Theoretical and Mathematical Physics

, Volume 178, Issue 2, pp 143–193 | Cite as

Landau-Lifshitz equation, uniaxial anisotropy case: Theory of exact solutions

  • R. F. Bikbaev
  • A. I. Bobenko
  • A. R. Its


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.


multisoliton solution ferromagnetism finite-gap integration theta function 


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  1. 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.Google Scholar
  2. 2.
    L. A. Takhtajan, Phys. Lett. A, 64, 235–237 (1977).ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    A. E. Borovik, JETP Lett., 28, 581–584 (1978).ADSGoogle Scholar
  4. 4.
    E. K. Sklyanin, “On complete integrability of the Landau-Lifshitz equation,” Preprint E-3-79, LOMI, Leningrad (1979).zbMATHGoogle Scholar
  5. 5.
    A. B. Shabat, Funct. Anal. Appl., 9, 244–247 (1975).CrossRefGoogle Scholar
  6. 6.
    V. E. Zakharov and S. V. Manakov, JETP Lett., 8, 243–245 (1973).ADSGoogle Scholar
  7. 7.
    I. A. Akhiezer and A. E. Borovik, JETP Lett., 52, 332–335 (1967).Google Scholar
  8. 8.
    I. A. Akhiezer, A. E. Borovik, JETP Lett., 52, 885–893 (1967).Google Scholar
  9. 9.
    A. M. Kosevich, Fiz. Met. Metalloved., 53, 420–446 (1982).Google Scholar
  10. 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.Google Scholar
  11. 11.
    I. V. Cherednik, Math. USSR-Izv., 22, 357–377 (1984).CrossRefzbMATHGoogle Scholar
  12. 12.
    E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, J. Phys. Soc. Japan, 52, 388–393 (1983).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    M. M. Bogdan and A. S. Kovalev, JETP Lett., 31, 424–427 (1980).ADSGoogle Scholar
  14. 14.
    A. V. Mikhailov, Phys. Lett. A, 92, 51–55 (1982).ADSCrossRefMathSciNetGoogle Scholar
  15. 15.
    Yu. L. Rodin, Lett. Math. Phys., 7, 3–8 (1983).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    A. I. Bobenko, Zap. Nauchn. Sem. LOMI, 123, 58–66 (1983).zbMATHMathSciNetGoogle Scholar
  17. 17.
    A. B. Borisov, Fiz. Met. Metalloved., 55, 230–234 (1983).Google Scholar
  18. 18.
    R. F. Bikbaev, A. I. Bobenko, and A. R. Its, Soviet Math. Dokl., 28, 512–516 (1983).zbMATHGoogle Scholar
  19. 19.
    A. I. Bobenko, Funct. Anal. Appl., 19, 5–17 (1985).CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    M. Jimbo, I. Miwa, and K. Ueno, Phys. D, 2, 306–352 (1981).CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    M. Jimbo and T. Miwa, Phys. D, 2, 407–448 (1981).CrossRefzbMATHMathSciNetGoogle Scholar
  22. 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).zbMATHGoogle Scholar
  23. 23.
    A. Hurwitz and R. Courant, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen, Springer, Berlin (1964).CrossRefzbMATHGoogle Scholar
  24. 24.
    E. I. Zverovich, Russ. Math. Surveys, 26, 117–192 (1971).ADSCrossRefGoogle Scholar
  25. 25.
    V. B. Matveev, “Abelian functions and solitons,” Preprint H 373, Univ. Wroclaw, Wroclaw (1976).Google Scholar
  26. 26.
    B. A. Dubrovin, Russ. Math. Surveys, 36, 11–92 (1981).ADSCrossRefMathSciNetGoogle Scholar
  27. 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).Google Scholar
  28. 28.
    B. A. Dubrovin and S. M. Natanzon, Funct. Anal. Appl., 16, 21–33 (1982).CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    A. I. Bobenko and C. Klein, eds., Computational Approach to Riemann Surfaces (Lect. Notes Math., Vol. 2013), Springer, Berlin (2011).zbMATHGoogle Scholar
  30. 30.
    H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 3, Elliptic and Modular Functions: Lame and Mathieu Functions, McGraw-Hill, New York (1955).Google Scholar
  31. 31.
    E. D. Belokolos and V. Z. Ènol’skii, Theor. Math. Phys., 53, 1120–1127 (1982).CrossRefGoogle Scholar
  32. 32.
    G. Forest and D.W. McLaughlin, J. Math. Phys., 23, 1248–1277 (1932).ADSCrossRefMathSciNetGoogle Scholar
  33. 33.
    E. D. Belokolos, A. I. Bobenko, V. B. Matveev, and V. Z. Ènol’skii, Russ. Math. Surveys, 41, 1–49 (1986).ADSCrossRefzbMATHGoogle Scholar
  34. 34.
    J. Zagrodzinski, J. Phys. A., 15, 3109–3118 (1982).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    L. A. Takhtadzhyan, JETP, 66, 228–233 (1974).ADSGoogle Scholar
  36. 36.
    J. D. Fay, Theta-Functions on Riemann Surfaces (Lect. Notes Math., Vol. 352), Springer, Berlin (1973).zbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • R. F. Bikbaev
    • 1
  • A. I. Bobenko
    • 2
  • A. R. Its
    • 3
  1. 1.St. Petersburg Department of the Steklov Institute of MathematicsSt. PetersburgRussia
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany
  3. 3.Department of Mathematical SciencesIndiana University-Purdue University IndianapolisIndianapolisUSA

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