The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes

TL;DR

This paper analytically describes the exact recurrence of rogue waves in the periodic NLS equation with one or two unstable modes using matched asymptotic expansions, linking initial data to rogue wave parameters.

Contribution

It introduces a novel analytical approach employing matched asymptotic expansions to precisely characterize rogue wave recurrence in the periodic NLS setting.

Findings

Exact rogue wave recurrence described analytically.

Nonlinear stages modeled by Akhmediev breathers.

Parameters of rogue waves depend explicitly on initial data.

Abstract

The focusing Nonlinear Schr\"odinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, the main physical mechanism for the generation of rogue (anomalous) waves (RWs) in Nature. In this paper we investigate the $x$-periodic Cauchy problem for NLS for a generic periodic initial perturbation of the unstable constant background solution, in the case of $N=1,2$ unstable modes. We use matched asymptotic expansion techniques to show that the solution of this problem describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI, and that the nonlinear RW stages are described by the N-breather solution of Akhmediev type, whose parameters, different at each RW appearence, are always given in terms of the initial data through elementary functions. This paper is motivated by a preceeding work of the authors in which a different approach, the finite gap method, was used to investigate periodic Cauchy problems giving rise to RW recurrence.