Solution of the Affine Toda Equations as an Initial Value Problem

Abstract

Affine Toda field is a multicomponent field in two space-time dimensions, satifying a generalisation of the sinh-Gordon equation. Solution of 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 al. using Kac-Moody algebraic method. This solution as shown by Olive et al. yields the soliton solutions provided one chooses the coupling parameter β of the affine Toda fields to be purely imaginary.