Abstract
A constructive method for constructing nonlocal symmetries of differential equations based on the Lie—Bäcklund theory of groups is developed. The concept of quasilocal symmetries is introduced. With the help of this method nonlocal symmetries of differential equations of the type of nonlinear thermal conductivity and gas dynamics are studied.
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Literature cited
V. K. Andreev and A. A. Rodionov, “Group analysis of equations of planar flows of an ideal fluid in Lagrangian coordinates”, Dokl. Akad. Nauk SSSR,298, No. 6, 1358–1361 (1988).
V. K. Andreev and A. A. Rodionov, “Group classification and exact solutions of equations of planar and rotationally-symmetric flow of an ideal fluid in Lagrangian coordinates”, Differents. Uravnen., No. 9, 1577–1586 (1988).
J. Astarita and J. Marucci, Foundations of Hydromechanics of Non-Newtonian Fluids [Russian translation], Mir, Moscow (1978).
I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, “Group classification of the equations of nonlinear filtration”, Dokl. Akad. Nauk SSSR,293, No. 5, 1033–1035 (1987).
I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, “Quasilocal symmetries of equations of the type of nonlinear thermal conductivity”, Dokl. Akad. Nauk SSSR,295, No. 1, 75 (1987).
I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, “Bäcklund transformations and nonlocal symmetries”, Dokl. Akad. Nauk SSSR,297, No. 1, 11–14 (1987).
I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, “Quasilocal symmetries of equations of mathematical physics”, in: Mathematical Modeling. Nonlinear Differential Equations of Mathematical Physics [in Russian], Nauka, Moscow (1987), pp. 22–56.
I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, “Group properties and exact solutions of equations of nonlinear filtration”, in: Numerical Methods of Solution of Problems of Filtration of a Multiphase Incompressible Fluid [in Russian], Novosibirsk (1987), pp. 24–27.
I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, “Basic types of invariant equations of one-dimensional gas-dynamics”, Preprint, Inst. Applied Math. of the Acad. of Sciences of the USSR, Moscow (1988).
G. I. Barenblatt, V. M. Entov, and V. M. Ryzhik, Theory of Nonstationary Filtration of a Fluid and Gas [in Russian], Nedra, Moscow (1972).
A. A. Berezovskii, Lectures on Nonlinear Boundary Problems of Mathematical Physics [in Russian], Parts I and II, Naukova Dumka, Kiev (1976).
N. N. Bogolyubov, “Question of model Hamiltonian in the theory of superconductivity”, in: Selected Works in Three Volumes [in Russian], Vol. 3, Naukova Dumka Kiev (1971), pp. 110–173.
A. M. Vinogradov and I. S. Krasil'shchik, “A method of calculation of higher symmetries of nonlinear evolution equations and nonlocal symmetries,” Dokl. Akad. Nauk SSSR,253, No. 6, 1289–1293 (1980).
A. M. Vinogradov and I. S. Krasil'shchik, “Theory of nonlocal symmetries of nonlinear partial differential equations”, Dokl. Akad. Nauk SSSR,275, No. 5, 1044–1049 (1984).
A. M. Vinogradov, I. S. Krasil'shchik, and V. V. Lychagin, Introduction to the Geometry of Nonlinear Differential Equations [in Russian], Nauka, Moscow (1986).
V. S. Vladimirov and I. V. Volovich, “Local and nonlocal flows for nonlinear equations”, Teor. Mat. Fiz.,62, No. 1, 3–29 (1985).
V. S. Vladimirov and I. V. Volovich, “Conservation laws for nonlinear equations”, Usp. Mat. Nauk,40, No. 4, 17–26 (1985).
R. K. Gazizov, “Contact transformations of equations of the type of nonlinear filtration”, in: Physicochemical Hydrodynamics. Inter-University Scientific Collection [in Russian], Izdat. Bashk. Gos. Univ., Ufa (1987), pp. 38–41.
N. Kh. Ibragimov, “Theory of Lie—Bäcklund transformation groups”, Mat. Sborn.,109, No. 2, 229–253 (1979).
N. Kh. Ibravimov, Transformation Groups in Mathematical Physics [in Russian], Nauka, Moscow (1983).
N. Kh. Ibragimov and A. B. Shabat, “Korteweg-de Vries equation from the group point of view”, Dokl. Akad. Nauk SSSR,244, No. 1, 57–61 (1979).
O. V. Kaptsov, “Extension of symmetries of evolutional equations”, Dokl. Akad. Nauk SSSR,262, No. 5, 1056–1059 (1982).
E. G. Kirnasov, “Wohlquist-Estabrook type coverings over the equation of thermal conductivity”, Mat. Zametki,42, No. 3, 422–434 (1987).
B. G. Konopel'chenko and V. G. Mokhnachev, “Group analysis of differential equations”, Yadernaya Fiz.,30, No. 2, 559–567 (1979).
V. V. Kornyak, “Application of a computer to study the symmetries of some equations of mathematical physics”, in: Group-Theoretic Studies of Equations of Mathematical Physics [in Russian], Inst. Mat. Akad. Nauk UkrSSR, Kiev (1985), pp. 114–119.
