Fully localised solitary waves are travelling-wave solutions of the three-dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence for water of finite depth has recently been established, and in this article we present an existence theory for water of infinite depth. The governing equations are reduced to a perturbation of the two-dimensional nonlinear Schrödinger equation, which admits a family of localised solutions. Two of these solutions are symmetric in both horizontal directions and an application of a suitable version of the implicit-function theorem shows that they persist under perturbations.
We prove that the 2D finite depth capillary water wave equations admit no solitary wave solutions. This closes the existence/non-existence problem for solitary water waves in 2D, under the classical assumptions of incompressibility and irrotationality, and with the physical parameters being gravity, surface tension and the fluid depth.
The KP-I equation \[ (u_t-2uu_x+\tfrac{1}{2}(β-\tfrac{1}{3})u_{xxx})_x -u_{yy}=0 \] arises as a weakly nonlinear model equation for gravity-capillary waves with strong surface tension (Bond number $β>1/3$). This equation admits --- as an explicit solution --- a `fully localised' or `lump' solitary wave which decays to zero in all spatial directions. Recently there has been interest in the \emph{full-dispersion KP-I equation} \[u_t + m({\mathrm D}) u_x + 2 u u_x = 0,\] where $m({\mathrm D})$ is the Fourier multiplier with symbol \[ m(k) = \left( 1 + β|k|^2|\right)^{\frac{1}{2}} \left( \frac{\ta…
Fully localised solitary waves are travelling-wave solutions of the three-dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as `lumps'), and a mathematically rigorous existence theory for strong surface tension (Bond number $β$ greater than $\frac{1}{3}$) has recently been given. In this article we present an existence theory for the physically more realistic case $0<β<\frac{1}{3}$. A classical vari…
A fundamental question in the study of water waves is the existence and stability of solitary waves. Solitary waves have been proved to exist and have been studied in many interesting situations, and often arise from the balance of different forces/factors influencing the fluid dynamics, e.g. gravity, surface tension or the fluid bottom. However, the existence of solitary waves has remained an open problem in two of the simplest cases, namely for either pure gravity waves or pure capillary waves in infinite depth. In this article we settle both of these questions in two space dimensions. Preci…
In this paper, we consider capillary-gravity waves propagating on the interface separating two fluids of finite depth and constant density. The flow in each layer is assumed to be incompressible and of constant vorticity. We prove the existence of small-amplitude solitary wave solutions to this system in the strong surface tension regime via a spatial dynamics approach. We then use a variant of the classical Grillakis--Shatah--Strauss (GSS) method to study the orbital stability/instability of these waves. We find an explicit function of the parameters (Froude number, Bond number, and the depth…
We study the existence of traveling wave solutions to a unidirectional shallow water model which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between $0$ and ${1}/{3}$) there exists small amplitude solitary waves that decay to asymptotically small periodic waves at spatial infinity. The size of the oscillations in the far field are shown to be small beyond all algebraic orders in the amplitude of the wave. We also present numeric…
Using a nonlocal version of the center manifold theorem and a normal form reduction, we prove the existence of small-amplitude generalized solitary-wave solutions and modulated solitary-wave solutions to the steady gravity-capillary Whitham equation with weak surface tension. Through the application of the center manifold theorem, the nonlocal equation for the solitary wave profiles is reduced to a four-dimensional system of ODEs inheriting reversibility. Along particular parameter curves, relating directly to the classical gravity-capillary water wave problem, the associated linear operator i…
Considered in this paper is a bi-directional model for the propagation of interfacial capillary-gravity waves in a two-layer system of fluids with rigid lid condition for the upper layer and lower layer with a much larger or infinite depth. The system is derived from a reformulation of the Euler equations for internal waves with nonnegligible surface tension effects in the interface and the corresponding asymptotic model under the Benjamin-Ono regime. Another unidirectional model, so-called regularized Benjamin equation, generalizing the Benjamin equation, is also introduced. Well-posedness of…
We prove existence and conditional energetic stability of solitary-wave solutions for the two classes of pseudodifferential equations $ u_t+\left(f(u)\right)_x-\left(L u\right)_x=0 $ and $ u_t+\left(f(u)\right)_x+\left(L u\right)_t=0, $ where $f$ is a nonlinear term, typically of the form $c|u|^p$ or $cu|u|^{p-1}$, and $L$ is a Fourier multiplier operator of positive order. The former class includes for instance the Whitham equation with capillary effects and the generalized Korteweg-de Vries equation, and the latter the Benjamin-Bona-Mahony equation. Existence and conditional energetic stabil…
We consider three-dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We use a spatial dynamics approach and formulate the steady Euler equations as an infinite-dimensional Hamiltonian system, where an unbounded spatial direction $x$ is considered as a time-like coordinate. In addition we consider wave motions that are periodic in another direction $z$. By analyzing the dispersion relation we detect several bifurcation scenarios,…
This paper studies the nonlinear stability of capillary-gravity waves propagating along the interface dividing two immiscible fluid layers of finite depth. The motion in both regions is governed by the incompressible and irrotational Euler equations, with the density of each fluid being constant but distinct. A diverse collection of small-amplitude solitary wave solutions for this system have been constructed by several authors in the case of strong surface tension (as measured by the Bond number) and slightly subcritical Froude number. We prove that all of these waves are (conditionally) orbi…
