Oscillatory spatially periodic weakly nonlinear gravity waves on deep water

A weakly nonlinear Hamiltonian model is derived from the exact water wave equations to study the time evolution of spatially periodic wavetrains. The model assumes that the spatial spectrum of the wavetrain is formed by only three free waves, i.e. a carrier and two side bands. The model has the same symmetries and invariances as the exact equations. As a result, it is found that not only the permanent form travelling waves and their stability are important in describing the time evolution of the waves, but also a new kind of family of solutions which has two basic frequencies plays a crucial role in the dynamics of the waves. It is also shown that three is the minimum number of free waves which is necessary to have chaotic behaviour of water waves

Special solutions to a compact equation for deep-water gravity waves

Recently, Dyachenko & Zakharov (2011) have derived a compact form of the well known Zakharov integro-differential equation for the third order Hamiltonian dynamics of a potential flow of an incompressible, infinitely deep fluid with a free surface. Special traveling wave solutions of this compact equation are numerically constructed using the Petviashvili method. Their stability properties are also investigated. Further, unstable traveling waves with wedge-type singularities, viz. peakons, are numerically discovered. To gain insights into the properties of singular traveling waves, we consider the academic case of a perturbed version of the compact equation, for which analytical peakons with exponential shape are derived. Finally, by means of an accurate Fourier-type spectral scheme it is found that smooth solitary waves appear to collide elastically, suggesting the integrability of the Zakharov equation.Comment: 17 pages, 14 figures, 41 references. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

Periodic travelling waves in a non-integrable one-dimensional lattice

The existence of a one-parameter family of periodic solutions representing longitudinal travelling waves is established for a one-dimensional lattice of identical particles with nearest-neighbour interaction. The potential is not given in closed form but is specified by only a few global properties. The lattice is either infinite or consists ofN particles on a circle with fixed circumference. Waves with low energy are sinusoidal and their properties are studied using bifurcation methods. Waves of high energy, however, are of solitary type, i.e. the excitation is strongly localized

The Discrete Nonlinear Schr\"odinger equation - 20 Years on

We review work on the Discrete Nonlinear Schr\"odinger (DNLS) equation over the last two decades.Comment: 24 pages, 1 figure, Proceedings of the conference on "Localization and Energy Transfer in Nonlinear Systems", June 17-21, 2002, San Lorenzo de El Escorial, Madrid, Spain; to be published by World Scientifi

On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation

We consider the Whitham equation $u_t + 2u u_x+Lu_x = 0$, where L is the nonlocal Fourier multiplier operator given by the symbol $m(\xi) = \sqrt{\tanh \xi /\xi}$. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travelling-wave solution. We find this wave as a limiting case at the end of the main bifurcation curve of $P$-periodic solutions, and give several qualitative properties of it, including its optimal $C^{1/2}$-regularity. An essential part of the proof consists in an analysis of the integral kernel corresponding to the symbol $m(\xi)$, and a following study of the highest wave. In particular, we show that the integral kernel corresponding to the symbol $m(\xi)$ is completely monotone, and provide an explicit representation formula for it.Comment: 40 pages, 3 figures. This version is identical to the one accepted for publication in Annales de l'Institut Henri Poincare, Analyse non lineair

Radiationless Travelling Waves In Saturable Nonlinear Schr\"odinger Lattices

The longstanding problem of moving discrete solitary waves in nonlinear Schr{\"o}dinger lattices is revisited. The context is photorefractive crystal lattices with saturable nonlinearity whose grand-canonical energy barrier vanishes for isolated coupling strength values. {\em Genuinely localised travelling waves} are computed as a function of the system parameters {\it for the first time}. The relevant solutions exist only for finite velocities.Comment: 5 pages, 4 figure