Constructing solutions of Hamilton-Jacobi equations for 2 D fields with one component by means of Baecklund transformations

Abstract

The Hamilton-Jacobi formalism generalized to 2-dimensional field theories according to Lepage's canonical framework is applied to several relativistic real scalar fields, e.g. massless and massive Klein-Gordon, Sinh and Sine-Gordon, Liouville and $\phi^4$ theories. The relations between the Euler-Lagrange and the Hamilton-Jacobi equations are discussed in DeDonder and Weyl's and the corresponding wave fronts are calculated in Carath\'eodory's formulation. Unlike mechanics we have to impose certain integrability conditions on the velocity fields to guarantee the transversality relations and especially the dynamical equivalence between Hamilton-Jacobi wave fronts and families of extremals embedded therein. B\"acklund Transformations play a crucial role in solving the resulting system of coupled nonlinear PDEs.