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Schrödinger maps and their associated frame systems. (English) Zbl 1142.35087
Schrödinger maps are maps from the space-time into a Kähler manifold with a metric $$h$$ and a complex structure $$Y$$ satisfying
$u: \mathbb{R}^\lambda\times \mathbb{R}\to (M,h,Y),\qquad \partial_t u= J\sum_\ell D_\ell\partial^\ell u,\tag{SM}$
where $$D$$ denotes the covariant derivative on $$u^{-1}TM$$. By using a pullback frame on $$u^{-1}TM$$, a gauge invariant nonlinear Schrödinger equation is associated to (SM); in the Coulomb gauge, this equation is given schematically by
$i\partial_t q=\Delta q+ \Delta^{-1}[\partial(0(|q|^2)]\partial q+ O(|q|^3).\tag{GNLS}$ The authors are interested in studying the correspondence between solutions $$u$$ of (SM) and solutions $$q$$ of (GNLS) for low-regularity data. They establish the equivalence when the target is $$S^2$$ or $$H^2$$. They also prove the existence of global weak solutions in $$H^2$$ for two space dimensions. The ideas are extended to the maps into compact Hermitian symmetric manifolds with trivial first cohomology.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 58H10 Cohomology of classifying spaces for pseudogroup structures (Spencer, Gelfand-Fuks, etc.)
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