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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: λ ×(M,h,Y), t u=J D u,( 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 t q=Δq+Δ -1 [(0(|q| 2 )]q+O(|q| 3 )·( 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:
35Q55NLS-like (nonlinear Schrödinger) equations
58H10Cohomology of classifying spaces for pseudogroup structures