Bilinear equations in Darboux transformations by BosonFermion correspondence
Abstract
Bilinear equation is an important property for integrable nonlinear evolution equation. Many famous research objects in mathematical physics, such as GromovWitten 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 (KupershmidtKiso version) and the BKP hierarchy, by the method of the BosonFermion 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.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 May 2022
 DOI:
 10.1016/j.physd.2022.133198
 arXiv:
 arXiv:2101.02520
 Bibcode:
 2022PhyD..43333198Y
 Keywords:

 Bilinear equations;
 Darboux transformations;
 BosonFermion correspondence;
 Tau functions;
 Squared eigenfunction potential;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Mathematical Physics;
 35Q51;
 37K10;
 37K40
 EPrint:
 47 pages