High-frequency averaging in semi-classical Hartree-type equations

TL;DR

This paper studies the asymptotic behavior of solutions to semi-classical Hartree-type equations, demonstrating a superposition principle due to high-frequency averaging effects that prevent resonant wave creation.

Contribution

It introduces a novel high-frequency averaging effect in semi-classical Hartree equations, establishing an asymptotic superposition principle for oscillatory pulses.

Findings

Superposition principle validated for weakly nonlinear regimes

High-frequency averaging inhibits resonant wave formation

Utilizes Wiener algebra framework for proof

Abstract

We investigate the asymptotic behavior of solutions to semi-classical Schroedinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a high-frequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new resonant waves. In the proof we make use of the framework of Wiener algebras.