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Jipeng Cheng

Publications and source records attributed to Jipeng Cheng.

34 records · Page 2Linked to original sources
arXiv2021

Bilinear equations in Darboux transformations by Boson-Fermion correspondence

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.

nlin.SI↗
arXiv2019

The extended D-Toda hierarchy

In a companion paper to this one, we proved that the Gromov--Witten theory of a Fano orbifold line of type $D$ is governed by a system of Hirota Bilinear Equations. The goal of this paper is to prove that every solution to the Hirota Bilinear Equations determines a solution to a new integrable hierarchy of Lax equations. We suggest the name extended D-Toda hierarchy for this new system of Lax equations, because it should be viewed as the analogue of Carlet's extended bi-graded Toda hierarchy, which is known to govern the Gromov--Witten theory of Fano orbifold lines of type $A$

math.AG↗
arXiv2019

Quantum torus symmetries of multicomponent modified KP hierarchy and reductions

In this paper, we construct the multicomponent modified KP hierarchy and its additional symmetries. The additional symmetries constitute an interesting multi-folds quantum torus type Lie algebra. By a reduction, we also construct the constrained multicomponent modified KP hierarchy and its Virasoro type additional symmetries.

nlin.SI↗
arXiv2018

The 2-component BKP Grassmanian and simple singularities of type D

It was proved in 2010 that the principal Kac--Wakimoto hierarchy of type $D$ is a reduction of the 2-component BKP hierarchy. On the other hand, it is known that the total descendant potential of a singularity of type $D$ is a tau-function of the principal Kac--Wakimoto hierarchy. We find explicitly the point in the Grassmanian of the 2-component BKP hierarchy (in the sense of Shiota) that corresponds to the total descendant potential. We also prove that the space of tau-functions of Gaussian type is parametrized by the base of the miniversal unfolding of the simple singularity of type $D$.

math.AG↗
arXiv2015

Ghost symmetry of the discrete KP hierarchy

In this paper, with the help of the $S$ function and ghost symmetry for the discrete KP hierarchy which is a semi-discrete version of the KP hierarchy, the ghost flow on its eigenfunction(adjoint eigenfunction) and the spectral representation of its Baker-Akhiezer function and adjoint Baker-Akhiezer function are derived. From these observations above, some important distinctions between the discrete KP hierarchy and KP hierarchy are shown. Also we give the ghost flow on the tau function and another kind of proof of the ASvM formula of the discrete KP hierarchy.

nlin.SI↗
arXiv2014

The "ghost" symmetry in the CKP hierarchy

In this paper, we systematically study the "ghost" symmetry in the CKP hierarchy through its actions on the Lax operator, dressing operator, eigenfunctions and the tau function. In this process, the spectral representation of the eigenfunction is developed and the squared eigenfunction potential is investigated.

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arXiv2013

The applications of the gauge transformation for the BKP hierarchy

In this paper, we investigated four applications of the gauge transformation for the BKP hierarchy. Firstly, it is found that the orbit of the gauge transformation for the constrained BKP hierarchy defines a special $(2 +1)$-dimensional Toda lattice equation structure. Then the tau function of the BKP hierarchy generated by the gauge transformation is showed to be the Pfaffian. And the higher Fay-like identities for the BKP hierarchy is also obtained through the gauge transformation. At last, the compatibility between the additional symmetry and the gauge transformation of the BKP hierarchy is proved.

nlin.SI↗
arXiv2013

Squared Eigenfunction Symmetries for the BTL and CTL Hierarchies

In this paper, the squared eigenfunction symmetries for the BTL and CTL hierarchies are explicitly constructed with the suitable modification of the ones for the TL hierarchy, by considering the BTL and CTL constraints. Also the connections with the corresponding additional symmetries are investigated: the squared eigenfunction symmetry generated by the wave function can be viewed as the generating function for the additional symmetries.

nlin.SI↗
arXiv2013

The gauge transformation of the constrained semi-discrete KP hierarchy

In this paper, the gauge transformation of the constrained semi-discrete KP(cdKP) hierarchy is constructed explicitly by the suitable choice of the generating functions. Under the $m$-step successive gauge transformation $T_m$, we give the transformed (adjoint) eigenfunctions and the $τ$-function of the transformed Lax operator of the cdKP hierarchy.

