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arXiv · 2508.01204

Global Well-posedness for the periodic fractional cubic NLS in 1D

Alexandre Megretski·Nikolaos Skouloudis
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Abstract

We consider the defocusing periodic fractional nonlinear Schrödinger equation $$ i \partial_t u +\left(-Δ\right)^αu=-\lvert u \rvert ^2 u, $$ where $\frac{1}{2}< α< 1$ and the operator $(-Δ)^α$ is the fractional Laplacian with symbol $\lvert k \rvert ^{2α}$. We establish global well-posedness in $H^s(\mathbb{T})$ for $s\geq \frac{1-α}{2}$ and we conjecture this threshold to be sharp as it corresponds to the pseudo-Galilean symmetry exponent. Our proof uses the $I$-method to control the $H^s(\mathbb{T})$-norm of solutions with infinite energy initial data. A key component of our approach is a set of improved long-time bilinear Strichartz estimates on the rescaled torus, which allow us to exploit the subcritical nature of the equation.

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Alexandre Megretski, Nikolaos Skouloudis. 2025-10-03. Global Well-posedness for the periodic fractional cubic NLS in 1D. https://arxiv.org/abs/2508.01204

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