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We derive various approximations for the solutions of nonlinear hyperbolicsystems with fastly oscillating initial data. We first provide error estimatesfor the so-called slowly varying envelope, full dispersion, and Schr\"odingerapproximations in a Wiener algebra; this functional framework allows us to giveprecise conditions on the validity of these models; we give in particular arigorous proof of the ``practical rule'' which serves as a criterion for theuse of the slowly varying envelope approximation (SVEA). We also discuss theextension of these models to short pulses and more generally to large spectrumwaves, such as chirped pulses. We then derive and justify rigorously a modifiedSchr\"odinger equation with improved frequency dispersion. Numericalcomputations are then presented, which confirm the theoretical predictions.

Short Pulses Approximations in Dispersive Media