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Schrödinger flow of maps into symplectic manifolds. (English) Zbl 0918.53017
Let \((N,\omega)\) be a symplectic manifold with an almost complex structure \(J\). Then \(h=\omega\circ J\) is a Riemannian metric. Let \((M,g)\) be a Riemannian manifold and consider a map \(u:M\to N\). The Schrödinger flow is the flow on \(N\) induced by the Hamiltonian vector field \(V_E\) of the energy functional \(E(u)=\int_M e(u) dv_g\) where, in local coordinates, \(e(u)=1/2 g^{\alpha\beta}h_{ij} \partial_\alpha u^i \partial_\beta u^j\). It is shown that the flow of a ferromagnetic chain can be represented as Schrödinger flow of maps into \(S^2\) and that there exists a unique local smooth solution for the initial value problem of the 1-dimensional Schrödinger flow of maps into a Kähler manifold. In the case that the targets are Kähler manifolds with constant curvature, it is proved that the 1-dimensional Schrödinger flow admits a unique global smooth solution.

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
37D99 Dynamical systems with hyperbolic behavior
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
Full Text: DOI
[1] Zhou., Y., Guo, Tan, S., Existence and uniqueness of smooth solution for system of ferromagnetic chain,Science in China Ser. A, 1991, 34: 257. · Zbl 0752.35074
[2] Kenig, C. E., Ponce G., Vega, L., Small solutions to nonlinear Schrödinger equations,Anal. Nonlinéare, 1993, 10(3):255. · Zbl 0786.35121
[3] Bourgain, J., Exponential sums and nonlinear Schrödinger equations,Geom. Funct. Anal., 1993, 3: 157. · Zbl 0787.35096
[4] Bourgain, J., Fourier transfrm restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I. Schrödinger equations,Geom. Funct. Anal., 1993, 3: 107. · Zbl 0787.35097
[5] Amann, H., Quasilinear parabolic systems under nonlinear boundary conditions,Arch. Rat. Mech. Anal., 1986, 92:153. · Zbl 0596.35061
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