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Norm inflation for a higher-order nonlinear Schr\"odinger equation with a derivative on the circle
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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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Well- and ill-posedness of the Cauchy problem for semi-linear Schr\"odinger equations on the torus
For s>5/2, a polynomial derivative Schrödinger nonlinearity is locally well-posed in H^s on the torus exactly when the imaginary part of the derivative of the nonlinearity with respect to ∂_x u has zero mean for every datum.
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