The finite gap method and the periodic Cauchy problem of $2+1$ dimensional anomalous waves for the focusing Davey-Stewartson 2 equation

TL;DR

This paper extends the finite gap method to analyze the periodic Cauchy problem for anomalous waves in the 2+1 dimensional focusing Davey-Stewartson 2 equation, revealing explicit solutions for small initial perturbations.

Contribution

It develops a leading-order solution framework for the 2+1D focusing DS2 equation using finite gap theory, generalizing periodic AW analysis from 1+1D NLS to higher dimensions.

Findings

Solution expressed in elementary functions of initial data

Extension of periodic AW theory to 2+1 dimensions

Leading-order approximation for small perturbations

Abstract

The focusing Nonlinear Schr\"odinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of $1+1$ dimensional 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 analogy with the recently developed analytic theory of periodic AWs of the focusing NLS equation, in this paper we extend these results to a $2+1$ dimensional context, concentrating on the focusing Davey-Stewartson 2 (DS2) equation, an integrable $2+1$ dimensional generalization of the focusing NLS equation. More precisely, we use the finite gap theory to solve, to leading order, the doubly periodic Cauchy problem of the focusing DS2 equation for small initial perturbations of the unstable background solution, what we call the periodic Cauchy problem of the AWs. As in the NLS case, we show that, to leading order, the solution of this Cauchy problem is expressed in terms of elementary functions of the initial data.