Solution of the affine Toda equations as an initial value problem

Affine Toda field is a multicomponent field in two space-time dimensions, satifying a generalisation of the sinh-Gordon equation partial derivative(2) phi + 4 root 2 mu(2)/beta sinh(root 2 beta phi) = 0. Solution oi the affine Toda field equations is presented as a path ordered exponential integrals of an initial value problem. This is in the same spirit as the work of Mansfield where one evolves the solution at the apex of a backward light-cone from the initial values at some arbitrary points along the legs of this light-cone. These two initial values are then connected by an arbitrary path. Selecting a particular path as the forward light-cone from a point at the infinite past, one obtains the solution to the field equation which reduces to the solution proposed by Olive et ad. using Kac-Moody algebraic method. This solution as shown by Olive et al. yields the soliton solutions provided one chooses the coupling parameter beta of the affine Toda fields to be purely imaginary.