Dam break problem for the focusing nonlinear Schr\"odinger equation and the generation of rogue waves

TL;DR

This paper introduces an analytically tractable scenario for rogue wave formation in the focusing NLS equation, using a box initial condition and advanced mathematical techniques to describe wave interactions and rogue wave emergence.

Contribution

It presents a novel analytical approach combining Whitham modulation theory and inverse scattering to model rogue wave generation from a rectangular barrier in the focusing NLS equation.

Findings

Interaction of dispersive dam break flows leads to large-amplitude breather lattices.

Analytical results closely match numerical simulations.

Modulated breather profiles resemble Akhmediev and Peregrine breathers.

Abstract

We propose a novel, analytically tractable, scenario of the rogue wave formation in the framework of the small-dispersion focusing nonlinear Schr\"odinger (NLS) equation with the initial condition in the form of a rectangular barrier (a "box"). We use the Whitham modulation theory combined with the nonlinear steepest descent for the semi-classical inverse scattering transform, to describe the evolution and interaction of two counter-propagating nonlinear wave trains --- the dispersive dam break flows --- generated in the NLS box problem. We show that the interaction dynamics results in the emergence of modulated large-amplitude quasi-periodic breather lattices whose amplitude profiles are closely approximated by the Akhmediev and Peregrine breathers within certain space-time domain. Our semi-classical analytical results are shown to be in excellent agreement with the results of direct numerical simulations of the small-dispersion focusing NLS equation.