Abstract
We propose methods for constructing functionally invariant solutions u(x, y, z, t) of the sine-Gordon equation with a variable amplitude in 3+1 dimensions. We find solutions u(x, y, z, t) in the form of arbitrary functions depending on either one (α(x, y, z, t)) or two (α(x, y, z, t), β(x, y, z, t)) specially constructed functions. Solutions f(α) and f(α, β) relate to the class of functionally invariant solutions, and the functions α(x, y, z, t) and β(x, y, z, t) are called the ansatzes. The ansatzes (α, β) are defined as the roots of either algebraic or mixed (algebraic and first-order partial differential) equations. The equations defining the ansatzes also contain arbitrary functions depending on (α, β). The proposed methods allow finding u(x, y, z, t) for a particular, but wide, class of both regular and singular amplitudes and can be easily generalized to the case of a space with any number of dimensions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Frenkel and T. Kontorova, Acad. Sci. USSR J. Phys., 1, 137–149 (1939).
E. L. Aero and A. N. Bulygin, Mech. Solids, 42, 807–822 (2007).
P. Guéret, IEEE Trans. Magnetics, 11, 751–754 (1975).
R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations, Acad. Press, London (1982).
P. G. de Gennes, The Physics of Liquid Crystals, Clarendon, Oxford (1974).
K. Lonngren and A. C. Scott, eds., Solitons in Action, Acad. Press, eds (1978).
W. L. McMillan, Phys. Rev. B, 14, 1496–1502 (1976).
V. G. Bykov, Nonlinear Wave Processes in Geologic Media [in Russian], Dal’nauka, Vladivostok (2000).
P. L. Chebyshev, Uspekhi Mat. Nauk, 1, No. 2(12), 38–42 (1946).
A. S. Davydov, Solitons in Bioenergetics [in Russian], Naukova Dumka, Kiev (1986).
L. A. Takhtadzhyan and L. D. Faddeev, Hamiltonian Approach in the Theory of Solitons [in Russian], Nauka, Moscow (1986)
L. A. Takhtadzhyan and L. D. Faddeev, English transl.: Hamiltonian Methods in the Theory of Solitons, Springer, Berlin (1987).
L. A. Takhtadzhyan and L. D. Faddeev, Theor. Math. Phys., 21, 1046–1057 (1974).
I. A. Garagash, V. N. Nikolaevskiy, Computational Continuum Mechanics, 2, 44–66 (2009).
H. Bateman, Electrical and Optical Wave Motion Cambridge Univ. Press, London (1914).
A. R. Forsyth, Messenger Math., 27, 99–118 (1898).
V. Smirnoff and S. Soboleff, “Sur une méthode nouvelle dans le problême plan des vibrations élastiques,” in: Trudy seismologichecskogo in-ta [Works of the Seismological Institute], No. 20, Acad. Sci. USSR, Leningrad (1932).
V. Smirnoff and S. Soboleff, C. R. Acad. Sci. Paris, 194, 1437–1439 (1932).
S. Sobolev, “Functionally invariant solutions of wave equation,” in: Travaux Inst. Physico-Math. Stekloff, Vol. 5, Acad. Sci. USSR, Leningrad (1934), pp. 259–2.
S. L. Sobolev, Selected Works [in Russian], Vol. 1, Equations of Mathematical Physics: Computational Mathematics and Cubature Formulas, Sobolev Inst. Math., Siberian Branch, Russ. Acad. Sci., Novosibirsk (2003)
S. L. Sobolev, Op. cit., Vol. 2, Functional Analysis: Partial Differential Equations, Sobolev Inst. Math., Siberian Branch, Russ. Acad. Sci., Novosibirsk (2006).
N. P. Erugin, Uchenye zap. Leningr. un-ta., 15, 101–134 (1948).
M. M. Smirnov, Dokl. AN SSSR, 67, 977–980 (1949).
E. L. Aero, A. N. Bulygin, and Yu. V. Pavlov, Theor. Math. Phys., 158, 313–319 (2009).
E. L. Aero, A. N. Bulygin, and Yu. V. Pavlov, Nelineinyi Mir, 7, 513–517 (2009).
E. L. Aero, A. N. Bulygin, and Yu. V. Pavlov, Differ. Equ., 47, 1442–1452 (2011).
E. L. Aero, A. N. Bulygin, and Yu. V. Pavlov, Appl. Math. Comput., 223, 160–166 (2013).
C. G. J. Jacobi, J. Reine Angew. Math., 36, 113–134 (1848)
C. G. J. Jacobi, “Über eine particulôre LÖsung der partiellen Differentialgleichung \(\frac{{{\partial ^2}V}}{{\partial {x^2}}} + \frac{{{\partial ^2}V}}{{\partial {y^2}}} + \frac{{{\partial ^2}V}}{{\partial {z^2}}}\),” in: C. G. J. Jacobi’s Gesammelte Werke (C. G. J. Jacobi and C. W. Borchardt, eds.), Vol. 2, Verlag von G.Reimer, Berlin(1882), pp. 191–2.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 184, No. 1, pp. 79–91, July, 2015.
Rights and permissions
About this article
Cite this article
Aero, E.L., Bulygin, A.N. & Pavlov, Y.V. Solutions of the sine-Gordon equation with a variable amplitude. Theor Math Phys 184, 961–972 (2015). https://doi.org/10.1007/s11232-015-0309-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11232-015-0309-8