We analyze a homogenization limit for the linear wave equation of second order. The spatial operator is assumed to be of divergence form with an oscillatory coefficient matrix aε that is periodic with characteristic length scale ε; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions uε well by a non-dispersive wave equation on fixed time inte...

In this work, we consider varying aspects of the stability of periodic traveling wave solutions to nonlinear dispersive equations. In particular, we are interested in deriving universal geometric criterion for the stability of particular third order nonlinear dispersive PDE’s. We begin by studying the spectral stability of such solutions to the generalized Korteweg-de Vries (gKdV) equation. Usi...

The notes serve as an introduction to the analysis of dispersive partial differential equations. They are organized as follows: • Part I focuses on basic theory for local and global analysis of the semilinear Schrödinger equation. • Part II concentrates on basic local and global theory for the Korteveg de Vries equation. • Part III gives a review of some recents results on a derivation of nonli...

Abstract. Homogenisation theory reveals that long one-dimensional waves propagating in a medium with spatially periodic wave speed can behave like dispersive waves in an appropriately homogenised medium. However the precise time and length scales over which this dispersive behaviour is a good approximation are not clear. This paper describes what happens in a specific case when the sound speed ...

We prove stability for arbitrarily long times of the zero solution for the so-called β-plane equation, which describes the motion of a two-dimensional inviscid, ideal fluid under the influence of the Coriolis effect. The Coriolis force introduces a linear dispersive operator into the 2d incompressible Euler equations, thus making this problem amenable to an analysis from the point of view of no...

where  and  are positive constants, was first proposed by Peregrine [74] for modeling the propagation of unidirectional weakly nonlinear and weakly dispersive water waves. Later on Benjamine et al. [9] proposed the use of the RLW equation as a preferred alternative to the more classical Korteweg de Vries (KdV) equation to model a large class of physical phenomena. These authors showed that RL...

We study the higher-order nonlinear dispersive equation ∂tu+ ∂ 2j+1 x u = ∑ 0≤j1+j2≤2j aj1,j2∂ j1 x u∂ j2 x u, x, t ∈ R. where u is a real(or complex-) valued function. We show that the associated initial value problem is well posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when a0,k 6= 0 for some k > j, in the sense that this equation cann...

ABSTRACT In this work, recently developed modified simple equation (MSE) method is applied to find exact traveling wave solutions of nonlinear evolution equations (NLEEs). To do so, we consider the (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation and coupled Klein-Gordon (cKG) equations. Two classes of explicit exact solutions-hyperbolic and trigonometric ...

We comment on traveling wave solutions and rational solutions to the 3+1 dimensional Kadomtsev–Petviashvili (KP) equations: (ut + 6uux + uxxx)x ± 3uyy ± 3uzz = 0. We also show that both of the 3+1 dimensional KP equations do not possess the three-soliton solution. This suggests that none of the 3+1 dimensional KP equations should be integrable, and partially explains why they do not pass the Pa...