MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS

By Zaher Hani, Benoit Pausader, Nikolay Tzvetkov, Nicola Visciglia. Forum of Mathematics, Pi, 2013

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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 $H^{s}$.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4e78a63b343312b923e81adccbb1be5b6d69f50e