E. V. Lenskii, “Group properties of equations of motion of a nonlinear viscoplastic medium”, Vestn. Mosk. Gos. Univ. (MGU), Mat., Mekhan., No. 5, 116–125 (1966).
L. V. Ovsyannikov, “Groups and group-invariant solutions of differential equations”, Dokl. Akad. Nauk SSSR,118, No. 3, 439–442 (1958).
L. V. Ovsyannikov, “Group properties of equations of thermal conductivity”, Dokl. Akad. Nauk SSSR,125, No. 3, 492–495 (1959).
L. V. Ovsyannikov, Group Properties of Differential Equations [in Russian], Izdat. Sib. Otd. (SO) Akad. Nauk SSSR, Novosibirsk (1962).
L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).
M. I. Petrashen' and E. D. Trifonov, Application of Group Theory to Quantum Mechanics [in Russian], Nauka, Moscow (1967).
V. V. Pukhnachev, “Evolution equations and Lagrangian coordinates”, in: Dynamics of Continuous Media in the Seventies [in Russian], Novosibirsk (1985), pp. 127–141.
V. V. Pukhnachev, “Equivalence transformations and hidden symmetries of evolution equations”, Dokl. Akad. Nauk SSSR,294, No. 3, 535–538 (1987).
B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and Their Application to Gas Dynamics [in Russian], Nauka, Moscow (1978).
S. R. Svirshchevskii, “Group properties of a model of thermal transport taking into account relaxation of thermal flow”, Preprint, Inst. Prikl. Mat. Akad. Nauk SSSR, No. 105 (1988).
V. A. Florin, “Some of the simplest nonlinear problems of consolidation water-saturated earth media”, Izv. Akad. Nauk SSSR, OTN, No. 9, 1389–1402 (1948).
V. A. Fok, “Hydrogen atom and non-Eulidean geometry”, Izv. Akad. Nauk SSSR, VII. Seriya Otdel. Mat. Estestv. Nauk, No. 2, 169–179 (1935).
V. I. Fushchich, “Supplementary invariances of relativistic equations of motion”, Teor. Mat. Fiz.,7, No. 1, 3–12 (1971).
V. I. Fushchich, “A new method of study of group properties of the equations of mathematical physics”, Dokl. Akad. Nauk SSSR,246, No. 4, 846–850 (1979).
V. I. Fushchich and V. V. Kornyak, “Realization on a computer of an algorithm for calculating nonlocal symmetries for Dirac type equations”, Preprint, Inst. Mat. Akad. Nauk UkrSSR, 85.20, No. 20, Kiev (1985).
N. G. Khor'kova, “Conservation laws and nonlocal symmetries”, Mat. Zametki,44, No. 1, 134–145 (1988).
W. F. Ames, Nonlinear Partial Differential Equations in Engineering, Academic Press, New York, Vol. 1 (1965), Vol. 2 (1972).
G. Bluman and S. Kumei, “On the remarkable nonlinear diffusion equation∂/∂x[a(u+b)−2∂u/∂x]−∂u/∂t=0,” J. Math. Phys.,21, No. 5, 1019–1023 (1980).
J. R. Burgan, A. Munier, M. R. Felix, and E. Fijalkow, “Homology and the nonlinear heat diffusion equation”, SIAM J. Appl. Math.,44, No. 1, 11–18 (1984).
J. D. Cole, “On a quasilinear parabolic equation used in aerodynamics”, Quart. Appl. Math.,9, 225–236 (1951).
E. Hopf, “The partial differential equation ut+uux=μuxx”, Commun. Pure. Appl. Math.,3, 201–230 (1950).
N. H. Ibragimov, “Sur l'equivalence des equations d'evolution qui admettent une algebre de Lie—Bäcklund infinie”, C. R. Acad. Sci. Ser. 1, Paris,293, No. 14, 657–660 (1981).
S. Lie, “Begründung einer Invariantentheorie der Berührungstransformationen”, Math. Ann.,8, No. 2, 215–228 (1874).
A. Munier, J. R. Burgan, J. Gutierrez, E. Fijalkow, and M. R. Feliz, “Group transformations and the nonlinear heat diffusion equation”, SIAM J. Appl. Math.,40, No. 2, 191–207 (1981).
A. Oron and P. Rosenau, “Some symmetries of the nonlinear heat and wave equations”, Phys. Lett. A,118, No. 4, 172–176 (1986).
L. V. Ovsjannikov, “Some problems arising in group analysis of differential equations”, in: Proc. Symp. Symmetry, Similarity and Group-Theory. Methods in Mechanics, Calgary (1974), pp. 181–202.
K. Kiso, “Pseudopotentials and symmetries of evolution equations”, Hokkaido Math. J.,18, No. 1, 125–136 (1989).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 34, pp. 3–83, 1989.
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Akhatov, I.S., Gazizov, R.K. & Ibragimov, N.K. Nonlocal symmetries. Heuristic approach. J Math Sci 55, 1401–1450 (1991). https://doi.org/10.1007/BF01097533
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DOI: https://doi.org/10.1007/BF01097533