Restricting affine Toda theory to the half-line

TL;DR

This paper investigates affine Toda field theory restricted to a half-line, analyzing boundary conditions, soliton reflections, and boundary excitations, with applications to vacuum solutions and semiclassical reflection matrix calculations.

Contribution

It introduces a method to analyze affine Toda theory on the half-line with specific boundary conditions, extending known solutions and reflection properties.

Findings

Solitons reflect as antisolitons at the boundary.

Boundary conditions can support boundary breathers.

Semiclassical calculations support conjectured reflection matrices.

Abstract

We restrict affine Toda field theory to the half-line by imposing certain boundary conditions at $x=0$. The resulting theory possesses the same spectrum of solitons and breathers as affine Toda theory on the whole line. The classical solutions describing the reflection of these particles off the boundary are obtained from those on the whole line by a kind of method of mirror images. Depending on the boundary condition chosen, the mirror must be placed either at, in front, or behind the boundary. We observe that incoming solitons are converted into outgoing antisolitons during reflection. Neumann boundary conditions allow additional solutions which are interpreted as boundary excitations (boundary breathers). For $a_n^{(1)}$ and $c_n^{(1)}$ Toda theories, on which we concentrate mostly, the boundary conditions which we study are among the integrable boundary conditions classified by Corrigan et.al. As applications of our work we study the vacuum solutions of real coupling Toda theory on the half-line and we perform semiclassical calculations which support recent conjectures for the $a_2^{(1)}$ soliton reflection matrices by Gandenberger.