We consider the long time behavior of solutions of the d-dimensional linear Boltzmann equation that arises in the weak coupling limit for the Schrödinger equation with a time-dependent random potential. We show that the intermediate mesoscopic time limit satisfies a Fokker-Planck type equation with the wave vector performing a Brownian motion on the (d − 1)-dimensional sphere of constant energy...

In this paper we study the interplay between short- and long-space scales in the context of conservative dispersive systems. We consider model systems in (1+1) dimensions that admit both long- and short-wavelength solutions in the linear regime. A nonlinear analysis of these systems is constructed, making use of multiscale expansions. We show that the equations governing the lowest order involv...

We consider the problem of wave propagation through one-dimensional spatio-temporal or dynamic laminates when the wavelength of the disturbance is large relative to the scale of the microstructure. Dynamic materials are heterogeneous formations assembled from materials which are distributed on a microscale in space and in time. Using the techniques of Floquet analysis and asymptotic expansions,...

The interface problem for the linear Schrödinger equation in one-dimensional piecewise homogeneous domains is examined by providing an explicit solution in each domain. The location of the interfaces is known and the continuity of the wave function and a jump in their derivative at the interface are the only conditions imposed. The problem of two semi-infinite domains and that of two finite-siz...

The Korteweg – de Vries (KdV) equation is a fundamental mathematical model for the description of weakly nonlinear long wave propagation in dispersive media. It is known to possess a number of families of exact analytic solutions. Two of them: solitons and nonlinear periodic travelling waves – are of particular interest from the viewpoint of fluid dynamics applications as they occur as typical ...

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).

We investigate dispersive estimates for the two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a t−1 decay rate as an operator from the Hardy space H to BMO, the space of functions of bounded mean oscillation. This estimate, along with the L conservation law allows one to deduce a family of Strichartz estimates. We classify the structure o...

The transformation of a solitary wave on an underwater step is studied analytically and numerically. The theoretical model includes the linear potential description of the wave transformation on a step and the weakly nonlinear theory of long waves based on the Korteweg–de Vries equation for reflected and transmitted waves far from a step. Numerical simulation of solitary wave transformation on ...

A technique for analyzing dispersion properties of numerical schemes is proposed. The method is able to deal with both non dispersive or dispersive waves, i.e. waves for which the phase speed varies with wavenumber. It can be applied to unstructured grids and to finite domains with or without periodic boundary conditions. We consider the discrete version L of a linear differential operator L. A...

  Nonlinear Dust lattice modes are studied in a hexagonal two-dimensional dusty plasma lattice, in presence of charge gradient of dust particles. In this lattice, such gradients affect nonlinear behavior of dust lattice waves. The amplitude modulation of off-plane transverse dust lattice wave packets is investigated considering the anisotropy of interactions, caused by the height-dependent char...