Twisted Affine Integrable Hierarchies and Soliton Solutions

TL;DR

This paper constructs integrable hierarchies based on twisted affine Lie algebras, classifies their solutions, and explicitly constructs soliton solutions using vertex operators, revealing new structures in integrable systems.

Contribution

It introduces a systematic method for building integrable hierarchies from twisted affine algebras and constructs soliton solutions with both zero and non-zero vacuum states.

Findings

Classified integrable models by algebraic structure and vacuum solutions.

Constructed soliton solutions using vertex operators.

Identified the role of vacuum parameters in solution structure.

Abstract

A systematic construction of a class of integrable hierarchy is discussed in terms of the twisted affine $A_{2r}^{(2)}$ Lie algebra. The zero curvature representation of the time evolution equations are shown to be classified according to its algebraic structure and according to its vacuum solutions. It is shown that a class of models admit both zero and constant (non zero) vacuum solutions. Another, consists essentially of integral non-local equations and can be classified into two sub-classes, one admitting zero vacuum and another of constant, non zero vacuum solutions. The two dimensional gauge potentials in the vacuum plays a crucial ingredient and are shown to be expanded in powers of the vacuum parameter $v_0$. Soliton solutions are constructed from vertex operators, which for the non zero vacuum solutions, correspond to deformations characterized by $v_0$.