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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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