arXiv subjects

Jipeng Cheng

Publications and source records attributed to Jipeng Cheng.

At least 19 recordsLinked to original sources

arXiv2026

The transformations of the mToda hierarchy in tau functions

In this paper, we investigate the modified Toda (mToda) hierarchy, which can be regarded as the 2-component first modified Kadomtsev-Petviashvili (mKP) hierarchy. We first investigate the connection between the Toda and mToda tau functions. Based on this, we construct the transformations for the mToda tau functions and Lax operators. Furthermore, we present the mToda squared eigenfunction symmetries and derive the Adler-Shiota-van Moerbeke (ASvM) formula, which plays a crucial role by connecting the actions of the additional symmetries on the wave functions with the Sato--B\"{a}cklund transformations of the tau functions. Finally, by establishing the equivalence between the actions of vertex operators on the mToda tau functions and the multi-step mToda transformations, we derive the mToda addition formulas, also known as the generalized Fay identities.

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arXiv2026

Tau functions of the constrained matrix KP hierarchy

The constrained matrix KP hierarchy $(L^{k})_{<0}=\sum_{i=1}^{m}Q_{i}\partial^{-1}R_{i}^{\intercal}$ is investigated from the aspects of tau functions. Firstly, the matrix KP hierarchy is viewed as one special reduction of the multi-component KP hierarchy. Then bilinear equations of the constrained matrix KP hierarchy as the multi-component KP hierarchy are given in terms of tau functions. Finally based upon these results, the tau functions for the constrained matrix KP hierarchy are constructed by using the multi-component boson-fermion correspondence. Notice that the solutions of the constrained matrix KP hierarchy are derived without using quasi-determinants.

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arXiv2026

The generalized Wronskian solutions of the constrained mKP hierarchy

In this paper, we investigate the $(k, m)$-constrained 1st modified Kadomtsev-Petviashvili (mKP) hierarchy $(L^k)_{\leq 0}= \sum_{i=1}^m q_i \partial^{-1} r_i \partial$. Here, we obtain the corresponding solutions in the form of generalized Wronskians, which include the Wronskians and Grammians as special cases. Most importantly, these generalized Wronskian solutions are proved to satisfy the bilinear equations of the $(k, m)$-constrained mKP hierarchy, which is generally nontrivial. Our results here will be helpful in the derivation of the more general addition formulae and polynomial solutions for the 1st mKP hierarchy.

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arXiv2026

The tau functions of the constrained CKP hierarchy

The CKP hierarchy is one important sub-hierarchy of the KP hierarchy, which is quite special due to its tau function. Here we construct the tau functions for the constrained CKP hierarchy $(L^k)_{<0}=\sum_{i=1}^{m}\big(q_{1,i}\partial^{-1}q_{2,i}-(-1)^kq_{2,i}\partial^{-1}q_{1,i}\big)$ with $k$ being odd or even positive integer by using the CKP Darboux transformations.

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arXiv2025

Transformations of the 2-component BKP tau functions

The 2-component BKP (2-BKP) hierarchy is an important integrable system corresponding to the infinite dimensional Lie algebras $b_{\infty}$ and $d_{\infty}$, which contains Novikov-Veselov equation and can be used to describe the total descendent potential of D type singularity. Here we firstly introduce the projections of the mixed pseudo-differential operators to rewrite the 2-BKP Lax equation in the Shiota construction, where the scalar Lax operators involving two differential operators $\partial_1$ and $\partial_2$ are used. Based upon this, the $(M_1,M_2)$-reduction of the 2-BKP hierarchy is given. After that, we give the most important result of this paper, i.e., the transformations of the 2-BKP tau functions, which are in fact the 2-BKP Darboux transformations. Here we further give the corresponding changes in the 2-BKP Lax operators. Also the corresponding results are investigated for the reduction case. Finally, the additional symmetries can be viewed as the special cases of the transformations of the 2-BKP tau functions. Besides, we discuss the Pfaffian identities of the 2-BKP tau functions by successive applications of the above transformations, which are closely related with the 2-BKP addition formulae.

