相关论文: Water Waves and Integrability

In this contribution we describe the role of several two-component integrable systems in the classical problem of shallow water waves. The starting point in our derivation is the Euler equation for an incompressible fluid, the equation of…

We review here the derivation of many of the most important models that appear in the literature (mainly in coastal oceanography) for the description of waves in shallow water. We show that these models can be obtained using various…

We prove a priori estimates for the compressible Euler equations modeling the motion of a liquid with moving physical vacuum boundary in an unbounded initial domain. The liquid is under influence of gravity but without surface tension. Our…

We consider a layer of an inviscid fluid with free surface which is subject to vertical high-frequency vibrations. We derive three asymptotic systems of equations that describe slowly evolving (in comparison with the vibration frequency)…

We derived here in a systematic way, and for a large class of scaling regimes, asymptotic models for the propagation of internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid…

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…

We show that horizontally symmetric water waves are traveling waves. The result is valid for the Euler equations, and is based on a general principle that applies to a large class of nonlinear partial differential equations, including some…

A new regularisation of the shallow water (and isentropic Euler) equations is proposed. The regularised equations are non-dissipative, non-dispersive and possess a variational structure. Thus, the mass, the momentum and the energy are…

Euler's equations govern the behavior of gravity waves on the surface of an incompressible, inviscid, and irrotational fluid of arbitrary depth. We investigate the spectral stability of sufficiently small-amplitude, one-dimensional Stokes…

In order to describe the dynamics of monochromatic surface waves in deep water, we derive a nonlinear and dispersive system of equations for the free surface elevation and the free surface velocity from the Euler equations in infinite…

Some classical and recent results on the Euler equations governing perfect (incompressible and inviscid) fluid motion are collected and reviewed, with some small novelties scattered throughout. The perspective and emphasis will be given…

We study traveling wave solutions of an equation for surface waves of moderate amplitude arising as a shallow water approximation of the Euler equations for inviscid, incompressible and homogenous fluids. We obtain solitary waves of…

A statistical method for calculating equilibrium solutions of the shallow water equations, a model of essentially 2-d fluid flow with a free surface, is described. The model contains a competing acoustic turbulent {\it direct} energy…

The problem of interest in this article are waves on a layer of finite depth governed by the Euler equations in the presence of gravity, surface tension, and vertical electric fields. Perturbation theory is used to identify canonical…

This article is a survey concerning the state-of-the-art mathematical theory of the Euler equations of incompressible homogenous ideal fluid. Emphasis is put on the different types of emerging instability, and how they may be related to the…

This manuscript concerns the dynamical interactions between wind and water waves, which are characterized through two-phase free interface problems for the Euler equations. We provide a comprehensive derivation on the linearized problems of…

We discuss several approaches to generalized solutions of problems describing the motion of inviscid fluids. We propose a new concept of dissipative solution to the compressible Euler system based on a careful analysis of possible…

Motivated by numerically modeling surface waves for inviscid Euler equations, we analyze linear models for damped water waves and establish decay properties for the energy for sufficiently regular initial configurations. Our findings give…

In this paper we propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in $\mathbb{R}^n$ with $n\geq 2$. We give a reformulation of the Euler equations as a differential…

The equations for waves on the surface of an irrotational incompressible fluid are derived in the coordinates of the velocity potential/stream function. The low frequency shallow water approximation for these waves is derived for a varying…