Solutions of the Hamilton-Jacobi equation for one component two dimensional field theories

The Hamilton-Jacobi formalism generalized to 2-dimensional field theories according to Lepage's canonical framework is applied to several covariant real scalar fields, e.g. massless and massive Klein-Gordon, Sine-Gordon, Liouville and Φ4 theories. For simplicity we use the Hamilton-Jacobi equation of DeDonder and Weyl. Unlike mechanics we have to impose certain integrability conditions on the velocity fields to guarantee the transversality relations between Hamilton-Jacobi wave fronts and the corresponding families of extremals embedded therein. Baecklund Transformations play a crucial role in solving the resulting system of coupled nonlinear PDEs. (orig.)