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arXiv2025

The two--component discrete KP hierarchy

The discrete KP hierarchy is also known as the $(l-l')$--th modified KP hierarchy. Here in this paper, we consider the corresponding two--component generalization, called the two--component discrete KP (2dKP) hierarchy. Firstly, starting from the bilinear equation of the 2dKP hierarchy, we derive the corresponding Lax equation by the Shiota method, this is using scalar Lax operators involving two difference operators $\Lambda_1$ and $\Lambda_2$. Then starting from the 2dKP Lax equation, we obtain the corresponding bilinear equation, including the existence of the tau function. From above discussions, we can determine which are essential in the 2dKP Lax formulation. Finally, we discuss the reduction of the 2dKP hierarchy corresponding to the loop algebra $\widehat{sl}_{M+N}=sl_{M+N}[\lambda,\lambda^{-1}]\oplus\mathbb{C}c \ (M,N\geq1)$.

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arXiv2025

One reduction of the modified Toda hierarchy

The modified Toda (mToda) hierarchy is a two-component generalization of the 1-st modified KP (mKP) hierarchy, which connects the Toda hierarchy via Miura links and has two tau functions. Based on the fact that the mToda and 1-st mKP hierarchies share the same fermionic form, we firstly construct the reduction of the mToda hierarchy $L_1(n)^M=L_2(n)^N+\sum_{l\in\mathbb{Z}}\sum_{i=1}^{m}q_{i,n}\Lambda^lr_{i,n+1}\Delta$ and $(L_1(n)^M+L_2(n)^N)(1)=0$, called the generalized bigraded modified Toda hierarchy, which can be viewed as a new two-component generalization of the constrained mKP hierarchy $\mathfrak{L}^k=(\mathfrak{L}^k)_{\geq 1}+\sum_{i=1}^m \mathfrak{q}_i\partial^{-1}\mathfrak{r}_i\partial$. Next the relation with the Toda reduction $\mathcal{L}_1(n)^M=\mathcal{L}_2(n)^{N}+\sum_{l\in \mathbb{Z}}\sum_{i=1}^{m}\tilde{q}_{i,n}\Lambda^l\tilde{r}_{i,n}$ is discussed. Finally we give equivalent formulations of the Toda and mToda reductions in terms of tau functions.

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arXiv2024

Toda Darboux transformations and vacuum expectation values

Determinant formulas for vacuum expectation values $\langle s+k+n-m,-s|e^{H(\mathbf{t})}\beta_m^{*}\cdots\beta_1^{*}\beta_n\cdots\beta_1g|k\rangle $ are given by using Toda Darboux transformations. Firstly notice that 2--Toda hierarchy can be viewed as the 2--component bosonizations of fermionic KP hierarchy, then two elementary Toda Darboux transformation operators $T_{+}(q)=\Lambda(q)\cdot\Delta\cdot q^{-1}$ and $T_{-}(r)=\Lambda^{-1}(r)^{-1}\cdot\Delta^{-1}\cdot r$ are constructed from the changes of Toda (adjoint) wave functions by using 2--component boson--fermion correspondence. Based on this, the above vacuum expectation values now can be realized as the successive applications of Toda Darboux transformations. So the corresponding determinant formulas can be derived from the determinant representations of Toda Darboux transformations. Finally by similar methods, we also give the determinant formulas for $\langle n-m|e^{\mathcal{H}(\mathbf{x})}\beta_m^{*}\cdots\beta_1^{*}\beta_n\cdots\beta_1g|k\rangle $ related with KP tau functions.

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arXiv2024

The Modified Toda Hierarchy

In this paper, modified Toda (mToda) equation is generalized to form an integrable hierarchy in the framework of Sato theory, which is therefore called mToda hierarchy. Inspired by the fact that Toda hierarchy is 2-component generalization of usual KP hierarchy, mToda hierarchy is constructed from bilinear equations of 2-component first modified KP hierarchy, where we provide the corresponding equivalence with Lax formulations. Then it is demonstrated that there are Miura links between Toda and mToda hierarchies, which means the definition of mToda hierarchy here is reasonable. Finally, Darboux transformations of the Toda and mToda hierarchies are also constructed by using the aforementioned Miura links.

