Exact solutions for the nonlinear Klein–Gordon and Liouville equations in four-dimensional Euclidean space
Y. Matsuno
Source abstract
A systematic method for constructing particular solutions of the nonlinear Klein–Gordon and Liouville equations in four-spatial dimensions is developed. The method of solution presented here first consists of reducing nonlinear partial differential equations to ordinary differential equations (ODE’s) by introducing symmetry variables and then seeking exact solutions for more tractable ODE’s. Various exact solutions are presented, in which new solutions with nonspherical symmetries are included. Furthermore, the exact method is applied to the above equations in general n-spatial dimensions. Among them, a conformally invariant nonlinear Klein–Gordon equation is particularly interesting from the viewpoint of field theories. The exact solutions for these equations are generalizations of those for the corresponding equations in four-spatial dimensions.
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