We consider the cubic nonlinear Schrödinger equation posed on the spatial domain ℝ x Tᵈ. We prove modified scattering and construct modified wave operators for small initial and final data respectively (1 ≤ d ≤ 4). The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when d ≥ 2. As a consequence, we obtain global strong solutions (for d ≥ 2) with infinitely growing Sobolev norms Hˢ.
Modified scattering for the cubic Schrödinger equation on product spaces and applications
VISCIGLIA, NICOLA;
2015-01-01
Abstract
We consider the cubic nonlinear Schrödinger equation posed on the spatial domain ℝ x Tᵈ. We prove modified scattering and construct modified wave operators for small initial and final data respectively (1 ≤ d ≤ 4). The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when d ≥ 2. As a consequence, we obtain global strong solutions (for d ≥ 2) with infinitely growing Sobolev norms Hˢ.File in questo prodotto:
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