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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.

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