The Euler equations for water waves in any depth have been shown to have solitary wave solutions when the effect of surface tension is included. This paper proposes three quadratic model equations for these types of waves in infinite depth with a two-dimensional fluid domain. One model is derived directly from the Euler equations. Two further simpler models are proposed, both having the full gr...

The effect of weak lateral dispersion of Zakharov-Kutznetsov-type on a Benjamin-Ono solitary wave is studied both asymptotically and numerically. The asymptotic solution is based on an approximate variational solution for the solitary wave, which is then modulated in time through the use of conservation equations. The effect of the dispersive radiation shed as the solitary wave evolves is also ...

Our concern in this talk is the problem of free surface water waves, the form of solitary wave solutions, and their behavior under collisions. Solitary waves for the Euler equations have been described since the time of Stokes. In a long wave perturbation regime they are well described by single soliton solutions of the Korteweg deVries equation (KdV). It is a famous result that multiple solito...

Nonlinear solitary structures of electron plasma waves have been investigated by using nonlinear quantum fluid equations for electrons with an arbitrary temperature. It is shown that the electron degeneracy parameter has significant effects on the linear and nonlinear properties of electron plasma waves. Depending on its value both compressive and rarefactive solitons can be excited in the mode...

This article presents a rigorous existence theory for small-amplitude three-dimensional travelling water waves. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable. Wave motions which are periodic in a second, different horizontal direction are detected using a centre-manifold reduction...

The interaction of solitary waves on shallow water is examined to fourth order. At first order the interaction is governed by the Korteweg–de Vries (KdV) equation, and it is shown that the unidirectional assumption, of right-moving waves only, is incompatible with mass conservation at third order. To resolve this, a mass conserving system of KdV equations, involving both rightand left-moving wa...

In both the ocean and the atmosphere, the interaction of a density stratified flow with topography can generate large-amplitude, horizontally propagating internal solitary waves. Often these waves are observed in regions where the waveguide properties vary in the direction of propagation. In this article we consider nonlinear evolution equations of the Kortewegde Vries type, with variable coeff...

The structure and stability of different types of multicolor optical spatial solitary waves created by interaction of light at a central frequency with two sideband waves both through cross-phase modulation and parametric four-wave mixing is presented. It is shown that a novel type of three-color spatial soliton appears above a power threshold when parametric coupling generates an instability o...

The theory for criticality presented in Part 1 is extended to the unsteady problem, and a new formulation of the Benjamin–Feir instability for Stokes waves in finite depth coupled to a mean flow, which takes the criticality matrix as an organizing centre, is presented. The generation of unsteady dark solitary waves at points of stability changes and their connection with the steady dark solitar...

[1] The evolution and breaking of internal solitary waves in a shallow upper layer as they approach a constant bottom slope is examined through laboratory experiments. The waves are launched in a two-layer fluid through the standard lock-release method. In most experiments, the wave amplitude is significantly larger than the depth of the shallow upper layer so that they are not well described b...