Real Hamiltonian forms of affine Toda field theories: spectral aspects

TL;DR

This paper investigates real Hamiltonian forms of affine Toda field theories linked to exceptional Lie algebras, analyzing their spectral properties through Lax operators, Riemann-Hilbert problems, and scattering data.

Contribution

It introduces new real Hamiltonian forms of affine Toda theories related to exceptional Kac-Moody algebras and studies their spectral and scattering properties.

Findings

Constructed Lax representations for the theories.

Formulated Riemann-Hilbert problems for spectral analysis.

Derived minimal scattering data sets for potential reconstruction.

Abstract

The paper is devoted to real Hamiltonian forms of 2-dimensional Toda field theories related to exceptional simple Lie algebras, and to the spectral theory of the associated Lax operators. Real Hamiltonian forms are a special type of "reductions" of Hamiltonian systems, similar to real forms of semi-simple Lie algebras. Examples of real Hamiltonian forms of affine Toda field theories related to exceptional complex untwisted affine Kac-Moody algebras are studied. Along with the associated Lax representations, we also formulate the relevant Riemann-Hilbert problems and derive the minimal sets of scattering data that determine uniquely the scattering matrices and the potentials of the Lax operators.