Some remarks about an effective description of high-frequency wave-packet propagation
Anna Logioti, Xin Meng, Guido Schneider
TL;DR
The paper studies the effective description of high-frequency wave-packet propagation for 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 $\varepsilon \ll 1$ and oscillatory initial data $\mathcal{U}(\xi,0)=\mathcal{U}_*(\xi) e^{ik_0 \xi/\varepsilon}+c.c.$ The authors employ a rescaling $\tau=\varepsilon t$, $\xi=\varepsilon x$, $U=\varepsilon \mathcal{U}$ and a Fourier-diagonalization of the linear part to derive higher-order NLS envelopes for the amplitude(s) $A_{n,1,0}$; under non-resonance conditions they prove rigorous error bounds between the NLS-based approximations and the true solution, extending results to quadratic nonlinearities and to non-polarized (multi-packet) data. The work develops a unified separation of internal and interaction dynamics, providing explicit phase and envelope corrections (e.g., $\Omega_{n,j}^{(1)}$, $\Psi_{n,j}^{(1)}$) and constructing higher-order approximations that remain valid on the long timescale $[0,T_0/\varepsilon^2]$. It further considers higher-dimensional settings with multidimensional NLS envelopes and discusses modulational issues and spatial separation that preserve decoupling, yielding comparable error estimates. Overall, the paper advances rigorous justification of high-order NLS approximations for dispersive systems with quadratic nonlinearities and multiple wave-packets, with practical implications for nonlinear optics and related wave problems.
Abstract
We consider systems of the form $ \partial_τ \mathcal U + \mathcal A(\partial_ξ) \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 (ξ,0) = \mathcal U_*(ξ) e^{ik_0 ξ/\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.