×

Landau-Lifshitz equation, uniaxial anisotropy case: theory of exact solutions. (English. Russian original) Zbl 1303.82038

Theor. Math. Phys. 178, No. 2, 143-193 (2014); translation from Teor. Mat. Fiz. 178, No. 2, 163-219 (2014).
Summary: Using the inverse scattering method, we study the XXZ Landau-Lifshitz equation well-known in the theory of ferromagnetism. We construct all elementary soliton-type excitations and study their interaction. We also obtain finite-gap solutions (in terms of theta functions) and select the real solutions among them.

MSC:

82D40 Statistical mechanics of magnetic materials
35C08 Soliton solutions
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35Q82 PDEs in connection with statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] L. D. Landau and E. M. Lifschitz, ”Toward a theory of the dispersion of magnetic permittivity in ferromagnetic bodies,” in: Collected Works of L. D. Landau in Two Volumes [in Russian], Vol. 1, Nauka, Moscow (1969), pp. 128–143.
[2] L. A. Takhtajan, Phys. Lett. A, 64, 235–237 (1977). · doi:10.1016/0375-9601(77)90727-7
[3] A. E. Borovik, JETP Lett., 28, 581–584 (1978).
[4] E. K. Sklyanin, ”On complete integrability of the Landau-Lifshitz equation,” Preprint E-3-79, LOMI, Leningrad (1979). · Zbl 0449.35089
[5] A. B. Shabat, Funct. Anal. Appl., 9, 244–247 (1975). · Zbl 0352.34020 · doi:10.1007/BF01075603
[6] V. E. Zakharov and S. V. Manakov, JETP Lett., 8, 243–245 (1973).
[7] I. A. Akhiezer and A. E. Borovik, JETP Lett., 52, 332–335 (1967).
[8] I. A. Akhiezer, A. E. Borovik, JETP Lett., 52, 885–893 (1967).
[9] A. M. Kosevich, Fiz. Met. Metalloved., 53, 420–446 (1982).
[10] N. N. Bogolyubov Jr. and A. K. Prikarpatskii, ”On finite-gap solutions of Heisenberg-type equations,” in: Mathematical Methods and Physico-Mathematical Fields (Ya. S. Podstrigach, ed.) [in Russian], Naukova Dumka, Kiev (1983), pp. 5–11.
[11] I. V. Cherednik, Math. USSR-Izv., 22, 357–377 (1984). · Zbl 0547.35109 · doi:10.1070/IM1984v022n02ABEH001448
[12] E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, J. Phys. Soc. Japan, 52, 388–393 (1983). · Zbl 0571.35105 · doi:10.1143/JPSJ.52.388
[13] M. M. Bogdan and A. S. Kovalev, JETP Lett., 31, 424–427 (1980).
[14] A. V. Mikhailov, Phys. Lett. A, 92, 51–55 (1982). · doi:10.1016/0375-9601(82)90289-4
[15] Yu. L. Rodin, Lett. Math. Phys., 7, 3–8 (1983). · Zbl 0539.35069 · doi:10.1007/BF00398705
[16] A. I. Bobenko, Zap. Nauchn. Sem. LOMI, 123, 58–66 (1983).
[17] A. B. Borisov, Fiz. Met. Metalloved., 55, 230–234 (1983).
[18] R. F. Bikbaev, A. I. Bobenko, and A. R. Its, Soviet Math. Dokl., 28, 512–516 (1983).
[19] A. I. Bobenko, Funct. Anal. Appl., 19, 5–17 (1985). · Zbl 0588.35015 · doi:10.1007/BF01086019
[20] M. Jimbo, I. Miwa, and K. Ueno, Phys. D, 2, 306–352 (1981). · Zbl 1194.34167 · doi:10.1016/0167-2789(81)90013-0
[21] M. Jimbo and T. Miwa, Phys. D, 2, 407–448 (1981). · Zbl 1194.34166 · doi:10.1016/0167-2789(81)90021-X
[22] E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin (1994). · Zbl 0809.35001
[23] A. Hurwitz and R. Courant, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen, Springer, Berlin (1964). · Zbl 0135.12101
[24] E. I. Zverovich, Russ. Math. Surveys, 26, 117–192 (1971). · Zbl 0226.30043 · doi:10.1070/RM1971v026n01ABEH003811
[25] V. B. Matveev, ”Abelian functions and solitons,” Preprint H 373, Univ. Wroclaw, Wroclaw (1976).
[26] B. A. Dubrovin, Russ. Math. Surveys, 36, 11–92 (1981). · Zbl 0549.58038 · doi:10.1070/RM1981v036n02ABEH002596
[27] V. E. Zaharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method [in Russian], Nauka, Moscow (1980); English transl., Plenum, New York (1984).
[28] B. A. Dubrovin and S. M. Natanzon, Funct. Anal. Appl., 16, 21–33 (1982). · Zbl 0554.35100 · doi:10.1007/BF01081805
[29] A. I. Bobenko and C. Klein, eds., Computational Approach to Riemann Surfaces (Lect. Notes Math., Vol. 2013), Springer, Berlin (2011).
[30] H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 3, Elliptic and Modular Functions: Lame and Mathieu Functions, McGraw-Hill, New York (1955).
[31] E. D. Belokolos and V. Z. Ènol’skii, Theor. Math. Phys., 53, 1120–1127 (1982). · Zbl 0547.35080 · doi:10.1007/BF01016682
[32] G. Forest and D.W. McLaughlin, J. Math. Phys., 23, 1248–1277 (1932). · Zbl 0498.35072 · doi:10.1063/1.525509
[33] E. D. Belokolos, A. I. Bobenko, V. B. Matveev, and V. Z. Ènol’skii, Russ. Math. Surveys, 41, 1–49 (1986). · Zbl 0612.35117 · doi:10.1070/RM1986v041n02ABEH003241
[34] J. Zagrodzinski, J. Phys. A., 15, 3109–3118 (1982). · Zbl 0502.35010 · doi:10.1088/0305-4470/15/10/015
[35] L. A. Takhtadzhyan, JETP, 66, 228–233 (1974).
[36] J. D. Fay, Theta-Functions on Riemann Surfaces (Lect. Notes Math., Vol. 352), Springer, Berlin (1973). · Zbl 0281.30013
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.