Non local Models for Envelope Waves in a Stratified Fluid

Abstract

A new, nonlocal evolution equation similar to the nonlinear Schrodinger equation is derived for envelope waves in a continuously stratified fluid by means of a multiscale perturbation technique. This new equation governs propagation of quasi-harmonic wave packets having length scales much longer than the depth of the density variations and much shorter than the total depth of fluid. Generalizations of the nonlocal evolution equation for a description of two-dimensional wave modulations are also presented. The modulational stability of small-amplitude waves is then investigated in the framework of the derived equations. It is shown that quasi-harmonic waves with the scales under consideration are unstable with respect to oblique perturbations at certain angles.

Periodic solutions for the Schrödinger equation with nonlocal smoothing nonlinearities in higher dimension

“…Nonlocal nonlinear Schrödinger equations appear in several contexts in physics, but not of the form (1.1) we are considering (see, for instance, [16,28,29,35,36]). Although such equations do not arise from any physical situation that we are aware of, the regularisation through a smoothing function was already considered in the literature (see [1,30] and [18]), as it provides some nice simplifications when looking for periodic or quasi-periodic or almost-periodic solutions in PDE problems.…”

The evolution equation for second-order internal solitary waves in stratified fluids of great depth

“…It has been proved that equation ( O" I =--b34clT +b35c I +-'7b31o7 , (2) The inhomogeneous linearized BO equation presented herein may be used for the study of both secondorder internal solitary waves and the quasi-harmonic wave packets (ref. [5,6]). …”

Well-posedness for the nonlocal nonlinear Schrödinger equation

“…So, the problem of deriving a kind of NLS equation modeling quasi-harmonic wave packets arises. Taking into account first-and second-order effects in the long-wave expansion, Pelinovsky and Grimshaw (see [18]) proposed the following universal evolution equation of the NLS type in the shallow-deep limit of a continuous stratified fluid:…”