Exact solutions of the multidimensional classical φ6-field equations obtained by symmetry reduction

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Abstract
The often used φ6 model of classical critical phenomena is studied in (3+1)‐dimensional Minkowski and Euclidean spaces. The Euler–Lagrange equations describing the kinetics of the scalar order parameter are in this case nonlinear Klein–Gordon equations. The method of symmetry reduction is systematically applied to derive all the solutions invariant under subgroups with generic orbits of codimension 1. Whenever the obtained ordinary differential equations have the Painlevé property, they can be transformed to one of two standard forms. These are then solved in terms of elliptic functions or elementary ones. This results in a large number of new exact solutions. Particularly interesting solutions are found in the immediate vicinity of the tricritical point. Our treatment of the φ6 theory is complete only for the four‐dimensional spaces M(3,1) and E(4), but many of the results are given for the more general cases of M(n,1) and E(n+1).