Some remarks about an effective description of high-frequency wave-packet propagation

Abstract

We consider systems of the form $ \partial_{\tau} \mathcal U + \mathcal A(\partial_{\xi}) \mathcal U + \frac{1}{\varepsilon} \mathcal E \mathcal U = \mathcal T_{2}( \mathcal U , \mathcal U ) + \varepsilon \mathcal T_3( \mathcal U , \mathcal U , \mathcal U ), $ with $ 0 < \varepsilon \ll 1 $ a small perturbation parameter. We are interested in an effective description of high-frequency wave-packet propagation associated to highly oscillatory initial conditions $ \mathcal U (\xi,0) = \mathcal U_*(\xi) e^{ik_0 \xi/\varepsilon} + c.c.. $ By classical perturbation analysis for polarized initial conditions NLS approximations up to an arbitrary order and for non-polarized initial conditions a system of decoupled NLS equations can be derived for the approximate description of the associated solutions. Under the validity of a number of non-resonance conditions we prove error estimates between these formal approximations and true solutions of the original system. The result improves results from the existing literature in at least two directions, firstly, the handling of higher order approximations in case of quadratic nonlinearities $ \mathcal T_2(\mathcal U,\mathcal U)$ and secondly, the handling of non-polarized initial conditions.