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We consider the problem of short-term prediction of rare, extreme water waves in unidirectional fields, a critical topic for ocean structures and naval operations. One possible mechanism for the occurrence of such rare, unusually-intense…

Pattern Formation and Solitons · Physics 2015-01-22 Will Cousins , Themistoklis P. Sapsis

Rogue waves are extreme and rare fluctuations of the wave field that have been discussed in many physical systems. Their presence substantially influences the statistical properties of an incoherent wave field. Their understanding is…

Pattern Formation and Solitons · Physics 2018-01-24 Rebecca El Koussaifi , Alexey Tikan , Alessandro Toffoli , Stephane Randoux , Pierre Suret , Miguel Onorato

We analytically study rogue-wave (RW) solutions and rational solitons of an integrable fifth-order nonlinear Schr\"odinger (FONLS) equation with three free parameters. It includes, as particular cases, the usual NLS, Hirota, and…

Pattern Formation and Solitons · Physics 2017-03-31 Yunqing Yang , Zhenya Yan , Boris A. Malomed

The nonlinear Schr\"odinger equation (NLSE) is a rich and versatile model, which in one spatial dimension has stationary solutions similar to those of the linear Schr\"odinger equation as well as more exotic solutions such as solitary waves…

Quantum Gases · Physics 2024-07-08 David B. Reinhardt , Dean Lee , Wolfgang P. Schleich , Matthias Meister

We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schr\"odinger equation (also other equations by natural generalization) formulated with nonzero boundary conditions at infinity. The purpose…

Exactly Solvable and Integrable Systems · Physics 2017-10-19 Deniz Bilman , Peter D. Miller

Using experimental data from three different rogue wave supporting systems, determinism and predictability of the underlying dynamics are evaluated with methods of nonlinear time series analysis. We included original records from the…

Atmospheric and Oceanic Physics · Physics 2015-03-03 Simon Birkholz , Carsten Brée , Ayhan Demircan , Günter Steinmeyer

Akhmediev and Kuznetsov-Ma breathers are rogue wave solutions of the nonlinear Schr\"odinger equation (NLSE). Talbot effect (TE) is an image recurrence phenomenon in the diffraction of light waves. We report the nonlinear TE of rogue waves…

Pattern Formation and Solitons · Physics 2014-03-11 Yiqi Zhang , Milivoj R. Belić , Huaibin Zheng , Haixia Chen , Changbiao Li , Jianping Song , Yanpeng Zhang

In the wind-driven wave regime, the Miles mechanism gives an estimate of the growth rate of the waves under the effect of wind. We consider the case where this growth rate, normalised with respect to the frequency of the carrier wave, is of…

Atmospheric and Oceanic Physics · Physics 2015-06-18 Maura Brunetti , Nadège Marchiando , Nicolas Berti , Jérôme Kasparian

We prove spatiotemporal algebraically decaying estimates for the density of the solutions of the linearly damped nonlinear Schr\"odinger equation with localized driving, when supplemented with vanishing boundary conditions. Their derivation…

Mathematical Physics · Physics 2019-12-10 G. Fotopoulos , N. I. Karachalios , V. Koukouloyannis , K. Vetas

We consider the discrete p-Schr\"odinger (DpS) equation, which approximates small amplitude oscillations in chains of oscillators with fully-nonlinear nearest-neighbors interactions of order alpha = p-1 >1. Using a mapping approach, we…

Pattern Formation and Solitons · Physics 2013-12-18 Guillaume James , Yuli Starosvetsky

In the present work, we revisit the so-called regularized short pulse equation (RSPE) and, in particular, explore the traveling wave solutions of this model. We theoretically analyze and numerically evolve two sets of such solutions. First,…

Pattern Formation and Solitons · Physics 2015-06-19 Y. Shen , T. P. Horikis , P. G. Kevrekidis , D. J. Frantzeskakis

We report water wave experiments performed in a long tank where we consider the evolution of nonlinear deep-water surface gravity waves with the envelope in the form of a large-scale rectangular barrier. Our experiments reveal that, for a…

In the present work, we explore the possibility of developing rogue waves as exact solutions of some nonlinear dispersive equations, such as the nonlinear Schr\"odinger equation, but also, in a similar vein, the Hirota, Davey-Stewartson,…

Pattern Formation and Solitons · Physics 2019-02-15 C. B. Ward , P. G. Kevrekidis

Supratransmission is a fascinating and counterintuitive nonlinear wave phenomenon that enables energy transmission through frequency band gaps. Recent studies have suggested that supratransmission in a damped-driven Klein-Gordon equation…

Pattern Formation and Solitons · Physics 2025-04-29 Abrari Noor Hasmi , Hadi Susanto

We show that the same special solution of the focusing nonlinear Schr\"{o}dinger equation that has been shown to arise in a certain near-field/large-order limit from soliton and Peregrine-like rogue wave solutions actually arises…

Exactly Solvable and Integrable Systems · Physics 2022-05-04 Deniz Bilman , Peter D. Miller

We develop a general analytical framework for determining the probability distribution of random nonlinear wave fields governed by the focusing nonlinear Schr\"odinger equation (fNLSE) in regimes where typical realizations are dominated by…

Pattern Formation and Solitons · Physics 2026-05-08 T. Congy , G. A. El

The nonlinear Schr\"odinger equation (NLSE) is a fundamental model for wave dynamics in nonlinear media ranging from optical fibers to Bose-Einstein condensates. Accurately estimating its parameters, which are often strongly correlated,…

Quantum Physics · Physics 2025-09-24 Louis Rossignol , Tangui Aladjidi , Myrann Baker-Rasooli , Quentin Glorieux

Different from statistical considerations on stochastic wave fields, this paper aims to contribute to the understanding of (some of) the underlying physical phenomena that may give rise to the occurrence of extreme, rogue, waves. To that…

Fluid Dynamics · Physics 2011-10-25 E. van Groesen , Andonowati , N. Karjanto

In a recent work we studied the first nonlinear stage of modulation instability (NLSMI) of x-periodic anomalous (rogue, freak, extreme) waves (AWs) of physically relevant multidimensional (generalizations of the focusing) nonlinear…

Pattern Formation and Solitons · Physics 2026-05-27 Francesco Coppini , Paolo Maria Santini

We present a brief overview on the existence/nonexistence of standing waves for the NonLinear Schr\"odinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. We first focus on the NLSE, both in the…

Analysis of PDEs · Mathematics 2019-02-06 William Borrelli , Raffaele Carlone , Lorenzo Tentarelli