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Exponential sums and nonlinear Schrödinger equations. (English) Zbl 0787.35096

A global result is given for the Cauchy problem, in space dimension \(n=4\), of the Schrödinger equation \[ i \partial_ tu+\Delta u+uF(| u |^ 2)=0,\;u \text{ periodic in } x,\;u(x,0)=\varphi(x), \] \(\varphi \in H^ 2(\mathbb{R}^ 4/ \mathbb{Z}^ 4)\), \(\| \varphi \|_ 2\) small, \(F(z)\leq cz^{1/2}\), \(| F'(z)| \leq Cz^{-1/2}\), \(| F''(z)| \leq Cz^{-3/2}\).
The global wellposedness is derived from the local wellposedness and the conservation laws. The local result is based on Picard’s fixed point method, using the associated integral equation.
Reviewer: L.Vazquez (Madrid)

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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References:

[1] [B]J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Preprint I.H.E.S., September 1992, and Part I in this issue.
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