This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variablecoefficient Korteweg–de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adia...

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 tends to infinity, their velocities ap...

We review recent results on global attractors of U(1)-invariant dispersive Hamiltonian systems. We study several models built upon the Klein-Gordon equation and sketch the proof that in these models, under certain generic assumptions, the weak global attractor is represented by the variety of all solitary waves. In general, the attractors may also contain multifrequency solitary waves; we give ...

We review recent results on global attractors of U(1)-invariant dispersive Hamiltonian systems. We study several models based on the Klein-Gordon equation and sketch the proof that in these models, under certain generic assumptions, the weak global attractor is represented by the set of all solitary waves. In general, the attractors may also contain multifrequency solitary waves; we give exampl...

We have studied the stress wave propagation in one-dimensional (1-D) nanoscopic buckyball (C60) system by molecular dynamics (MD) simulation and quantitative modeling. Simulation results have shown that solitary waves are generated and propagating in the buckyball system through impacting one buckyball at one end of the buckyball chain. We have found the solitary wave behaviors are closely depe...

EEG signals are records of nonlinear solitary waves in human brains. The waves have several types (e.g., α, β, γ, θ, δ) in response to different levels of consciousness. They are classified into two groups: Group-1 consists of complex storm-like waves (α, β, and γ); Group-2 is composed of simple quasilinear waves (θ and δ). In order to elucidate the mechanism of EEG wave formation and propagati...

the thomas-fermi (tf) equation has proved to beuseful for the treatment of many physical phenomena. in this pa-per, the traveling wave solutions of the kdv equation is investi-gated by the simplest equation method. also, the effect of differentparameters on these solitary waves is considered. the numericalresults is conformed the good accuracy of presented method.

We investigate linear stability of solitary waves of a Hamiltonian system. Unlike weakly nonlinear water wave models, the physical system considered here is nonlinearly dispersive, and contains nonlinearity in its highest derivative term. This results in more detailed asymptotic analysis of the eigenvalue problem in presence of a large parameter. Combining the technique of singular perturbation...

We investigate multi-hump spatial solitary waves and multi-soliton patterns generated by an incoherent interaction of two optical beams in a medium with saturable (e.g. photorefractive) nonlinearity. Applying a bifurcation analysis and numerical relaxation technique, we reveal different scenarios of creating the multi-hump solitons and find the families of solitary waves composed of two mutuall...