New Classical Solutions to F4 Theory

Abstract

The partial differential equation 02rf>+m2c/>+Ac/>3 = 0 (where 0 2 = 0;-\72) is converted to the ordinary differential equation 2¢>/du2 + (Nlu)d¢>ldu +m2c/> +).¢>3 = O. In this conversion, 02U and (OU)2 are assumed to depend only on u, where u is a Fnction of x,y, Z, t, and further, Nlu is the normal curvature of the hypersurface u = constant. In order to solve for u, the analogous Euclidean 4-space problem with 0 2 = 0; +\72 is examined initially. The only possible values of N are then 0, 1, 2, 3 corresponding to hypersurfaces that are a hyperplane, a hypercylinder (one curved face), a hypercylinder (two curved faces) and a ypersphererespectively. These hypersurfaces are transformed to Minkowski space via xk ~ ixk , k = 1,2,3. Then by solving the ordinary differential equation new solutions to the original partial differential equation are found, one of which has a closed form.