Bilinear equations in Darboux transformations by Boson-Fermion correspondence

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 $(l-l^{\prime })$ -th modified KP hierarchy. The corresponding examples of these new bilinear equations are given.

arXiv comment47 pages

References in corpus (18)

• General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions
• Free fermions and tau-functions
• Vertex operator solutions to the discrete KP-hierarchy
• On additional symmetries of the KP hierarchy and Sato's Bäcklund transformation
• On the BKP hierarchy: additional symmetries, Fay identity and Adler-Shiota-van Moerbeke formula
• Method of Squared Eigenfunction Potentials in Integrable Hierarchies of KP Type
• Polynomial tau-functions of BKP and DKP hierarchies
• Supersymmetric BKP systems and their symmetries
• The "Ghost" Symmetry of the BKP hierarchy
• Pfaffian and determinantal tau functions I
• Gromov--Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda Hierarchies
• Hirota equations for the extended bigraded Toda hierarchy and the total descendent potential of CP^1 orbifolds
• Solving bi-directional soliton equations in the KP hierarchy by gauge transformation
• Open intersection numbers, matrix models and MKP hierarchy
• Coupled Modified KP Hierarchy and Its Dispersionless Limit
• BKP hierarchy and Pfaffian point process
• Modified KP and Discrete KP
• Hirota Quadratic Equations for the Gromov--Witten Invariants of ${\mathbb{P}}_{n-2,2,2}^1$

Cited by in corpus (1)

• CKP hierarchy and free Bosons