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arXiv2024

Solutions of generalized constrained discrete KP hierarchy

Solutions of a generalized constrained discrete KP (gcdKP) hierarchy with constraint on Lax operator $L^k=(L^k)_{\geq m}+\sum_{i=1}^lq_i\Delta^{-1}\Lambda^mr_i$, are invesitigated by Darboux transformations $T_D(f)=f^{[1]}\cdot\Delta\cdot f^{-1}$ and $T_I(g)=(g^{[-1]})^{-1}\cdot\Delta^{-1}\cdot g$. Due to this special constraint on Lax operator, it is showed that the generating functions $f$ and $g$ of the corresponding Darboux transformations, can only be chosen from (adjoint) wave functions or $(L^k)_{<m}=\sum_{i=1}^lq_i\Delta^{-1}\Lambda^mr_i$. Then successive applications of Darboux transformations for gcdKP hierarchy are discussed. Finally based upon above, solutions of gcdKP hierarchy are obtained from $L^{\{0\}}=\Lambda$ by Darboux transformations.

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arXiv2024

Lax formulation of 3--component KP hierarchy by Shiota construction

It is quite basic in integrable systems to deriving Lax equations from bilinear equations. For multi--component KP theory, corresponding Lax structures are mainly constructed by matrix pseudo-differential operators for fixed discrete variables, or by matrix difference operators for even-component cases. Here we use Shiota method to construct Lax structure of 3-component KP hierarchy and its reduction by introducing two shift operators $\Lambda_1$ and $\Lambda_2$, where relations among different discrete variables can be easily found. We believe the results here are quite typical for general multi-component KP theory, which may be helpful for general cases.

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arXiv2024

Generalized Bigraded Toda Hierarchy

Bigraded Toda hierarchy $L_1^M(n)=L_2^N(n)$ is generalized to $L_1^M(n)=L_2^{N}(n)+\sum_{j\in \mathbb Z}\sum_{i=1}^{m}q^{(i)}_n\Lambda^jr^{(i)}_{n+1}$, which is the analogue of the famous constrained KP hierarchy $L^{k}= (L^{k})_{\geq0}+\sum_{i=1}^{m}q_{i}\partial^{-1}r_i$. It is known that different bosonizations of fermionic KP hierarchy will give rise to different kinds of integrable hierarchies. Starting from the fermionic form of constrained KP hierarchy, bilinear equation of this generalized bigraded Toda hierarchy (GBTH) are derived by using 2--component boson--fermion correspondence. Next based upon this, the Lax structure of GBTH is obtained. Conversely, we also derive bilinear equation of GBTH from the corresponding Lax structure.

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arXiv2024

Lax structure and tau function for large BKP hierarchy

In this paper, we mainly investigate Lax structure and tau function for the large BKP hierarchy, which is also known as Toda hierarchy of B type, or Hirota--Ohta--coupled KP hierarchy, or Pfaff lattice. Firstly, the large BKP hierarchy can be derived from fermionic BKP hierarchy by using a special bosonization, which is presented in the form of bilinear equation. Then from bilinear equation, the corresponding Lax equation is given, where in particular the relation of flow generator with Lax operator is obtained. Also starting from Lax equation, the corresponding bilinear equation and existence of tau function are discussed. After that, large BKP hierarchy is viewed as sub--hierarchy of modified Toda (mToda) hierarchy, also called two--component first modified KP hierarchy. Finally by using two basic Miura transformations from mToda to Toda, we understand two typical relations between large BKP tau function $\tau_n(\mathbf{t})$ and Toda tau function $\tau_n^{\rm Toda}(\mathbf{t},-\mathbf{t})$, that is, $\tau_n^{{\rm Toda}}(\mathbf{t},-{\mathbf{t}})=\tau_n(\mathbf{t})\tau_{n-1}(\mathbf{t})$ and $\tau_n^{{\rm Toda}}(\mathbf{t},-{\mathbf{t}})=\tau_n^2(\mathbf{t})$. Further we find $\big(\tau_n(\mathbf{t})\tau_{n-1}(\mathbf{t}),\tau_n^2(\mathbf{t})\big)$ satisfies bilinear equation of mToda hierarchy.

