Solitary wave propagation in elastic plate on the Winkler foundation

Abstract

A new problem to find the solitary solution for interaction of elastic plate with the Winkler foundation is considered. It is assumed that the plate is characterized by elastically nonlinear properties which are described by the geometrically nonlinear deformations. It is assumed that longitudinal deformations are not taken into account, the Lagrangian corresponding to the Kirchhoff classical model of transverse bending vibrations of plate is obtained. For corresponding resolving equation a solution is found in the class of the amplitude modulated signals. By means of regularizations of formal asymptotic decompositions, the Schrodinger nonlinear equation, determining the amplitude in the first order of smallness, is obtained. We obtain the solitory solution in correspondence with considerations and results obtained by Abblowitz et al., Newel, Zakharov and Shabat. There are a lot of systems of extensive applications in various fields which include such elements. In connection with it our communication considers the problem of the interaction of an elastic plate with the Winkler foundation. A linear problem was first very clearly considered by S. P. Timoshenko as STRENGTH OF MATERIALS, vol. 1 and 2, during his work at the KPI. Since then no course, better than this one, has been developed.