Prolongation Loop Algebras for a Solitonic System of Equations

TL;DR

This paper studies an integrable electromagnetic system in a two-level medium, demonstrating how soliton solutions are preserved and structured within an infinite-dimensional algebraic framework, revealing its Hamiltonian properties.

Contribution

It introduces a novel algebraic approach to analyze solitonic solutions in a Maxwell-Bloch system, including the preservation of reality and pole structure under transformations.

Findings

Backlund transformation preserves soliton reality

Pole structure characterized with broadening parameter

Hamiltonian formulation established for the system

Abstract

We consider an integrable system of reduced Maxwell-Bloch equations that describes the evolution of an electromagnetic field in a two-level medium that is inhomogeneously broadened. We prove that the relevant Backlund transformation preserves the reality of the n-soliton potentials and establish their pole structure with respect to the broadening parameter. The natural phase space of the model is embedded in an infinite dimensional loop algebra. The dynamical equations of the model are associated to an infinite family of higher order Hamiltonian systems that are in involution. We present the Hamiltonian functions and the Poisson brackets between the extended potentials.