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arXiv2024

Bosonic construction of CKP tau function

The CKP tau function has been an important topic in mathematical physics. In this paper, the inverse of vacuum expectation value of exponential of certain bosonic fields, is showed to be the CKP tau function given by Chang and Wu, in the language of CKP Darboux transformation. In fact, computation of the above vacuum expectation value is usually quite difficult, since the square of bosonic fields is usually not zero. Here the corresponding vacuum expectation value is understood as successive application of CKP Darboux transformations, so that we can compute it by using the methods of integrable systems, where a useful formula is given. For applications, we construct solutions of KdV hierarchy by vacuum expectation value of bosonic fields, by the fact that KdV hierarchy is the 2-reduction of CKP hierarchy.

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arXiv2024

On the c-k constrained KP and BKP hierarchies: the Fermionic pictures, solutions and additional symmetries

In this paper, we study two generalized constrained integrable hierarchies, which are called the $c$-$k$ constrained KP and BKP hierarchies. The Fermionic picture of the $c$-$k$ constrained KP hierarchy is given. We give some solutions for the $c$-$k$ constrained KP hierarchy by using the free Fermion operators and define its additional symmetries. Its additional flows form a subalgebra of the Virasoro algebra. Furthermore, the additional flows acting on eigenfunctions $q_{i}(t)$ and adjoint eigenfunctions $r_{i}(t)$ of the $c$-$k$ constrained KP hierarchy are presented. Next, we define the $c$-$k$ constrained BKP hierarchy and obtain its bilinear identity and solutions. The algebra formed by the additional symmetric flow of the $c$-$k$ constrained BKP hierarchy that we defined is still a subalgebra of the Virasoro algebra and it is a subalgebra of the algebra formed by the additional flows of the $c$-$k$ constrained KP hierarchy.

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arXiv2023

Equivalence of two constructions for $\widehat{sl}_2$--integrable hierarchies

In this paper, we investigate the equivalence of Date--Jimbo--Kashiwara--Miwa (DJKM) construction and Kac--Wakimoto (KW) construction for $\widehat{sl}_2$--integrable hierarchies. DJKM method has gained great success in constructions of integrable hierarchies corresponding to classical ABCD affine Lie algebras, while the KW method is more applicable, which can be even used in exceptional EFG affine Lie algebras. But in KW construction, it is quite difficult to obtain Lax equations for the corresponding integrable hierarchies, while in DJKM construction, one can derive Lax structures for many integrable hierarchies. It is still an open problem for the derivation of Lax equations from bilinear equations in KW construction. Therefore if we can show the equivalent DJKM construction for the integrable hierarchies derived by the KW construction, then it will be helpful to get corresponding Lax structures. Here the equivalence of DJKM and KW methods is showed in the $\widehat{sl}_2$--integrable hierarchy for principal and homogeneous representations by using the language of the lattice vertex algebras.

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arXiv2021

Hirota Quadratic Equations for the Gromov--Witten Invariants of $\mathbb{P}_{n-2,2,2}^1$

Fano orbifold lines are classified by the Dynkin diagrams of type $A,D,$ and $E$. It is known that the corresponding total descendant potential is a tau-function of an appropriate Kac--Wakimoto hierarchy. It is also known that in the A-case the Kac--Wakimoto hierarchies admit an extension and that the total descendant potential is a tau-function of an extended Kac--Wakimoto hierarchy. The goal of this paper is to prove that in the D-case the total descendent potential is also a tau-function of an extended Kac--Wakimoto hierarchy.

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arXiv2021

CKP hierarchy and free Bosons

In this paper, free Bosons are used to study some integrable properties of the CKP hierarchy, from the aspects of tau functions. Firstly, 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 with 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 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 upon these results, we finally obtain the actions of the CKP additional symmetries on the CKP tau functions constructed by free Bosons.

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