Connection between the Affine and Conformal Affine Toda Models and their Hirota's Solution

TL;DR

This paper demonstrates the relationship between affine and conformal affine Toda models, showing how solutions of one can be mapped to the other, and constructs soliton solutions using Hirota's method for specific algebraic cases.

Contribution

It establishes a gauge fixing connection between AT and CAT models and introduces Hirota's tau-functions to generate soliton solutions for these models.

Findings

Solutions of AT models can be mapped to solutions of CAT models.

Hirota's tau-functions are effective in constructing soliton solutions.

Explicit soliton solutions are obtained for models based on t and p algebras.

Abstract

It is shown that the Affine Toda models (AT) constitute a ``gauge fixed'' version of the Conformal Affine Toda model (CAT). This result enables one to map every solution of the AT models into an infinite number of solutions of the corresponding CAT models, each one associated to a point of the orbit of the conformal group. The Hirota's $\tau$-function are introduced and soliton solutions for the AT and CAT models associated to $\hat {SL}(r+1)$ and $\hat {SP}(r)$ are constructed.