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arXiv · 1311.2275

Modified scattering for the cubic Schr\"odinger equation on product spaces and applications

Zaher Hani·Benoit Pausader·Nikolay Tzvetkov·Nicola Visciglia
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Abstract

We consider the cubic nonlinear Schr\"odinger 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 ($1\leq d\leq 4)$. 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 $d\geq 2$. As a consequence, we obtain global solutions to the defocusing and focusing problems on $\mathbb{R}\times \mathbb{T}^d$ (for any $d\geq 2$) with infinitely growing high Sobolev norms $H^s$.

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Zaher Hani, Benoit Pausader, Nikolay Tzvetkov, Nicola Visciglia. 2013-11-10. Modified scattering for the cubic Schr\"odinger equation on product spaces and applications. https://arxiv.org/abs/1311.2275

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