The finite gap method and the analytic description of the exact rogue wave recurrence in the periodic NLS Cauchy problem. 1

Abstract

The focusing Nonlinear Schrödinger (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 appearence 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 1-breather Akhmediev solution, whose parameters, different at each RW appearence, 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 1 arXiv:1707.05659v2 [nlin.SI] 22 Jul 2017 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. Since the solution of the Cauchy problem is given in terms of different elementary functions in different time intervals, obviously matching in the corresponding overlapping regions, an alternative approach, based on matched asymptotic expansions, is suggested and presented in a separate paper in which the RW recurrence, in the case of a finite number of unstable modes and of a generic initial perturbation exciting all of them, is again described in term of elementary functions.

Effect of a small loss or gain in the periodic nonlinear Schrödinger anomalous wave dynamics

“…• If the initial condition is the highly non generic Akhmediev breather (14), as in the experiments in [58], then all the NLS spectral gaps are initially closed [39,42], β = 0, and Q m = −(ν/ 2 )(σ 1 /|a| 4 )m is real. It follows that we are basically as in the case ν 2 , and, after the first AW appearance, essentially not affected by loss/gain, the solution enters immediately the SVAs (29) (see Figure 3).…”

“…To do it, we make use of the following ingredients. i) Some aspects of the deterministic theory of periodic AWs recently developed in [39,40] using the finite-gap method; ii) few basic aspects of the theory of perturbations of soliton PDEs (developed in the infinite line case in [47,48]), and in the finite-gap case in [32]); iii) the classical Darboux transformations for NLS [91,68].…”

“…Here we summarize the aspects of the finite gap theory and of the results obtained in [39,40] used in this work. If Ψ(λ, x, t) is a fundamental matrix solution of (31), (32) such that Ψ(λ, 0, 0) is the identity, then the monodromy matrix T (λ) is the entire function of λ…”

“…If 1 ≤ n ≤ N , where N is the number of unstable modes, λ + n splits into the pair of branch points (E 2n−1 , E 2n ), and λ − n into the pair of branch points (Ē 2n−1 ,Ē 2n ); if n > N , each λ n splits into a pair of complex conjugate eigenvalues. In the simplest case in which N = 1, E 1 , E 2 are the branch points obtained through the splitting of the excited mode λ + 1 = i|a| sin φ 1 =: λ 1 , and [39]…”

“…In [39,40] it was suggested to approximate the solution of the generic Cauchy problem, corresponding to infinitely many gaps (see Figure 5 (left picture)), by the finite-gap solution obtained closing all gaps near the real line, since they correspond to the stable modes and contribute to the solution to O( ) (see Figure 5 (right picture)). If N = 1, we obtain a genus 2 spectral curve with two O( ) handles (see Figure 5 (right picture)).…”