×

The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps. (English) Zbl 0798.35139

Let \(M\) be a Riemannian manifold without boundary and \(N\) be \(S^ 2\). The aim of the paper is to solve the Landau-Lifshitz (LL) type equations of \(M\) into \(S^ 2\): \[ \partial_ t u= -\alpha_ 1 u(u\Delta_ M u)+ \alpha_ 2 u\Delta_ M u, \] \((\alpha_ 1>0\), \(\alpha_ 2=\text{const})\). A global existence of solutions for the LL-equation is proved and some new links between harmonic maps and the solutions of the LL-equation of the ferromagnetic spin chain are established.

MSC:

35Q58 Other completely integrable PDE (MSC2000)
58E20 Harmonic maps, etc.
58Z05 Applications of global analysis to the sciences
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amann, H.: Quasilinear parabolic systems under nonlinear boundary conditions. Arch. Rat. Mech. Anal.92, 153-192 (1986) · Zbl 0596.35061 · doi:10.1007/BF00251255
[2] Chang, K.C.: Heat flow and boundary value problem for harmonic maps. Anal. Non-Lineaire6(5), 363-395 (1989) · Zbl 0687.58004
[3] Chen, Y.: Weak solutions to the evolution problem of harmonic maps. Math. Z.201, 69-74 (1989) · Zbl 0685.58015 · doi:10.1007/BF01161995
[4] Chen, Y., Ding, W.: Blow-up and global existence for heat flows of harmonic maps. Invent. Math.99, 567-578 (1990) · Zbl 0674.58019 · doi:10.1007/BF01234431
[5] Chen, Y., Hong, M.C.: Heat flow ofp-harmonic maps with values into spheres (to appear)
[6] Chen, Y., Struwe, M.: Existence and partial regularity results for the heat flow for harmonic maps. Math. Z.201, 83-103 (1989) · Zbl 0652.58024 · doi:10.1007/BF01161997
[7] Coron, J.M.: Nonuniqueness for the heat flow of harmonic maps. Anal. Non-Lineaire7, 335-344 (1990) · Zbl 0707.58017
[8] Coron, J.M., Ghidaglia, J.M.: Explosion en temps fini pour le flot des applications harmoniques. C.R. Acad. Sci. Paris308, 339-344 (1989) · Zbl 0679.58017
[9] Eells, J., Lemaire, L.: Another report on harmonic maps. Bull. London Math. Soc.20, 385-524 (1988) · Zbl 0669.58009 · doi:10.1112/blms/20.5.385
[10] Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math.86, 109-160 (1964) · Zbl 0122.40102 · doi:10.2307/2373037
[11] Eidel’man, S.D.: Parabolic systems. Amsterdam, London: North-Holland. Groningen: Noord-hoff 1969
[12] Friedman, A.: Partial differential equations. New York: Holt, Rinehart and Winston 1969 · Zbl 0224.35002
[13] Fogedby, H.C.: Theoretical aspects of mainly low dimensional magnetic systems (Lect. Notes Phys., vol. 131) Berlin, Heidelberg, New York: Springer 1980
[14] Hamilton, R.: Harmonic maps of manifold with boundary. (Lect. Notes Math., vol. 471) Berlin, Heidelberg, New York: Springer 1975 · Zbl 0308.35003
[15] Hungerbuhler, N.: private communication
[16] Klainerman, S.: Global existence for nonlinear wave equations. Commun. Pure Appl. Math.33, 43-101 (1980) · Zbl 0414.35054 · doi:10.1002/cpa.3160330104
[17] Landau, L.D., Lifshitz, E.M.: On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowj.8, 153 (1935); terHaar, D. (eds.) Reproduced in: Collected Papers of L. D. Landau, pp. 101-114. New York: Pergamon Press 1965 · Zbl 0012.28501
[18] Lakshmanan, M., Nakamura, K.: Landau-Lifshitz equation of ferromagnetism: exact treatment of the Gilbert damping. Phys. Rev. Lett.53(26), 2497-2499 (1984) · doi:10.1103/PhysRevLett.53.2497
[19] Ladyzenskaja, O.A., Solonnikov, V.A., Urel’ceva, N.N.: Linear and quasilinear equations of parabolic type. (23) Translations of Mathematical Mongraphs, 1968
[20] Nakamura, K., Sasada, T.: Solition and wave trains in ferromagnets. Phys. Lett.48(A), 321-322 (1974) · doi:10.1016/0375-9601(74)90447-2
[21] Struwe, M.: On the evolution of harmonic maps of Riemannian surfaces. Commun. Math. Helv.60, 558-581 (1985) · Zbl 0595.58013 · doi:10.1007/BF02567432
[22] Struwe, M.: On the evolution of harmonic maps in higher dimensions. J. Differ. Geom.28, 485-502 (1988) · Zbl 0631.58004
[23] Takhtalian, L.A.: Integration of the continuous Heisenberg spin chain through the inverse scattering method. Phys. Lett.15B, 3470-3476 (1977)
[24] Tjon, J., Wright, J.: Soliton in the continuous Heisenberg chain. Phys. Rev.15B, 3470-3476 (1977)
[25] Zhou, Y., Guo, B.: Weak solution of system of ferromagnetic chain with several variables. Scientia Sinica, Ser. A27, 1251-1266 (1987) · Zbl 0656.35123
[26] Zhou, Y., Guo, B., Tan, S.: Existence and Uniqueness of smooth solution for system of ferromagnetic chain. Scientia Sinica, Ser. A34, 257-266 (1991) · Zbl 0752.35074
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.