The finite-gap method and the periodic NLS Cauchy problem of anomalous waves for a finite number of unstable modes

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, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of NLS (what we call the Cauchy problem of the AWs), in the case of a finite number N of unstable modes. We show how the finite gap method adapts to this specific Cauchy problem through three basic simplifications, allowing one to construct the solution, at the leading and relevant order, in terms of elementary functions of the initial data. More precisely, we show that, at the leading order, i) the initial data generate a partition of the time axis into a sequence of finite intervals; ii) in each interval I of the partition, only a subset of N (I) ≤ N unstable modes are "visible", and iii) the NLS solution is approximated, for t ∈ I, by the N (I)-soliton solution of Akhmediev type, describing the nonlinear interaction of these visible unstable modes, whose parameters are expressed in terms of the initial data through elementary functions. This result explains the relevance of the m-soliton solutions of Akhmediev type, with m ≤ N , in the generic periodic Cauchy problem of the AWs, in the case of a finite number N of unstable modes. 1 arXiv:1810.09247v2 [math-ph]

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

“…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…”

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

“…Universal behavior of the modulationally unstable constant background was studied asymptotically in [11]. The finite-gap method was employed to relate the unstable modes on the constant background with the occurrence of rogue waves [24,25]. Rogue waves of infinite order were constructed in [10] based on recent developments in the inverse scattering method [9].…”

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

“…Recent study by Grinevich and Santini [19] (equations (48)-(49) ) shows finite-gap soliton solutions of genius-1 or genius-2 may be developed from a Cauchy problem of the NLSE with a very small perturbation. For the genius-1 case, a nonzero background expressed by elliptic function is formed before the rogue wave solution or the Akhmediev breather solution is formed.…”