Bilinear equations in Darboux transformations by Boson-Fermion correspondence

TL;DR

This paper explores bilinear equations related to Darboux transformations in integrable hierarchies using Boson-Fermion correspondence, revealing new equations and their spectral properties in mathematical physics.

Contribution

It introduces a Fermionic approach to derive bilinear equations for Darboux transformations in KP, modified KP, and BKP hierarchies, including new equations in the Darboux chain.

Findings

Derived new bilinear equations in Darboux chains.

Connected Fermionic and Bosonic pictures of eigenfunctions.

Provided explicit examples of the new bilinear equations.

Abstract

Bilinear equation is an important property for integrable nonlinear evolution equation. Many famous research objects in mathematical physics, such as Gromov-Witten invariants, can be described in terms of bilinear equations to show their connections with the integrable systems. Here in this paper, we mainly discuss the bilinear equations of the transformed tau functions under the successive applications of the Darboux transformations for the KP hierarchy, the modified KP hierarchy (Kupershmidt-Kiso version) and the BKP hierarchy, by the method of the Boson-Fermion correspondence. The Darboux transformations are considered in the Fermionic picture, by multiplying the different Fermionic fields on the tau functions. Here the Fermionic fields are corresponding to the (adjoint) eigenfunctions, whose changes under the Darboux transformations are showed to be the ones of the squared eigenfunction potentials in the Bosonic picture, used in the spectral representations of the (adjoint) eigenfunctions. Then the successive applications of the Darboux transformations are given in the Fermionic picture. Based upon this, some new bilinear equations in the Darboux chain are derived, besides the ones of $(l-l')$ -th modified KP hierarchy. The corresponding examples of these new bilinear equations are given.