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arXiv · 2501.04205

Well- and ill-posedness of the Cauchy problem for semi-linear Schr\"odinger equations on the torus

Toshiki Kondo·Mamoru Okamoto
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Abstract

We consider the Cauchy problem for semi-linear Schr\"odinger equations on the torus $\mathbb T$. We establish a necessary and sufficient condition on the polynomial nonlinearity for the Cauchy problem to be well-posed in the Sobolev space $H^s(\mathbb T)$ for $s>\frac 52$. For the well-posedness, we use the energy estimates and the gauge transformation. For the ill-posedness, we prove the non-existence of solutions to the Cauchy problem.

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Toshiki Kondo, Mamoru Okamoto. 2025-01-08. Well- and ill-posedness of the Cauchy problem for semi-linear Schr\"odinger equations on the torus. https://arxiv.org/abs/2501.04205

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