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- ItemHigh-frequency averaging in semi-classical Hartree-type equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Giannoulis, Johannes; Mielke, Alexander; Sparber, ChristofWe investigate the asymptotic behavior of solutions to semi-classical Schröodinger 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.
- ItemInteraction of modulated pulses in the nonlinear Schrödinger equation with periodic potential(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Giannoulis, Johannes; Mielke, Alexander; Sparber, ChristofWe consider a cubic nonlinear Schrödinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic dynamics of these pulses.