It is now well known that the focussing nonlinear Schrödinger equation allows plane waves to be modulationally unstable, and at the same time supports breather solutions which are often invoked as models for rogue waves. This suggests a direct connection between modulation instability and the existence of rogue waves. In this chapter we review this connection for a suite of long wave models, su...

Rogue waves are extraordinarily high and steep isolated waves, which appear suddenly in a calm sea and disappear equally fast. However, though the rogue waves are localized surface waves, their theoretical models and experimental observations are available mostly in one dimension, with the majority of them admitting only limited and fixed amplitude and modular inclination of the wave. We propos...

In this paper, a quasi-rogue wave solution of the Camassa-Holm equation is obtained by discrete method and Hirota’s bilinear transform method. The wave’s amplitude suddenly changes high and then suddenly disappears. The whole process experiences a very short time. Waveform’s characteristics are similar to rogue waves in shallow water. This obtained solution can be used to describe the possible ...

Rogue waves [1]—extreme ocean waves with heights up to thirty meters—have been common elements in stories from people working at sea, as well as in popular fiction. It is not hard to imagine why: being on a small boat in a stormy sea swell and suddenly encountering a wave maybe three times as high as the surrounding waves is a terrifying experience (Fig. 1). However, up to a few decades ago, oc...

Experimental observations of rare giant pulses or rogue waves were done in the output intensity of an optically injected semiconductor laser. The long-tailed probability distribution function of the pulse amplitude displays clear non-Gaussian features that confirm the rogue wave character of the intensity pulsations. Simulations of a simple rate equation model show good qualitative agreement wi...

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the ...

Article history: Received 7 December 2010 Accepted 10 December 2010 Available online 17 December 2010 Communicated by V.M. Agranovich We study the spectra of the Peregrine soliton and higher-order rational solutions of the nonlinear Schrödinger equation (NLSE), which we use as a model of the rogue waves in optics and in the deep ocean. We show that these solutions have specific triangular spect...

Rogue waves observed in the ocean and elsewhere are often modelled by certain solutions of the nonlinear Schrodinger equation, describing the modulational instability of a plane wave and the subsequent development of multi-phase nonlinear wavetrains. In this paper, we describe how integrability and application of the inverse scattering transform can be used to construct a class of explicit asym...

We present a method for finding the hierarchy of rational solutions of the self-focusing nonlinear Schrödinger equation and present explicit forms for these solutions from first to fourth order. We also explain their relation to the highest amplitude part of a field that starts with a plane wave perturbed by random small amplitude radiation waves. Our work can elucidate the appearance of rogue ...

In the presence of many waves, giant events can occur with a probability higher than expected for random dynamics. By studying linear light propagation in a glass fiber, we show that optical rogue waves originate from two key ingredients: granularity, or a minimal size of the light speckles at the fiber exit, and inhomogeneity, that is, speckles clustering into separate domains with different a...