Related papers: On Whitham and related equations
Encyclopedic article covering shallow water wave models used in oceanography and atmospheric science. Sections: Definition of the Subject; Introduction and Historical Perspective; Completely Integrable Shallow Water Wave Equations; Shallow…
When speaking of unsolved problems in physics, this is surprising at first glance to discuss the case of fluid mechanics. However, there are many deep open questions that come with the theory of fluid mechanics. In this paper, we discuss…
The infinite-genus limit of the KdV-Whitham equations is derived. The limit involves special scaling for the associated spectral surface such that the integrated density of states remains finite as $N \to \infty$ (the thermodynamic type…
The routes mechanical energy takes from injection to dissipation in the ocean are currently an open problem. The goals of this project are to: 1) Study the energy exchanges between near-inertial waves and the balanced flow using idealized…
The wave kinetic equation has become an important tool in different fields of physics. In particular, for surface gravity waves, it is the backbone of wave forecasting models. Its derivation is based on the Hamiltonian dynamics of surface…
This work presents a tentative discussion of certain aspects of energy behavior in the context of mathematical fluid dynamics. While some observations are made regarding certain patterns in energy behavior under particular conditions, the…
Our aim is to study the effect of a continuous prescribed density variation on the propagation of ocean waves. More precisely, we derive KdV-type shallow water model equations for unidirectional flows along the Equator from the full…
A forced KdV equation is derived to describe weakly nonlinear, shallow water surface wave propagation over non trivial bottom boundary condition. We show that different functional forms of bottom boundary conditions self-consistently…
The so-called Whitham equation arises in the modeling of free surface water waves, and combines a generic nonlinear quadratic term with the exact linear dispersion relation for gravity waves on the free surface of a fluid with finite depth.…
Understanding what happens inside the rippling and dancing surface of a liquid remains one of the great challenges of fluid dynamics. Using molecular dynamics (MD) we can pick apart the interface structure and understand surface tension. In…
This paper establishes a sharp, expanded wave-breaking criterion for a class of nonlinear nonlocal Whitham-type equations, significantly generalizing the classical threshold introduced by Seliger. While the system of inequalities governing…
In the paper a new nonlinear equation describing shallow water waves with the topography of the bottom directly taken into account is derived. This equation is valid in the weakly nonlinear, dispersive and long wavelength limit. Some…
Tidal disruption events involve numerous physical processes (fluid dynamics, magnetohydrodynamics, radiation transport, self-gravity, general relativistic dynamics) in highly nonlinear ways, and, because TDEs are transients by definition,…
I review recent and not so recent progress on formulating and numerically implementing a consistent set of relativistic equations which describe the space-time evolution of viscous relativistic fluids without violating causality.
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 consider wave models with lower order terms and recollect some recent results on energy and dispersive estimates for their solution based on symbolic type estimates for coefficients and partly stabilisation conditions. The exposition is…
In this article we reconsider the problem of the propagation of waves in a random medium in a kinetic regime. The final aim of this program would be the understanding of the conditions which allow to derive a kinetic or radiative transfer…
In this article we shall go over recent work in proving dispersive and Strichartz estimates for the Dirichlet-wave equation. We shall discuss applications to existence questions outside of obstacles and discuss open problems.
Whitham theory of modulations is developed for periodic waves described by nonlinear wave equations integrable by the inverse scattering transform method associated with $2\times2$ matrix or second order scalar spectral problems. The theory…
We revisit and derive the shock-change equations relating the dynamics of a shock wave with the partial derivatives describing the motion of a reactive fluid with general equation of state in a stream-tube with arbitrary area variation. We…