Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation

Details Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation

2005-08-11

arxiv.org/abs/math/0508210v4

Mathematics

Analysis of PDEs

28 pages, no figures, to appear, J.Func. Anal. Some minor gaps filled in

Scientific paper

We establish that the quadratic non-linear Schr\"odinger equation $$ iu_t + u_{xx} = u^2$$ where $u: \R \times \R \to \C$, is locally well-posed in $H^s(\R)$ when $s \geq -1$ and ill-posed when $s < -1$. Previous work of Kenig, Ponce and Vega had established local well-posedness for $s > -3/4$. The local well-posedness is achieved by an iteration using a modification of the standard $X^{s,b}$ spaces. The ill-posedness uses an abstract and general argument relying on the high-to-low frequency cascade present in the non-linearity, and a computation of the first non-linear iterate.

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