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

DOI10.1070/RM9863zbMath1454.35340arXiv1810.09247WikidataQ128014773 ScholiaQ128014773MaRDI QIDQ5220224

Paolo Maria Santíni, Petr G. Grinevich

Publication date: 11 March 2020

Published in: Russian Mathematical Surveys (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1810.09247

zbMATH Keywords

focusing nonlinear Schrödinger equationasymptotics in terms of elementary functionsfinite-gap approximationperiodic Cauchy problem for anomalous wavesRiemann surfaces close to degenerate ones

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) Nonlinear waves in solid mechanics (74J30) Water waves, gravity waves; dispersion and scattering, nonlinear interaction (76B15) Solitary waves for incompressible inviscid fluids (76B25) NLS equations (nonlinear Schrödinger equations) (35Q55) Lasers, masers, optical bistability, nonlinear optics (78A60) Soliton solutions (35C08)

Related Items (10)

Solitons, breathers and rational solutions for a (2+1)-dimensional dispersive long wave system ⋮ The effect of loss/gain and Hamiltonian perturbations of the Ablowitz—Ladik lattice on the recurrence of periodic anomalous waves ⋮ Modulation instability, periodic anomalous wave recurrence, and blow up in the Ablowitz–Ladik lattices ⋮ The periodic N breather anomalous wave solution of the Davey–Stewartson equations; first appearance, recurrence, and blow up properties ⋮ Inverse scattering transform for nonlinear Schrödinger systems on a nontrivial background: a survey of classical results, new developments and future directions ⋮ The periodic Cauchy problem for PT-symmetric NLS, I: the first appearance of rogue waves, regular behavior or blow up at finite times ⋮ Sato Grassmannian and degenerate sigma function ⋮ On closed finite gap curves in spaceforms. I ⋮ Periodic standing waves in the focusing nonlinear Schrödinger equation: rogue waves and modulation instability ⋮ The linear and nonlinear instability of the Akhmediev breather

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