Modulation non linéaire d'un train d'ondes dans une structure élastique

Abstract
Composite structures formed by an infinite elastic thin plate placed on elastic foundation are uniform waveguides enables to focus a high energy density in order that nonlinearities can be excited. This class of elastic structures is an interesting candidate for the real observation of two-dimensional wave trains or packets with a soliton-shape envelope in elastic solids. The purpose of this contribution is to study the influences of the geometric dispersion and material nonlinearities of the elastic substrat on the modulation of flexural waves in this simple test structure. The analysis is restricted to excitations which consist of slowly varying envelope in space and time modulating a harmonic carrier wave. In the case of small perturbations the problem thus posed is solved by using the technique of multiple scales. It is shown at the lowest of the secularity conditions, the complex amplitude of the envelope satisfies a 2-D nonlinear Schrödinger equation with the nonlinear coefficient as a function of the wave number and the sign of the nonlinearity. Particular solutions of this equation namely gray or dark solitons are given and illustrated for a circurlar frequency of the carrier wave close of the linear spectrum gap.