KP Solitons are Bispectral

TL;DR

This paper proves that all KP solitons exhibit bispectrality when including spectral translations, confirming a conjecture and highlighting potential links to integrable particle systems.

Contribution

It verifies the conjecture that all KP solitons are bispectral with spectral translations, extending previous results to a broader class.

Findings

All KP solitons are bispectral with spectral translation.

Supports the conjecture linking KP solitons and bispectrality.

Discusses implications for duality in integrable particle systems.

Abstract

It is by now well known that the wave functions of rational solutions to the KP hierarchy which can be achieved as limits of the pure $n$-soliton solutions satisfy an 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 a recent preprint (math-ph/9806001) evidence was presented to support the conjecture that all KP solitons (including their rational degenerations) are bispectral if one also allows translation operators in the spectral parameter. In this note, the conjecture is verified, and thus it is shown that all KP solitons have a form of bispectrality. The potential significance of this result to the duality of the classical Ruijsenaars and Sutherland particle systems is briefly discussed.