Multiscale Expansions, Symmetries and the Nonlinear Schrödinger Hierarchy

We study the propagation of quasi-monocromatic, nondissipative and weakly nonlinear waves, modelled by partial differential equations in 1 + 1 dimensions, using a multitime expansion. In the case of pure radiation, we show that the asymptotic character of this expansion is guaranted by requiring that the modulation of the leading amplitude of the waves satisfy the nonlinear Schrodinger hierarchy of evolution equations with respect to the slow space-time variables characteristic of the problem. The theory of the symmetries of integrable systems plays a crucial role in the derivation of this result.