×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.