Bispectrality of KP Solitons

TL;DR

This paper extends the concept of bispectrality to certain non-rational KP soliton solutions, showing they satisfy eigenvalue equations for non-local operators, potentially linking to dualities in integrable systems.

Contribution

It introduces a new form of bispectrality for non-rational KP solitons involving non-local operators, expanding the understanding of spectral properties in integrable systems.

Findings

Non-rational KP solitons satisfy eigenvalue equations with non-local operators.

This form of bispectrality may relate to the duality of the Ruijsenaars system.

Extends bispectrality beyond rational solutions to a broader class of KP solitons.

Abstract

It is by now well known that the wave functions of rational solutions to the KP hierarchy (those which can be achieved as limits of the pure n-soliton solutions) satisfy an additional eigenvalue equation for ordinary differential operators in the spectral parameter. This property is known as ``bispectrality'' and has proved to be both interesting and useful. In this note, it is shown that certain (non-rational) soliton solutions of the KP hierarchy satisfy an eigenvalue equation for a non-local operator constructed by composing ordinary differential operators in the spectral parameter with translation operators in the spectral parameter, and therefore have a form of bispectrality as well. Considering the results relating ordinary bispectrality to the self-duality of the rational Calogero-Moser particle system, it seems likely that this new form of bispectrality should be related to the duality of the Ruijsenaars system.