Related papers: Higher-order Hamiltonian Model for Unidirectional …
The classical problem of irrotational long waves on the surface of a shallow layer of an ideal fluid moving under the influence of gravity as well as surface tension is considered. A systematic procedure for deriving an equation for surface…
There is a tendency to write the equations of general relativity as a first order symmetric system of time dependent partial differential equations. However, for numerical reasons, it might be advantageous to use a second order formulation…
The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of…
A relaxed notion of displacement convexity is defined and used to establish short time existence and uniqueness of Wasserstein gradient flows for higher order energy functionals. As an application, local and global well-posedness of…
We calculate certain estimates for the solution of the characteristic problem of the wave equation reduced to first order, in terms of the free data prescribed on two transverse surfaces, one of which is characteristic. Estimates of such…
An alternative way for the derivation of the new KdV-type equation is presented. The equation contains terms depending on the bottom topography (there are six new terms in all, three of which are caused by the unevenness of the bottom). It…
The classical approach to linking lattice dynamics properties to continuum equations of motion, the "method of long waves," is extended to include higher order terms. The additional terms account for non-local and non-linear effects. In the…
Extended shallow water wave equations are derived, using the method of asymptotic expansions, from the Euler (or water wave) equations. These extended models are valid one order beyond the usual weakly nonlinear, long wave approximation,…
In this paper, we consider the one-dimensional generalized Benjamin--Bona--Mahony (gBBM) equation \[(1-\partial_x^2)u_t+(u+u^p)_x=0,\qquad p=2,3,4,\dots,\] posed either on the real line $\mathbb R$ or on the torus $\mathbb T$. This equation…
In this paper we derive a new formulation of the water waves equations with vorticity that generalizes the well-known Zalkarov-Craig-Sulem formulation used in the irrotational case. We prove the local well-posedness of this formulation, and…
We determine the phase portrait of a Hamiltonian system of equations describing the motion of the particles in linear deep-water waves. The particles experience in each period a forward drift which decreases with greater depth.
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…
We consider the one-dimensional shallow water problem with capillary surfaces and moving contact {lines}. An energy-based model is derived from the two-dimensional water wave equations, where we explicitly discuss the case of a stationary…
We consider the gravity-capillary water waves problem in a domain $\Omega_t \subset \mathbb{T} \times \mathbb{R}$ with substantial geometric features. Namely, we consider a variable bottom, smooth obstacles in the flow and a constant…
Many equations that arise in a physical context can be posed in the form of a Hamiltonian system, meaning that there is a symplectic structure on an appropriate phase space, and a Hamiltonian functional with respect to which time evolution…
This article provides a survey on some main results and recent developments in the mathematical theory of water waves. More precisely, we briefly discuss the mathematical modeling of water waves and then we give an overview of local and…
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…
A single incompressible, inviscid, irrotational fluid medium bounded by a free surface and varying bottom is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet-Neumann operators. The equations for the…
Since the pioneering work of Kelvin on Laplace tidal equations, a zoology of trapped waves have been found in the context of coastal dynamics. Among them, the one originally computed by Kelvin plays a particular role, as it is an…
We describe the physical hypotheses underlying the derivation of an approximate model of water waves. For unidirectional surface shallow water waves moving over an irrotational flow as well as over a non-zero vorticity flow, we derive the…