Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation

Abstract

Rogue waves appearing on deep water or in optical fibres are often modelled by certain breather solutions of the focusing nonlinear Schrödinger (fNLS) equation which are referred to as solitons on finite background (SFBs). A more general modelling of rogue waves can be achieved via the consideration of multiphase, or finite-band, fNLS solutions of whom the standard SFBs and the structures forming due to their collisions represent particular, degenerate, cases. A generalized rogue wave notion then naturally enters as a large-amplitude localized coherent structure occurring within a finite-band fNLS solution. In this paper, we use the winding of real tori to show the mechanism of the appearance of such generalized rogue waves and derive an analytical criterion distinguishing finite-band potentials of the fNLS equation that exhibit generalized rogue waves.

Rogue waves on the double-periodic background in the focusing nonlinear Schrödinger equation

“…as in physical experiments (see [6]), then M 2 exceeds the triple factor for all rogue waves constructed on Figures 5 and 7 as shown in Table 3. Thus, all rogue waves constructed here correspond to physically acceptable rogue waves on the double-periodic background.…”

Multi‐breather and high‐order rogue waves for the nonlinear Schrödinger equation on the elliptic function background

“…The rogue waves for the focusing NLSE on the cn and dn background are obtained by combining the Darboux transformation with nonlinearization of the Lax pair [12]. Meanwhile, there are some relevant works on the rogue waves on the higher genus solutions [4], in which the analytical criterion to generalized rogue waves are obtained by the Riemann-Hilbert method. The bounded ultra-elliptic solutions for the defocusing NLSE are considered by the effective integration method [38].…”

Formation of rogue waves from a locally perturbed condensate

“…They concluded that only initial perturbations of significant amplitude can contain spatially localized breathers, that is consistent with the theory suggested here (the minimal amplitude of SLCP generated by breathers is determined by the perturbation width, see the second paragraph). Meanwhile S. Randoux, P. Suret and G. El have suggested that the key role in the development of periodic perturbations should be attributed to the finite-band solutions of the NLSE [31] (see also [41]). The problems of localized and periodic condensate perturbations are complimentary and both have fundamental importance.…”