Singular sector of the KP hierarchy, $\bar{\partial}$-operators of non-zero index and associated integrable systems

TL;DR

This paper explores integrable hierarchies linked to the singular sector of the KP hierarchy, focusing on $ar{ ext{d}}$-operators with non-zero index, revealing new multidimensional equations and constraints.

Contribution

It introduces a novel approach to study integrable hierarchies via $ar{ ext{d}}$-operators of non-zero index and constructs these hierarchies using the $ar{ ext{d}}$-dressing method.

Findings

Identified hierarchies as restrictions of KP to finite codimension submanifolds.

Constructed hierarchies with multidimensional equations and constraints.

Analyzed hidden KdV, Boussinesq, and Gelfand-Dikii hierarchies.

Abstract

Integrable hierarchies associated with the singular sector of the KP hierarchy, or equivalently, with $\dbar$-operators of non-zero index are studied. They arise as the restriction of the standard KP hierarchy to submanifols of finite codimension in the space of independent variables. For higher $\dbar$-index these hierarchies represent themselves families of multidimensional equations with multidimensional constraints. The $\dbar$-dressing method is used to construct these hierarchies. Hidden KdV, Boussinesq and hidden Gelfand-Dikii hierarchies are considered too.