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.


35A30 Geometric theory, characteristics, transformations in context of PDEs
35L71 Second-order semilinear hyperbolic equations
Full Text: DOI


[1] Ames, W.F.; Lohner, R.J.; Adams, E., Group properties of \(u_{\mathit{tt}} = (f(u) u_x)_x\), Int J non-linear mech, 16, 439, (1981) · Zbl 0503.35058
[2] Baumann, G., Symmetry analysis of differential equations with Mathematica, (2000), Springer New York · Zbl 0898.34003
[3] Bluman, G.W.; Kumei, S., Symmetries and differential equations, (1989), Springer New York · Zbl 0698.35001
[4] Euler, N.; Steeb, W.H., Continuous symmetries Lie algebras and differential equations, (1992), Bibliographisches Institut Mannheim · Zbl 0764.35098
[5] Hydon, P.E., Symmetry methods for differential equations, (2000), Cambridge University Press Cambridge · Zbl 1035.35005
[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] ()
[9] ()
[10] ()
[11] Ibragimov, N.H., Elementary Lie group analysis and ordinary differential equations, (1999), Wiley Chichester · Zbl 1047.34001
[12] Clarkson, PeterE.; Mansfield, E.L., Symmetry reductions and exact solutions of a class of nonlinear heat equations, Physica D, 70, 3, 250, (1994) · Zbl 0812.35017
[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), Springer New York · Zbl 0588.22001
[15] Ovsiannikov, L.V., Group analysis of differential equations, (1982), Academic Press New York · Zbl 0485.58002
[16] Rudra, P., Symmetry group of the non-linear Klein Gordon equation, J phys A, 19, 2499, (1986) · Zbl 0621.35083
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.