Related papers: Overhanging solitary water waves
In this paper we construct periodic capillarity-gravity water waves with an arbitrary bounded vorticity distribution. This is achieved by reexpressing, in the height function formulation of the water wave problem, the boundary condition…
We are interested in the system of gravity water waves equations without surface tension. Our purpose is to study the optimal regularity thresholds for the initial conditions. In terms of Sobolev embeddings, the initial surfaces we consider…
In geophysical environments, wave motions that are shaped by the action of gravity and global rotation bear the name of gravito-inertial waves. We present a geometrical description of gravito-inertial surface waves, which are low-frequency…
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…
Two-dimensional nonlinear gravity waves travelling in shallow water on a vertically sheared current of constant vorticity are considered. Using Euler equations, in the shallow water approximation, hyperbolic equations for the surface…
The purpose of this article is to clarify the Cauchy theory of the water waves equations as well in terms of regularity indexes for the initial conditions as for the smoothness of the bottom of the domain (namely no regularity assumption is…
We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial…
We study stationary capillary-gravity waves in a two-dimensional body of water that rests above a flat ocean bed and below vacuum. This system is described by the Euler equations with a free surface. Our main result states that there exist…
We construct small-amplitude solitary traveling gravity-capillary water waves with a finite number of point vortices along a vertical line, on finite depth. This is done using a local bifurcation argument. The properties of the resulting…
In this paper we consider the steady water wave problem for waves that possess a merely $L_r-$integrable vorticity, with $r\in(1,\infty)$ being arbitrary. We first establish the equivalence of the three formulations--the velocity…
Starting with the periodic waves earlier constructed for the gravity Whitham equation, we parameterise the solution curves through relative wave height, and use a limiting argument to obtain a full family of solitary waves. The resulting…
For the problem describing steady, gravity waves with vorticity on a two-dimensional, unidirectional flow of finite depth the following results are obtained. (i) Bounds for the free-surface profile and for Bernoulli's constant. (ii) If only…
We consider Euler's equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is…
It is well known that the water-wave problem with weak surface tension has small-amplitude line solitary-wave solutions which to leading order are described by the nonlinear Schr\"odinger equation. The present paper contains an existence…
In this paper, we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves. We construct such spacetimes as solutions to the characteristic initial value problem of the Einstein vacuum equations…
We show that the governing equations for two-dimensional gravity water waves with constant non-zero vorticity have a nearly-Hamiltonian structure, which becomes Hamiltonian for steady waves.
We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the…
The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. It is proved that no small-amplitude waves are supported by a horizontal shear flow whose free surface is still…
In this paper we present a characterization of the symmetric rotational periodic gravity water waves of finite depth and without stagnation points in terms of the underlying flow. Namely, we show that such a wave is symmetric and has a…
Surface waves on a stationary flow of water are considered, in a linear model that includes the surface tension of the fluid. The resulting gravity-capillary waves experience a rich array of horizon effects when propagating against the…