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The modified BKP (mBKP) hierarchy is an important integrable hierarchy related with BKP hierarchy. Darboux transformation is a powerful method to obtain various solutions of integrable systems. But for mBKP hierarchy, it is usually very difficult to construct the corresponding Darboux transformations, since it has a more complicated constraint on the Lax operator, compared with the BKP hierarchy. In this paper, we use the fermionic approach to obtain the explicit Darboux transformation operators of the mBKP hierarchy starting from the changes in tau functions. Also we investigate the relations among these new Darboux transformation operators. It is usually very difficult to convert the changes in tau functions to the ones in the Lax operator, while here we provide one successful example. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some topics involving the modified D-type Kadomtsev-Petviashvili (DKP) hierarchy. Note that the modified DKP (mDKP) hierarchy is just the modified B-type Kadomtsev-Petviashvili (BKP) hierarchy in the bosonic forms. Based on this fact, we first proved the conjecture in You [Physica D 50, 429–462 (1991)], that is, the product of the two tau functions of the mDKP hierarchy is some tau function of the KP hierarchy. Then, we investigate the Darboux transformations of the DKP and mDKP hierarchy. Finally, the solutions of the constrained BKP hierarchy are given in the context of the fermionic representation of infinite Lie algebra d∞. [ABSTRACT FROM AUTHOR]

In this paper, the modified KP hierarchy in the Kupershmidt–Kiso version is extended to the super case by Kac–van de Leur construction, that is, using highest weight representations of the even part in the tensor product of the infinite-dimensional Lie superalgebra gl∞|∞ with Grassmann algebra G. First, the super modified KP (SmKP) hierarchy is constructed in terms of superfermionic bilinear equations. Then, the superbosonic form of the SmKP hierarchy is given by super boson–fermion correspondence. With the help of super Hirota bilinear operators, the corresponding super Hirota bilinear equations of the SmKP hierarchy are obtained. Next, the Darboux transformations of this new SmKP hierarchy are expressed in the form of free superfermions and various solutions are derived. Finally, the super bilinear equations in the form of super wave functions are also constructed from the superbosonic ones, which is hoped to be helpful to obtain the corresponding Lax structures. [ABSTRACT FROM AUTHOR]

In this paper, free bosons are used to study some integrable properties of Kadomtsev-Petviashvili (KP) hierarchy of C type (CKP hierarchy), from the aspects of tau functions. First, the modified CKP hierarchy is constructed by using free bosons, and the corresponding Lax structure is given. Then, the constrained CKP hierarchy is found to be related to the modified CKP hierarchy, and the corresponding solutions are derived by using free bosons. Next, by using the relations between the Darboux transformations and the squared eigenfunction symmetries, we express the Darboux transformations of the CKP hierarchy in terms of free bosons, by which one can better understand the essential properties of the CKP Darboux transformations. In particular, the additional symmetries of the CKP hierarchy can be viewed as the infinitesimal generator of the CKP Darboux transformations. Based on these results, we finally obtain the actions of the CKP additional symmetries on the CKP tau functions constructed by free bosons. [ABSTRACT FROM AUTHOR]

There are two ways to choose the generating functions of the gauge transformations T D (Φ) = (Φ − 1) x − 1 ∂ Φ − 1 and T I (Ψ) = Ψ − 1 ∂ − 1 Ψ x , when dicussing the gauge transformations for the constrained modified KP hierarchy. The first is to select the (adjoint) eigenfunctions, while the second is the (adjoint) wave functions. In this paper, we will mainly discuss the gauge transformations obtained by the second method. The corresponding successive applications are considered. Also, we investigate the results of the gauge transformation derived through the union of these two methods. [ABSTRACT FROM AUTHOR]

The Miura links between the KP and modified KP hierarchies are extended to the SUSY KP (SKP) and SUSY modified KP hierarchies (SmKP) of Manin Radul and Jacobian types. The corresponding Darboux transformations in the SUSY case are constructed by using Miura links. In this sense, one can better understand why one step Darboux transformation of the SKP hierarchy can not keep the odd flows. In addition, the Darboux transformations in the SmKP case can be obtained naturally by the Miura links. Then the squared eigenfunction symmetry of the SmKP hierarchy is obtained by using Miura link from the SKP hierarchy. At last, starting from the results in the SKP hierarchy, the constrained SmKP hierarchy of Manin Radul type can also be constructed by Miura link, that is, by adding appropriate squared eigenfunction symmetry in odd flow parts. The Darboux transformations of the constrained SKP and SmKP hierarchies are also discussed by using the Miura links. [ABSTRACT FROM AUTHOR]

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. • Darboux transformations for KP, mKP and BKP hierarchies are formulated in the Fermionic picture. • Successive applications of the Darboux transformations in the Fermionic picture are given. • New bilinear equations in the Darboux chains are obtained. [ABSTRACT FROM AUTHOR]