Solutions of the Hamilton--Jacobi equation for one component two dimensional Field Theories

TL;DR

This paper extends the Hamilton--Jacobi formalism to two-dimensional scalar field theories, demonstrating how Bäcklund transformations facilitate solving complex nonlinear PDEs within this framework.

Contribution

It applies Lepage's canonical framework to 2D scalar fields and highlights the role of Bäcklund transformations in solving the resulting equations.

Findings

Successful application to various scalar field models

Identification of integrability conditions for velocity fields

Use of Bäcklund transformations to solve nonlinear PDEs

Abstract

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 $\phi^4$ 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. B\"acklund Transformations play a crucial role in solving the resulting system of coupled nonlinear PDEs.