Vertex Operator Representation of the Soliton Tau Functions in the $A_n^{(1)}$ Toda Models by Dressing Transformations

TL;DR

This paper establishes the equivalence between the group-algebraic approach and dressing symmetry method for constructing soliton solutions in the $A_n^{(1)}$ Toda field theory, connecting two different frameworks for understanding these solutions.

Contribution

It demonstrates the equivalence of the group-algebraic and dressing symmetry methods for soliton construction in $A_n^{(1)}$ Toda models.

Findings

Proves the equivalence of two soliton construction methods.

Links algebraic and geometric approaches to Toda solitons.

Provides a unified understanding of soliton solutions.

Abstract

We study the relation between the group-algebraic approach and the dressing symmetry one to the soliton solutions of the $A_n^{(1)}$ Toda field theory in 1+1 dimensions. Originally solitons in the affine Toda models has been found by Olive, Turok and Underwood. Single solitons are created by exponentials of elements which ad-diagonalize the principal Heisenberg subalgebra. Alternatively Babelon and Bernard exploited the dressing symmetry to reproduce the known expressions for the fundamental tau functions in the sine-Gordon model. In this paper we show the equivalence between these two methods to construct solitons in the $A_n^{(n)}$ Toda models.