Abstract
In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the Schrödinger flow for maps from a compact Riemannian manifold into a complete Kähler manifold, or from a Euclidean space Rm into a compact Kähler manifold. As a consequence, we prove that Heisenberg spin system is locally well-posed in the appropriate Sobolev spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ding, W. Y., Wang, Y. D., Schröinger flows of maps into symplectic manifolds, Science in China, Ser. A, 1998, 41 (7): 746.
Landau, L. D., Lifshitz, E. M., On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowj., 1935, 8: 153; reproduced in Collected Papers of L. D. Landau, New York: Pergaman Press, 165, 101–114.
Faddeev, L., Takhtajan, L. A., Hamiltonian Methods in the Theory of Solitons, Berlin-Heidelberg-New York: Springer-Verlag, 1987.
Nakamura, K., Sasada, T., Soliton and wave trains in ferromagnets, Phys. Lett. A, 1974, 48: 321.
Zhou, Y., Guo, B., Tan, S., Existence and uniqueness of smooth solution for system of ferromagnetic chain, Science in China, Ser. A, 1991, 34(3): 257.
Pang, P., Wang, H., Wang, Y. D., Schrödinger flow of map into Kähler manifolds, Asian J. of Math., inpress.
Wang, H., Wang, Y. D., Global inhomogeneous Schrödinger flow, Int. J. Math., 2000, 11: 1079.
Pang, P., Wang, H., Wang, Y. D., Local existence for inhomogeneous Schrödinger flow of maps into Kähler manifolds, Acta Math. Sinica, English Series, 2000, 16: 487.
Terng, C. L., Uhlenbeck, K., Schrödinger flows on Grassmannians, in Integrable Systems, Geometry and Topology, Somerville, MA: International Press, in press.
Chang, N., Shatah, J., Uhlenbeck, K., Schrödinger maps, Commun. Pure Appl. Math., 2000, 53: 157.
Wang, Y. D., Ferromagnetic chain equation from a closed Riemannian manifold into S2, Int. J. Math., 1995, 6: 93.
Wang, Y. D., Heisenberg chain systems from compact manifolds into S2, J. Math. Phys., 1998, 39(1): 363.
Sulem, P., Sulem, C., Bardos, C., On the continuous limit for a system of classical spins, Commun. Math. Phys., 1986, 107: 431.
Aubin, T., Nonlinear Analysis on Manifolds, Monge-Ampère Equations, Berlin-Heidelberg-New York: Springer-Verlag, 1982.
Eells, I., Lemaire, L., Another report on harmonic maps, Bull. London Math. Soc., 1988, 20: 385.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ding, W., Wang, Y. Local Schrödinger flow into Kähler manifolds. Sci. China Ser. A-Math. 44, 1446–1464 (2001). https://doi.org/10.1007/BF02877074
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02877074