Among a lot of fascinating nonlinear effects, there is a great deal of interest in self modulation of a plane wave, or modulational instability (MI), which occurs in nonlinear dispersive media. Qualitatively a MI is the tendency for amplitude of a modulated carrier wave to break into isolated structures or solitons. Energy originally resident in the carrier of wave packets or long pulse is gradually transferred by nonlinear interaction, in the medium, into the spectrum side bands. As the side-band energy grows in amplitude, the modulated wave breaks up into a series of localized objects or envelope solitons. Composite structures formed by a singly or doubly-curved shallow shell resting on a nonlinear elastic foundation are wave guides that enable one to focus a high energy density in order that nonlinearities be excited. This class of elastic structures is an interesting candidate for the real observation of two-dimensional localized modes in elastodynamics. The purpose of the present work is to study the influences of the geometric dispersion, the prestress on the shallow shell and the material nonlinearity of the elastic foundation on the vibrations modes of the structure. The basic equations which govern the dynamics of the elastic structure are deduced from a variational principle. The analysis is restricted to signals which consist of a slowly varying envelope in space and time modulating a harmonic carrier wave. In the limit of low amplitude the coupled equations are solved by means of a reductive perturbation method. It is shown that the complex amplitude of the envelope is governed by a two-dimensional nonlinear SchrΓΆdinger equation. This equation allows us to study the modulational instability conditions leading to different zones of instability. The mechanism of the self generated nonlinear waves in the elastic structure beyond the birth of modulational instability is numerically investigated on the original equations.