Solutions of the three-dimensional sine-Gordon equation

Article

Abstract

We obtain exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation in a form that Lamb previously proposed for integrating the two-dimensional sine-Gordon equation. The three-dimensional solutions depend on arbitrary functions F(α) and ϕ(α,β), whose arguments are some functions α(x, y, z, t) and β(x, y, z, t). The ansatzes must satisfy certain equations. These are an algebraic system of equations in the case of one ansatz. In the case of two ansatzes, the system of algebraic equations is supplemented by first-order ordinary differential equations. The resulting solutions U(x, y, z, t) have an important property, namely, the superposition principle holds for the function tan(U/4). The suggested approach can be used to solve the generalized sine-Gordon equation, which, in contrast to the classical equation, additionally involves first-order partial derivatives with respect to the variables x, y, z, and t, and also to integrate the sinh-Gordon equation. This approach admits a natural generalization to the case of integration of the abovementioned types of equations in a space with any number of dimensions.

Keywords

sine-Gordon equation wave equation Hamilton-Jacobi equation superposition principle 

References

  1. 1.
    J. Frenkel and T. Kontorova, J. Phys. Acad. Sci. USSR, 1, 137–149 (1939).MathSciNetGoogle Scholar
  2. 2.
    E. L. Aero and A. N. Bulygin, Izv. Ross. Akad. Nauk. Ser. Mekh. Tverd. Tela, No. 5, 170–187 (2007).Google Scholar
  3. 3.
    E. L. Aero, J. Engrg. Math., 55, 81–95 (2006).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    P. Gueret, IEEE Trans. Magnetics, 11, 751–754 (1975).CrossRefADSGoogle Scholar
  5. 5.
    K. Lonngren and A. Scott, eds., Solitons in Action, Acad. Press, New York (1978).MATHGoogle Scholar
  6. 6.
    W. L. McMillan, Phys. Rev. B, 14, 1496–1502 (1976).CrossRefADSGoogle Scholar
  7. 7.
    G. L. Lamb Jr., Phys. Lett. A, 25, 181–182 (1967).CrossRefADSGoogle Scholar
  8. 8.
    R. K. Bullough and P. J. Caudrey, eds., Solitons (Topics Current Phys., Vol. 17), Springer, Berlin (1980).MATHGoogle Scholar
  9. 9.
    P. G. de Gennes, The Physics of Liquid Crystals, Clarendon, Oxford (1974).Google Scholar
  10. 10.
    V. G. Bykov, Nonlinear Wave Processes in Geological Media [in Russian], Dalnauka, Vladivostok (2000).Google Scholar
  11. 11.
    A. S. Davydov, Solitons in Bioenergetics [in Russian], Naukova Dumka, Kiev (1986).MATHGoogle Scholar
  12. 12.
    L. A. Takhtadzhyan and L. D. Faddeev, Hamiltonian Approach in the Theory of Solitons [in Russian], Nauka, Moscow (1986); English transl.: Hamiltonian Methods in the Theory of Solitons, Springer, Berlin (1987).Google Scholar
  13. 13.
    L. A. Takhtadzhyan and L. D. Faddeev, Theor. Math. Phys., 21, 1046–1057 (1974).CrossRefGoogle Scholar
  14. 14.
    G. L. Lamb Jr., Rev. Modern Phys., 43, 99–124 (1971).CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    R. M. Miura, ed., Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications (Lect. Notes Math., Vol. 515), Springer, Berlin (1976).MATHCrossRefGoogle Scholar
  16. 16.
    M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM Stud. Appl. Math., Vol. 4), SIAM, Philadelphia (1981).MATHGoogle Scholar
  17. 17.
    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
  18. 18.
    R. Hirota, J. Phys. Soc. Japan, 33, 1459–1463 (1972).ADSCrossRefGoogle Scholar
  19. 19.
    R. Steuerwald, Abh. Bayer. Akad. Wiss., N. F., No. 40, 1–106 (1936).Google Scholar
  20. 20.
    H. Bateman, The Mathematical Analysis of Electrical and Optical Wave-Motion: On the Basis of Maxwell’s Equations, Dover, New York (1955).MATHGoogle Scholar

Copyright information

© MAIK/Nauka 2009

Authors and Affiliations

  1. 1.Institute for Machine Science ProblemsRASSt. PetersburgRussia

Personalised recommendations