On Schrodinger equations with modified dispersion

Details On Schrodinger equations with modified dispersion On Schrodinger equations with modified dispersion

2011-05-25

arxiv.org/abs/1105.4954v3

Dynamics of Partial Differential Equations 8, 3 (2011) 173-184

Mathematics

Analysis of PDEs

10 pages. References and comments added

Scientific paper

We consider the nonlinear Schrodinger equation with a modified spatial dispersion, given either by an homogeneous Fourier multiplier, or by a bounded Fourier multiplier. Arguments based on ordinary differential equations yield ill-posedness results which are sometimes sharp. When the Fourier multiplier is bounded, we infer that no Strichartz-type estimate improving on Sobolev embedding is available. Finally, we show that when the symbol is bounded, the Cauchy problem may be ill-posed in the case of critical regularity, with arbitrarily small initial data. The same is true when the symbol is homogeneous of degree one, where scaling arguments may not even give the right critical value.

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