Norm inflation for a higher-order nonlinear Schrödinger equation with a derivative on the circle

Toshiki Kondo () and Mamoru Okamoto ()
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Toshiki Kondo: Graduate School of Science, Osaka University
Mamoru Okamoto: Graduate School of Science, Osaka University

Partial Differential Equations and Applications, 2025, vol. 6, issue 2, 1-14

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

Keywords: Schrödinger equation; Ill-posedness; Norm inflation; Unconditional uniqueness (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s42985-025-00315-4

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