On Schrödinger equations with modified dispersion

We consider the nonlinear Schrödinger 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.

Keywords

nonlinear Schrödinger equation, Fourier multiplier, ill-posed

2010 Mathematics Subject Classification

Primary 35Q55. Secondary 35A01, 35B33, 35B45.

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Published 15 September 2011