The Charged Free Boson Integrable Hierarchy.

摘要

Fully integrable PDEs, those with an infinite number of conservation laws, are of particular interest to the modeling community as exact solutions can often be found. There are several methods currently used to study integrable systems. Constructing integrable hierarchies using the language of vertex algebras provides a simple formulation of the structure of the hierarchy. It also allows us to easily find desirable solutions, such as soliton solutions to the KP hierarchy. For classical hierarchies, such as the KP and Toda hierarchies, this construction relies on the boson-fermion correspondence.;The vertex algebra formed by two charged free bosons does not have a fermionic Fock space. To investigate the charged free boson integrable hierarchy, Friedan-Martinec-Shenker bosonization is used in place of the boson-fermion correspondence. The charged free bosons were studied by Kac and Radul; while the hierarchy was mentioned it was not studied in any detail. We thoroughly investigate this hierarchy and several of its reductions. This gives a generalization of the vertex algebra methods currently used to study integrable hierarchies. We also find several interesting PDEs, including the Euler beam equation which was known to be integrable but had not been studied using an algebraic approach.