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

TL;DR

This paper extends the Hamilton-Jacobi formalism to 2D field theories, using Bäcklund transformations to construct solutions for various scalar fields, and discusses the relation between Euler-Lagrange and Hamilton-Jacobi equations.

Contribution

It introduces a method employing Bäcklund transformations within the Hamilton-Jacobi framework for 2D scalar field theories, addressing integrability and transversality conditions.

Findings

Bäcklund transformations effectively solve coupled nonlinear PDEs in this context.

The formalism clarifies the relation between wave fronts and extremals in 2D fields.

Integrability conditions are essential for the equivalence of Hamilton-Jacobi wave fronts and extremals.

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.