Integrable boundary conditions for classical sine-Gordon theory

TL;DR

This paper classifies all boundary conditions that preserve integrability in the classical sine-Gordon model on a half-line, identifying the most general form and demonstrating the existence of an infinite set of conserved quantities.

Contribution

It derives the most general local boundary condition maintaining integrability for the classical sine-Gordon equation, expanding understanding of boundary effects in integrable systems.

Findings

Identified the most general integrable boundary condition for sine-Gordon

Constructed an infinite set of conserved quantities including the Hamiltonian

Confirmed consistency with recent quantum boundary results

Abstract

The possible boundary conditions consistent with the integrability of the classical sine-Gordon equation are studied. A boundary value problem on the half-line $x\leq 0$ with local boundary condition at the origin is considered. The most general form of this boundary condition is found such that the problem be integrable. For the resulting system an infinite number of involutive integrals of motion exist. These integrals are calculated and one is identified as the Hamiltonian. The results found agree with some recent work of Ghoshal and Zamolodchikov.