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.
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)
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)
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.
The regularized Benjamin–Ono equation appears in the modeling of long-crested interfacial waves in two-fluid systems. For this equation, Fourier–Galerkin and colloca-tion semi-discretizations are proved to be spectrally convergent. A new exact solution is found and used for the experimental validation of the numerical algorithm. The scheme is then used to… (More)
a r t i c l e i n f o a b s t r a c t Keywords: Whitham KdV Modulational instability Fourier–Floquet–Hill method Dispersion Water waves The Whitham equation was proposed as an alternate model equation for the simplified description of unidirectional wave motion at the surface of an inviscid fluid. An advantage of the Whitham equation over the KdV equation… (More)
A spectral semi-discretization of the Camassa-Holm equation is defined. The Fourier-Galerkin and a de-aliased Fourier-collocation method are proved to be spectrally convergent. The proof is supplemented with numerical explorations which illustrate the convergence rates and the use of the dealiasing method.
We derive a new model for the description of large amplitude internal waves in a two-fluid system. The displacement of the interface between the two fluids is assumed to be of small slope, but no smallness assumption is made on the wave amplitude. The derivation of the model is based on the perturbation theory for Hamiltonian systems. In the case of a… (More)
The generalized Korteweg–de Vries equation has the property that solutions with initial data that are analytic in a strip in the complex plane continue to be analytic in a strip as time progresses. Established here are algebraic lower bounds on the possible rate of decrease in time of the uniform radius of spatial analyticity for these equations. Previously… (More)
It is shown that in water of ÿnite depth, the surface proÿle Á of a periodic traveling wave uniquely determines the corresponding ow in the body of the uid. This holds for rotational ow as long as the vorticity function () satisÿes the condition () max x∈R Á 2 (x) ¡ ¡ 2. This condition is also shown to be sharp.