On the constrained KP hierarchy II

TL;DR

This paper explores a specific constrained KP hierarchy, providing explicit formulas and a new proof of its compatibility, along with discussing its Poisson structure, thereby advancing the mathematical understanding of integrable systems.

Contribution

It introduces explicit formulas for vector field actions on differential operators and offers a new constructive proof of the hierarchy's compatibility, enhancing the theoretical framework.

Findings

Explicit formulas for vector fields on differential operators

A new constructive proof of hierarchy compatibility

Discussion of the Poisson structure of the constrained hierarchy

Abstract

A constrained KP hierarchy is discussed that was recently suggested by Aratyn et al. and by Bonora et al. This hierarchy is a restriction of the KP to a submanifold of operators which can be represented as a ratio of two purely differential operators of prescribed orders. Explicit formulas for action of vector fields on these two differential operators are written which gives a new description of the hierarchy and provides a new, more constructive, proof of compatibility of the constraint with the hierarchy. Also the Poisson structure of the constrained hierarchy is discussed.