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Group classification, optimal system and optimal reductions of a class of Klein Gordon equations. (English) Zbl 1221.35023
Summary: Complete symmetry analysis is presented for non-linear Klein Gordon equations $u_{tt}=u_{xx}+f(u)$. A group classification is carried out by finding $f(u)$ that give larger symmetry algebra. One-dimensional optimal system is determined for symmetry algebras obtained through group classification. The subalgebras in one-dimensional optimal system and their conjugacy classes in the corresponding normalizers are employed to obtain, up to conjugacy, all reductions of equation by two-dimensional subalgebras. This is a new idea which improves the computational complexity involved in finding all possible reductions of a PDE of the form $F(x,t,u,u_{x},u_{t},u_{xx},u_{tt},u_{xt})=0$ to a first order ODE. Some exact solutions are also found.

MSC:
35A30Geometric theory for PDE, characteristics, transformations
35L71Semilinear second-order hyperbolic equations
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References:
[1] Ames, W. F.; Lohner, R. J.; Adams, E.: Group properties of utt=$(f(u)$ux)x, Int J non-linear mech 16, 439 (1981) · Zbl 0503.35058 · doi:10.1016/0020-7462(81)90018-4
[2] Baumann, G.: Symmetry analysis of differential equations with Mathematica, (2000) · Zbl 0946.35002
[3] Bluman, G. W.; Kumei, S.: Symmetries and differential equations, (1989) · Zbl 0698.35001
[4] Euler, N.; Steeb, W. H.: Continuous symmetries Lie algebras and differential equations, (1992) · Zbl 0755.35112
[5] Hydon, P. E.: Symmetry methods for differential equations, (2000) · Zbl 0951.34001
[6] Ibragimov NH. Classification of the invariant solutions to the equations of the two-dimensional transient state flow of gas, Zh Prikl Mekh Tekhn Fiz 1966;7(4): 19. English translation: J Appl Math Tech Phys 1966;7(4):11.
[7] Ibragimov NH. Optimal systems of subgroups and classification of invariant solutions of equations for planar non-stationary gas flows, Master of Science Thesis in Mathematics, Institute of Hydrodynamics, USSR Academic Science, Novosibirsk State University (1965). English translation: Paper 1 in N.H. Ibragimov, Selected Works, vol. II. Karlskrona: ALGA Publications; 2006.
[8] , Symmetries exact, solutions and conservation laws 1 (1994)
[9] , Applications in engineering and physical sciences 2 (1995)
[10] , New trends in theoretical developments and computational methods 3 (1996)
[11] Ibragimov, N. H.: Elementary Lie group analysis and ordinary differential equations, (1999) · Zbl 1047.34001
[12] Clarkson, Petere.; Mansfield, E. L.: Symmetry reductions and exact solutions of a class of nonlinear heat equations, Physica D 70, No. 3, 250 (1994) · Zbl 0812.35017 · doi:10.1016/0167-2789(94)90017-5
[13] Mansfield EL. Differential Gröbner bases, Ph.D. thesis, University of Sydney, Australia; 1992.
[14] Olver, P. J.: Applications of Lie groups to differential equations, (1986) · Zbl 0588.22001
[15] Ovsiannikov, L. V.: Group analysis of differential equations, (1982) · Zbl 0485.58002
[16] Rudra, P.: Symmetry group of the non-linear Klein Gordon equation, J phys A 19, 2499 (1986) · Zbl 0621.35083 · doi:10.1088/0305-4470/19/13/015