplitude and modular inclination reachable by this class of solutions are fixed owing to the absence of relevant free tunable parameters, making it difficult to adjust to the continuously varied range of shape and sizes of the observed oceanic RWs. However, recently, higher order rational solutions to the NLS equation allowing free parameters have been discovered [26,27,32], though they seem to represent multi-peak waves in the x-t plane for the non-trivial choice of parameters [27]. The single-peak solution, which is suitable for describing RWs having a single appearance in time, is obtained unfortunately for a trivial choice of the free parameters.…”