Two-dimensional modulation and instabilities of flexural waves of a thin plate on nonlinear elastic foundation

In the present paper we intend to examine in detail the formation of localized modes and waves mediated by modulational instability in an elastic structure. The elastic composite structure consists of a nonlinear foundation coated with an elastic thin plate. The problem deals with flexural waves traveling on the plate. The attention is devoted to the behavior of nonlinear waves in the small-amplitude limit in view of deducing criteria of instability which produce localized waves. It is shown that, in the small-amplitude limit, the basic equation which governs the plate deflection is approximated by a two-dimensional nonlinear Schrödinger equation. The latter equation allows us to study the modulational instability conditions leading to different zones of instability. The examination of the instability provides useful information about the possible selection mechanism of the modulus of the carrier wave vector and growth rate of the instabilities taking place in both (longitudinal and transverse) directions of the plate. The mechanism of the self-generated nonlinear waves on the plate beyond the birth of modulational instability is numerically investigated. The numerics show that an initial plane wave is then transformed, through the instability process, into nonlinear localized waves which turn out to be particularly stable. In addition, the influence of the prestress on the nature of localized structures is also examined. At length, in the conclusion some other wave problems and extensions of the work are evoked.