We establish the local well-posedness for a new nonlinearly dispersive wave equation and we show that the equation has solutions that exist for indefinite times as well as solutions which blowup in finite time. Furthermore, we derive an explosion criterion for the equation and we give a sharp estimate from below for the existence time of solutions with smooth initial data.

We generalize the approach first proposed by Manton [Nucl. Phys. B 150, 397 (1979)] to compute solitary wave interactions in translationally invariant, dispersive equations that support such localized solutions. The approach is illustrated using as examples solitons in the Korteweg-de Vries equation, standing waves in the nonlinear Schrödinger equation, and kinks as well as breathers of the sin...

The dynamics of solitons of the nonlinear Schrödinger equation under the influence of dissipative and dispersive perturbations is investigated. In particular a coupling to a long-wave mode is considered using extended GinzburgLandau equations. The study is motivated by the experimental observation of localized wave trains (‘pulses’) in binary-liquid convection. These pulses have been found to d...

We classify 2+1 dimensional integrable systems with nonlocality of the intermediate long wave type. Links to waterbag system are established. Dimensional reductions constructed in this paper provide dispersive regularisations hydrodynamic equations governing propagation nonlinear waves a shear flow piecewise linear velocity profile (for special values vorticities).

The KdV equation with small dispersion is a model for the formation and propagation of dispersive shock waves. Dispersive shock waves are characterized by the appearance of modulated oscillations nearby the breaking point. The modulation in time and space of the amplitude, the frequencies and the wave-numbers of these oscillations is described by the g-phase Whitham equations. We study the init...

The KdV equation models the propagation of long waves in dispersive media, while the NLS equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. A system that couples the two equations to model the interaction of long and short waves is mathematically attractive and such a system has been studied over the last decades. We evaluate the validity of this...

Dispersive effects induced by weak hydrostatic imbalance in the presence of topography and stratification are incorporated into a new model of barotropic (vertically integrated) mesoscale ocean dynamics. This barotropic model is obtained by first expanding the solutions of three dimensional Euler-Boussinesq equations in a regular perturbation expansion in terms of the several small dimensionles...