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arXiv · 2407.17782

Norm inflation for a higher-order nonlinear Schr\"odinger equation with a derivative on the circle

Toshiki Kondo·Mamoru Okamoto
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Abstract

We consider a periodic higher-order nonlinear Schr\"odinger equation with the nonlinearity $u^k \partial_x u$, where $k$ is a natural number. We prove the norm inflation in a subspace of the Sobolev space $H^s(\mathbb{T})$ for any $s \in \mathbb{R}$. In particular, the Cauchy problem is ill-posed in $H^s(\mathbb{T})$ for any $s \in \mathbb{R}$.

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Toshiki Kondo, Mamoru Okamoto. 2024-07-25. Norm inflation for a higher-order nonlinear Schr\"odinger equation with a derivative on the circle. https://doi.org/10.1007/s42985-025-00315-4

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