Classifying the hierarchy of nonlinear-Schrödinger-equation rogue-wave solutions

Abstract

We present a systematic classification for higher-order rogue-wave solutions of the nonlinear Schrödinger equation, constructed as the nonlinear superposition of first-order breathers via the recursive Darboux transformation scheme. This hierarchy is subdivided into structures that exhibit varying degrees of radial symmetry, all arising from independent degrees of freedom associated with physical translations of component breathers. We reveal the general rules required to produce these fundamental patterns. Consequently, we are able to extrapolate the general shape for rogue-wave solutions beyond order 6, at which point accuracy limitations due to current standards of numerical generation become non-negligible. Furthermore, we indicate how a large set of irregular rogue-wave solutions can be produced by hybridizing these fundamental structures.

Collision of Akhmediev Breathers in Nonlinear Fiber Optics

“…The richer family of multirogue wave solutions of the NLSE is also very attractive for future works. These more general higher-order solutions exhibit multiple maxima, i.e., multiple rogue waves, with a complex spatiotemporal arrangement [27,[33][34][35][36].…”

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].…”

Patterns of deformations of Peregrine breather of order 3 and 4 solutions to the NLS equation with multi parameters

“…The extrapolation was done until the order N ¼ 9 in [36]. Ohta and Yang [37] presented the study of the case cas N ¼ 3 with rings and triangles.…”