Abstract for the talk on 05.03.2019 (15:15 h)
Oberseminar ANALYSIS - PROBABILITYTomáš Dohnal (Martin-Luther-Universität Halle-Wittenberg)
Wavepackets and Nearly Solitary Waves in d-dimensional Periodic Media
In nonlinear periodic media of arbitrary dimension d we consider the small amplitude asymptotics of wavepackets. The wavepackets have N ∈ ℕ carrier Bloch waves of equal frequency. We use the cubic Gross-Pitaevskii equation (GP) as a prototype of the governing equation. We discuss two classical asymptotic scalings, one (for N = 1) leading to the nonlinear Schroedinger equation and one (for N > 1) leading to first order coupled mode equations (CMEs) as effective amplitude equations. Both of these models can support solitary waves - thus predicting nearly solitary waves of the GP. In particular, the CMEs for d = 1 for the case of the coupling of two counter-propagating Bloch waves support a family of solitary waves parametrized by the velocity v ∈ (−1,1). Can this be generalized to d dimensions such that in the CMEs a solitary wave family parametrized by
∈ (−1,1)d exists? Solitary waves are typically found in spectral gaps. For d ≥ 2 at least four (N = 4) carrier waves are needed to produce CMEs with a spectral gap that supports solitary waves. However, only standing solitary waves have been found so far. We also provide a validity result of the d−dimensional NLS-asymptotics as well as the CME-asymptotics over asymptotically large time intervals.