In 1968 V.E. Zakharov derived the Nonlinear Schrödinger equation for the 2D water wave problem in the absence of surface tension, i.e., for the evolution of gravity driven surface water waves, in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet. In this paper we give a rigorous proof that the wave packets in the two-dimensional water...

Explicit formulas are given for the multi-peakon-antipeakon solutions of the Camassa– Holm equation, and a detailed analysis is made of both short-term and long-term aspects of the interaction between a single peakon and single anti-peakon. The strongly nonlinear equation ( 1 − 1 4 D ) ut = 3 2 ( u ) x − 1 8 ( u2x ) x − 1 4 (uuxx)x, (1) introduced by Camassa and Holm [5] as a possible model for...

Modulated deep-water 1D Stokes waves are considered experimentally and theoretically. Wave trains are modulated in a controlled fashion and their evolution is recorded. Data from repeated laboratory experiments are reproducible near the wave maker, but diverge away from the wave maker. Numerical integration of a perturbed nonlinear Schrodinger equation and an associated linear spectral problem ...

Nonlinear differential equations are known to describe a wide variety of phenomena not only in physics, in which applications extend over magnetofluid dynamics, water surface gravity waves, electromagnetic radiation reactions, and ion acoustic waves in plasmas, just to name a few, but also in biology and chemistry. It is one of the important tasks in the study of nonlinear differential equation...

In the classical water-wave problem, fully localized nonlinear waves of permanent form, commonly referred to as lumps, are possible only if both gravity and surface tension are present. While much attention has been paid to shallow-water lumps, which are generalizations of Korteweg-de Vries solitary waves, the present study is concerned with a distinct class of gravity-capillary lumps recently ...

We discuss physical and statistical properties of rogue wave generation in deep water from the perspective of the focusing Nonlinear Schrödinger equation and some of its higher order generalizations. Numerical investigations and analytical arguments based on the inverse spectral theory of the underlying integrable model, perturbation analysis, and statistical methods provide a coherent picture ...

We study the stability of spatially periodic solutions to the Kawahara equation, a fifth order, nonlinear partial differential equation. The equation models the propagation of nonlinear water-waves in the long-wavelength regime, for Weber numbers close to 1/3 where the approximate description through the Korteweg-de Vries (KdV) equation breaks down. Beyond threshold, Weber number larger than 1/...

In problems of dispersive wave propagation governed by two distinct restoring-force mechanisms, the phase speed of linear sinusoidal wavetrains may feature a minimum, cmin, at non-zero wavenumber, kmin. Examples include waves on the surface of a liquid in the presence of both gravity and surface tension, flexural waves on a floating ice sheet, in which case capillarity is replaced by the flexur...

The transformation of a weakly nonlinear interfacial solitary wave in an ideal twolayer flow over a step is studied. In the vicinity of the step the wave transformation is described in the framework of the linear theory of long interfacial waves, and the coefficients of wave reflection and transmission are calculated. A strong transformation arises for propagation into shallower water, but a we...