$A_n^{(1)}$ affine Toda field theories with integrable boundary conditions revisited

TL;DR

This paper revisits integrable boundary conditions in affine Toda field theories, specifically analyzing the algebraic structures and conserved quantities for different boundary types, with new results on soliton preserving conditions.

Contribution

It introduces novel soliton preserving boundary conditions for $A_n^{(1)}$ affine Toda theories and systematically derives their conserved quantities, expanding understanding of boundary integrability.

Findings

SNP boundary conditions recover known results.

SP boundary conditions lead to new conserved quantities.

Number of integrals doubles for SP boundary conditions.

Abstract

Generic classically integrable boundary conditions for the $A_{n}^{(1)}$ affine Toda field theories (ATFT) are investigated. The present analysis rests primarily on the underlying algebra, defined by the classical version of the reflection equation. We use as a prototype example the first non-trivial model of the hierarchy i.e. the $A_2^{(1)}$ ATFT, however our results may be generalized for any $A_{n}^{(1)}$ ($n>1$). We assume here two distinct types of boundary conditions called some times soliton preserving (SP), and soliton non-preserving (SNP) associated to two distinct algebras, i.e. the reflection algebra and the ($q$) twisted Yangian respectively. The boundary local integrals of motion are then systematically extracted from the asymptotic expansion of the associated transfer matrix. In the case of SNP boundary conditions we recover previously known results. The other type of boundary conditions (SP), associated to the reflection algebra, are novel in this context and lead to a different set of conserved quantities that depend on free boundary parameters. It also turns out that the number of local integrals of motion for SP boundary conditions is `double' compared to those of the SNP case.