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The theory of internal waves between two bodies of immiscible fluid is important both for its interest to ocean engineering and as a source of numerous interesting mathematical model equations that exhibit nonlinearity and dispersion. In this paper we derive a Hamiltonian formulation of the problem of a dynamic free interface (with rigid lid upper boundary… (More)

We study properties of solitary wave solutions of three evolution equations arising in the modeling of internal waves Our experiments indicate that broad classes of initial data resolve into solitary waves but also suggest that solitary waves do not interact exactly thus suggesting two of these equa tions are not integrable In the course of our numerical… (More)

Solitary-wave solutions of a nonlinearly dispersive equation are considered. It is found that solitary waves are peaked or smooth waves, depending on the wave speed. The stability of the smooth solitary waves also depends on the wave speed. Orbital stability is proved for some wave speeds, while instability is proved for others.

We explore numerically different aspects of periodic traveling-wave solutions of the Camassa–Holm equation. In particular, the time evolution of some recently found new traveling-wave solutions and the interaction of peaked and cusped waves is studied. 2005 Elsevier Ltd. All rights reserved.

Article history: Received 12 February 2009 Accepted after revision 2 February 2010 Available online 6 March 2010 Presented by Gérard Iooss

The existence of traveling waves for the original Whitham equation is investigated. This equation combines a generic nonlinear quadratic term with the exact linear dispersion relation of surface water waves of finite depth. It is found that there exist small-amplitude periodic traveling waves with sub-critical speeds. As the period of these traveling waves… (More)

- Maria T. Elenius, Jan M. Nordbotten, Henrik Kalisch
- 2011

In geological storage of carbon dioxide (CO2), the buoyant CO2 plume eventually accumulates under the caprock. Due to interfacial tension between the CO2 phase and the water phase, a capillary transition zone develops in the plume. This zone contains supercritical CO2 as well as water with dissolved CO2. Under the plume, a diffusive boundary layer forms. We… (More)

The Korteweg-de Vries (KdV) equation is widely recognized as a simple model for unidirectional weakly nonlinear dispersive waves on the surface of a shallow body of fluid. While solutions of the KdV equation describe the shape of the free surface, information about the underlying fluid flow is encoded into the derivation of the equation, and the present… (More)

Existence and admissibility of δ-shock solutions is discussed for the non-convex strictly hyperbolic system of equations ∂tu+ ∂x( 2 (u 2 + v2)) = 0, ∂tv + ∂x(v(u− 1)) = 0. The system is fully nonlinear, i.e. it is nonlinear with respect to both unknowns, and it does not admit the classical Lax-admissible solution for certain Riemann problems. By introducing… (More)

The method of weak asymptotics is used to find singular solutions of the shallow-water system which can contain Dirac-δ distributions (Espinosa & Omel’yanov, 2005). Complex-valued approximations which become real-valued in the distributional limit are shown to extend the range of possible singular solutions. It is shown, in this paper, how this approach can… (More)