High-frequency averaging in semi-classical Hartree-type equations
- Giannoulis, Johannes
- Mielke, Alexander
- Sparber, Christof
2010 Mathematics Subject Classification
- 35B40 35C20 81Q20
- Nonlinear Schrödinger equation, Hartree-type nonlinearity, Wiener space, propagation of pulses, justification of amplitude equations, high-frequency asymptotics, WKB approximation
We 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.
- Asymptot. Anal., 70 (2010) pp. 87--100.