MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS. (14th August 2015)

• Record Type: Journal Article
• Authors: HANI, ZAHER; PAUSADER, BENOIT; TZVETKOV, NIKOLAY; VISCIGLIA, NICOLA
• Is Part Of: Forum of mathematics. Volume 3(2015)
• Publication Date: 2015-08-14
• Subjects: 35Q55; Mathematics -- Periodicals; 510
• DOI: 10.1017/fmp.2015.5
• Languages: English
• ISSNs: 2050-5086

Abstract

Abstract : We consider the cubic nonlinear Schrödinger equation posed on the spatial domain $\mathbb{R}\times \mathbb{T}^{d}$ . We prove modified scattering and construct modified wave operators for small initial and final data respectively ( $1\leqslant d\leqslant 4$ ). 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 $d\geqslant 2$ . As a consequence, we obtain global strong solutions (for $d\geqslant 2$ ) with infinitely growing high Sobolev norms $H^{s}$ .