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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 $$u: \bbfR^\lambda\times \bbfR\to (M,h,Y),\qquad \partial_t u= J\sum_\ell D_\ell\partial^\ell u,\tag SM$$ 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.

35Q55NLS-like (nonlinear Schrödinger) equations
58H10Cohomology of classifying spaces for pseudogroup structures
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