nlin.SI↗
arXiv2012

On the squared eigenfunction symmetry of the Toda lattice hierarchy

The squared eigenfunction symmetry for the Toda lattice hierarchy is explicitly constructed in the form of the Kronecker product of the vector eigenfunction and the vector adjoint eigenfunction, which can be viewed as the generating function for the additional symmetries when the eigenfunction and the adjoint eigenfunction are the wave function and the adjoint wave function respectively. Then after the Fay-like identities and some important relations about the wave functions are investigated, the action of the squared eigenfunction related to the additional symmetry on the tau function is derived, which is equivalent to the Adler-Shiota-van Moerbeke (ASvM) formulas.

nlin.SI↗
arXiv2012

The Recursion operators of the BKP hierarchy and the CKP Hierarchy

In this paper, under the constraints of the BKP(CKP) hierarchy, a crucial observation is that the odd dynamical variable $u_{2k+1}$ can be explicitly expressed by the even dynamical variable $u_{2k}$ in the Lax operator $L$ through a new operator $B$. Using operator $B$, the essential differences between the BKP hierarchy and the CKP hierarchy are given by the flow equations and the recursion operators under the $(2n+1)$-reduction. The formal formulas of the recursion operators for the BKP and CKP hierarchy under $(2n+1)$-reduction are given. To illustrate this method, the two recursion operators are constructed explicitly for the 3-reduction of the BKP and CKP hierarchies. The $t_7$ flows of $u_2$ are generated from $t_1$ flows by the above recursion operators, which are consistent with the corresponding flows generated by the flow equations under 3-reduction.

math-ph↗
arXiv2011

The Additional Symmetries for the BTL and CTL Hierarchies

The Toda lattice (TL) hierarchy was first introduced by K.Ueno and K.Takasaki in \cite{uenotaksasai} to generalize the Toda lattice equations\cite{toda}. Along the work of E. Date, M. Jimbo, M. Kashiwara and T. Miwa \cite{DJKM} on the KP hierarchy, K.Ueno and K.Takasaki in \cite{uenotaksasai} develop the theory for the TL hierarchy: its algebraic structure, the linearization, the bilinear identity, $τ$ function and so on. Also the analogues of the B and C types for the TL hierarchy, i.e. the BTL and CTL hierarchies, are considered in \cite{uenotaksasai}, which are corresponding to infinite dimensional Lie algebras $\textmd{o}(\infty)$ and $\textmd{sp}(\infty)$ respectively. In this paper, we will focus on the study of the additional symmetries for the BTL and CTL hierarchies.

nlin.SI↗
arXiv2010

Recursion operators for KP, mKP and Harry-Dym Hierarchies

In this paper, we give a unified construction of the recursion operators from the Lax representation for three integrable hierarchies: Kadomtsev-Petviashvili (KP), modified Kadomtsev-Petviashvili (mKP) and Harry-Dym under $n$-reduction. This shows a new inherent relationship between them. To illustrate our construction, the recursion operator are calculated explicitly for $2$-reduction and $3$-reduction.

nlin.SI↗
arXiv2010

Additional symmetries of constrained CKP and BKP hierarchies

The additional symmetries of the constrained CKP (cCKP) and BKP (cBKP) hierarchies are given by their actions on the Lax operators, and their actions on the eigenfunction and adjoint eigenfunction $\{Φ_i,Ψ_i \}$ are presented explicitly. Furthermore, we show that acting on the space of the wave operator, $\partial_k^*$ forms new centerless $W^{cC}_{1+\infty}$ and $W^{cB}_{1+\infty}$-subalgebra of centerless $W_{1+\infty}$ respectively. In order to define above symmetry flows $\partial_k^*$ of the cCKP and cBKP hierarchies, two vital operators $Y_k$ are introduced to revise the additional symmetry flows of the CKP and BKP hierarchies.

nlin.SI↗
arXiv2010

The "Ghost" Symmetry of the BKP hierarchy

In this paper, we systematically develop the "ghost" symmetry of the BKP hierarchy through its actions on the Lax operator $L$, the eigenfunctions and the $τ$ function. In this process, the spectral representation of the eigenfunctions and a new potential are introduced by using squared eigenfunction potential(SEP) of the BKP hierarchy. Moreover, the bilinear identity of the constrained BKP hierarchy and Adler-Shiota-van-Moerbeke formula of the BKP hierarchy are re-derived compactly by means of the spectral representation and "ghost" symmetry.

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