In the past three decades, traveling wave solutions to the Korteweg–de Vries equation have been studied extensively and a large number of theoretical issues concerning the KdV equation have received considerable attention. These wave solutions when they exist can enable us to well understand the mechanism of the complicated physical phenomena and dynamical processes modeled by these nonlinear e...

Solitary water waves are long nonlinear waves that can propagate steadily over long distances. They were first observed by Russell in 1837 in a now famous report [26] on his observations of a solitary wave propagating along a Scottish canal, and on his subsequent experiments. Some forty years later theoretical work by Boussinesq [8] and Rayleigh [25] established an analytical model. Then in 189...

Considering the fractal structure of space-time, a Burgers – Korteweg – de Vries (BKdV) type equation is obtained. Particularly, if the motions of the “non-differentiable fluid” are irrotational, the BKdV type equation is reduced to a non-linear Schrödinger type equation. In this case, the scalar complex velocity field simultaneously becomes wave function.

We study persistence properties of solutions to some canonical dispersive models, namely the semi-linear Schrödinger equation, the k-generalized Korteweg-de Vries equation and the Benjamin-Ono equation, in weighted Sobolev spaces Hs(Rn) ∩ L2(|x|ldx), s, l > 0.

The transformation of a weakly nonlinear interfacial solitary wave in an ideal twolayer flow over a step is studied. In the vicinity of the step the wave transformation is described in the framework of the linear theory of long interfacial waves, and the coefficients of wave reflection and transmission are calculated. A strong transformation arises for propagation into shallower water, but a we...

The Ostrovsky equation is a modification of the Korteweg-de Vries equation which takes account of the effects of background rotation. It is well known that then the usual Korteweg-de Vries solitary wave decays and is replaced by radiating inertia-gravity waves. Here we show through numerical simulations that after a long-time a localized wave packet emerges as a persistent and dominant feature....