The finite gap method and the analytic description of the exact rogue wave recurrence in the periodic NLS Cauchy problem. 1
Abstract
The focusing NLS equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, considered the main physical mechanism for the appearance of rogue (anomalous) waves (RWs) in Nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for periodic initial perturbations of the unstable background solution of NLS exciting just one of the unstable modes. We distinguish two cases. In the case in which only the corresponding unstable gap is theoretically open, the solution describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI, and the nonlinear RW stages are described by the 1breather Akhmediev solution, whose parameters, different at each RW appearance, are always given in terms of the initial data through elementary functions. If the number of unstable modes is >1, this uniform in t dynamics is sensibly affected by perturbations due to numerics and/or real experiments, provoking O(1) corrections to the result. In the second case in which more than one unstable gap is open, a detailed investigation of all these gaps is necessary to get a uniform in $t$ dynamics, and this study is postponed to a subsequent paper. It is however possible to obtain the elementary description of the first nonlinear stage of MI, given again by the Akhmediev 1breather solution, and how perturbations due to numerics and/or real experiments can affect this result.
 Publication:

arXiv eprints
 Pub Date:
 July 2017
 arXiv:
 arXiv:1707.05659
 Bibcode:
 2017arXiv170705659G
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Mathematical Physics;
 Physics  Atmospheric and Oceanic Physics;
 Physics  Optics
 EPrint:
 68 pages, Remark at page 14 and formula (32) of version 1 have been removed