arXiv subjects

Wenchuang Guan

Publications and source records attributed to Wenchuang Guan.

6 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--Bäcklund 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.

nlin.SI↗

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.

nlin.SI↗

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 $L_1(n)^M=L_2(n)^N+\sum_{l\in\mathbb{Z}}\sum_{i=1}^{m}q_{i,n}Λ^lr_{i,n+1}Δ$ 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 $\mathcal{L}_1(n)^M=\mathcal{L}_2(n)^{N}+\sum_{l\in \mathbb{Z}}\sum_{i=1}^{m}\tilde{q}_{i,n}Λ^l\tilde{r}_{i,n}$ is discussed. Finally we give equivalent formulations of the Toda and mToda reductions in terms of tau functions.

nlin.SI↗

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.

nlin.SI↗

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 $τ_n(\mathbf{t})$ and Toda tau function $τ_n^{\rm Toda}(\mathbf{t},-\mathbf{t})$, that is, $τ_n^{\rm Toda}(\mathbf{t},-{\mathbf{t}})=τ_n(\mathbf{t})τ_{n-1}(\mathbf{t})$ and $τ_n^{\rm Toda}(\mathbf{t},-{\mathbf{t}})=τ_n^2(\mathbf{t})$. Further we find $\big(τ_n(\mathbf{t})τ_{n-1}(\mathbf{t}),τ_n^2(\mathbf{t})\big)$ satisfies bilinear equation of mToda hierarchy.

nlin.SI↗

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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