相关论文: Variational derivation of the Camassa-Holm shallow…

We describe the physical hypothesis in which an approximate model of water waves is obtained. For an irrotational unidirectional shallow water flow, we derive the Camassa-Holm equation by a variational approach in the Lagrangian formalism.

By a variational approach in the Lagrangian formalism, we derive the nonlinear integrable two-component Camassa-Holm system (1). We show that the two-component Camassa-Holm system (1) with the plus sign arises as an approximation to the…

A reduced dynamical model is derived which describes the interaction of weak inertia-gravity waves with nonlinear vortical motion in the context of rotating shallow-water flow. The formal scaling assumptions are (i) that there is a…

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…

In this short note, we derive a system of two nonlocal equations for the water-wave problem following the work of [AFM06]. Specifically, we consider a fluid with a one-dimensional free surface for an irrotational fluid both with, and…

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 study traveling wave solutions of the nonlinear variational wave equation. In particular, we show how to obtain global, bounded, weak traveling wave solutions from local, classical ones. The resulting waves consist of monotone and…

In our paper we show that the Camassa-Holm equation does not represent a long wave asymptotic due to a major inconsistency with the theory of shallow water waves. We state that any solution of the Camassa-Holm equation, which is not…

The classical equations of irrotational water waves have recently been reformulated as a system of two equations, one of which is an explicit non-local equation for the wave height and for the velocity potential evaluated on the free…

In the present study a mathematical model of long-crested water waves propagating mainly in one direction with the effect of Earth's rotation is derived by following the formal asymptotic procedures. Such a model equation is analogous to…

In this paper we apply the approach of formal asymptotic expansions and perturbation theory to derive a new highly nonlinear shallow-water model from the full governing equations for two dimensional incompressible fluid with constant…

In this paper we provide a formal derivation of both the Camassa-Holm equation and the fractional Camassa-Holm equation for the propagation of small-but-finite amplitude long waves in a nonlocally and nonlinearly elastic medium. We first…

We modify the nonlinear shallow water equations, the Korteweg-de Vries equation, and the Whitham equation, to permit constant vorticity, and examine wave breaking, or the lack thereof. By wave breaking, we mean that the solution remains…

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 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)…

A non-local evolution equation of the Camassa-Holm type with dissipation is considered. The local well-posedness of the solutions of the Cauchy problem involving the equation is established via Kato's approach and the wave breaking scenario…

This paper is concerned with the derivation of a two-component system modelling shallow-water waves with constant vorticity under the Camassa-Holm scaling from our newly established Green-Naghdi equations with a linear shear. It is worth…

We derive the Whitham equations from the water waves equations in the shallow water regime using two different methods, thus obtaining a direct and rigorous link between these two models. The first one is based on the construction of…

We study here the water-waves problem for uneven bottoms in a highly nonlinear regime where the small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is known, that for such regimes, a generalization of the KdV…

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…