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Zbl 1142.35087
Nahmod, Andrea; Shatah, Jalal; Vega, Luis; Zeng, Chongchun
Schrödinger maps and their associated frame systems.
(English)
[J] Int. Math. Res. Not. 2007, No. 21, Article ID rnm088, 29 p. (2007). ISSN 1073-7928; ISSN 1687-0247/e

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.
[Viorel Iftimie (Bucureşti)]
MSC 2000:
*35Q55 NLS-like (nonlinear Schroedinger) equations
58H10 Cohomology of classifying spaces for pseudogroup structures

Keywords: nonlinear Schrödinger equations; Schrödinger map; global solutions; Kähler manifold; compact Hermitian manifolds; cohomology

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