Affine Lie Algebraic Origin of Constrained KP Hierarchies

TL;DR

This paper develops an affine Lie algebraic framework for constrained KP hierarchies, revealing their connection to generalized NLS and Toda lattice structures, and introduces new hierarchies via algebraic and Lax approaches.

Contribution

It provides an affine $sl(n+1)$ algebraic construction of constrained KP hierarchies, establishing their equivalence through different methods and uncovering their relation to Toda lattice structures.

Findings

Hierarchies are equivalent via eigenvalue and Lax approaches.

Constrained KP hierarchies interpolate between GNLS and multi-boson KP-Toda.

The Toda lattice structure underpins the hierarchy's origin.

Abstract

We present an affine $sl (n+1)$ algebraic construction of the basic constrained KP hierarchy. This hierarchy is analyzed using two approaches, namely linear matrix eigenvalue problem on hermitian symmetric space and constrained KP Lax formulation and we show that these approaches are equivalent. The model is recognized to be the generalized non-linear Schr\"{o}dinger ($\GNLS$) hierarchy and it is used as a building block for a new class of constrained KP hierarchies. These constrained KP hierarchies are connected via similarity-B\"{a}cklund transformations and interpolate between $\GNLS$ and multi-boson KP-Toda hierarchies. Our construction uncovers origin of the Toda lattice structure behind the latter hierarchy.