Dam break problem for the focusing nonlinear Schrödinger equation and the generation of rogue waves

El, Gennady, Khamis, Eduardo and Tovbis, Alex (2016) Dam break problem for the focusing nonlinear Schrödinger equation and the generation of rogue waves. Nonlinearity, 29 (9). pp. 2798-2836. ISSN 0951-7715

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Official URL: https://doi.org/10.1088/0951-7715/29/9/2798

Abstract

We propose a novel, analytically tractable, scenario of the rogue wave formation in the framework of the small-dispersion focusing nonlinear Schrödinger (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.

Item Type: Article
Uncontrolled Keywords: nonlinear Schrödinger equation, rogue waves, modulation theory, semi-classical limit, Riemann–Hilbert problem
Subjects: G100 Mathematics
Department: Faculties > Engineering and Environment > Mathematics, Physics and Electrical Engineering
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Depositing User: Becky Skoyles
Date Deposited: 30 Nov 2018 16:06
Last Modified: 11 Oct 2019 10:01
URI: http://nrl.northumbria.ac.uk/id/eprint/37004

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