What question did this study set out to answer?

The aim is to explore solutions of the KP hierarchy using boundary values on the unit circle within Banach spaces.

April 7, 2026Lobachevskii Journal of Mathematics0 citations

Boundary Values on the Circle and Solutions of KP and Strict KP

GHG. F. HelminckUniversity of Amsterdam EPE. A. PanasenkoTambov State University

Key Points

The aim is to explore solutions of the KP hierarchy using boundary values on the unit circle within Banach spaces.
Introduced a collection of Banach spaces with a countable Schauder basis.
Constructed open subsets of each Banach space to derive solutions of the KP hierarchy.
Developed a fiber bundle over the selected open subsets for strict KP hierarchy solutions.
Analyzed the nonvanishing of a Fredholm determinant to describe the open subsets.
Demonstrated that each constructed solution of the KP hierarchy is expressible in terms of the Fredholm determinant.
Established a connection between the geometry of Banach spaces and the solutions of the KP hierarchies.

Abstract

In this paper we present a wide collection of Banach spaces B of boundary values on the unit circle around the origin. A common property of the set is that each B has a countable Schauder basis. In each GL (B) we present an open subset (B) from which we can construct a solution of the KP hierarchy and we introduce a fiber bundle over (B) that yields solutions of the strict KP hierarchy. Finally we show that the set (B) can be described by the nonvanishing of a Fredholm determinant. The solutions of the KP hierarchy constructed from (B) can be expressed in this Fredholm determinant.

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