arXiv subjects

Chuanzhong Li

Publications and source records attributed to Chuanzhong Li.

At least 19 recordsLinked to original sources

arXiv2026

Regularizations for shock and rarefaction waves in the perturbed solitons of the KP equation

Using an asymptotic perturbation method, we study the initial value problem for the KP equation with initial data consisting of parts of exact line-soliton solutions. We consider a slow modulation of the soliton parameters, described by a dynamical system obtained via the perturbation method. {The dynamical system is given by a $2$-component quasi-linear system.} In particular, we show that a singular solution (\emph{shock wave}) {of the system} leads to the generation of a new soliton as a result of the resonant interaction of solitons. We also show that a regular solution corresponding to a rarefaction wave {of the system} can be described by a parabola (which we call a \emph{parabolic soliton}). We then perform numerical simulations of the initial value problem and show that they are in excellent agreement with the results obtained by the perturbation method.

nlin.PS↗

arXiv2025

BiHom-Lie brackets and the Toda equation

We introduce a BiHom-type skew-symmetric bracket on $\mathfrak{gl}(V)$ built from two commuting inner automorphisms $α=Ad_ψ$ and $β=Ad_ϕ$ with $ψ,ϕ\in \mathfrak{gl}(V)$ and integers $i,j$. We prove that $(\mathfrak{gl}(V),[\cdot,\cdot]^{(i,j)}_{(ψ,ϕ)},α,β)$ is a BiHom--Lie algebra, and we study the Lax equation obtained by replacing the commutator in the finite nonperiodic Toda lattice by this bracket. For the symmetric choice $ϕ=ψ$ with $(i,j)=(0,0)$, the deformed flow is equivariant under conjugation and becomes gauge-equivalent, via $\widetilde L=ψ^{-1}Lψ$, to a Toda-type Lax equation with a conjugated triangular projection. In particular, scalar deformations amount to a constant rescaling of time. On embedded $2\times2$ blocks, we derive explicit trigonometric and hyperbolic formulas that make symmetry constraints (e.g. tracelessness) transparent. In the asymmetric hyperbolic case, we exhibit a trace obstruction showing that the right-hand side is generically not a commutator, which amounts to symmetry breaking of the isospectral property. We further extend the construction to the weakly coupled Toda lattice with an indefinite metric and provide explicit $2\times2$ solutions via an inverse-scattering calculation, clarifying and correcting certain formulas in the literature. The classical Toda dynamics are recovered at special parameter values.

nlin.SI↗

arXiv2025

Exact $S$-duality Map for Rigid Surface Operators

Surface operators in four-dimensional gauge theories are two-dimensional defects, serving as natural generalizations of Wilson lines and 't Hooft line operators. They act as ideal probes for exploring the non-perturbative structure of the theory. Rigid surface operators are a specific class of surface operators characterized by the absence of continuous deformation parameters. It is expected that a closed $S$-duality map should exist among these rigid operators. While progress has been made on specific examples or subclasses by leveraging invariants and empirical conjectures, a complete picture remains elusive. A significant challenge arises when multiple rigid surface operators share identical invariants, making the determination of $S$-duality relations difficult. More critically, a mismatch exists in the number of rigid surface operators between dual theories when classified by invariants; this is referred to as the \textit{mismatch problem}. This discrepancy suggests the necessity of extending the scope of consideration beyond strictly rigid operators. In this paper, we propose a direct, natural, and precise $S$-duality map for rigid surface operators. Our map is realized by moving the longest row in the pair of partitions defining a surface operator from one factor to the other, with an additional box appended or deleted to balance the total number of boxes. This mapping naturally incorporates non-rigid surface operators, thereby resolving the mismatch problem. The proposed map is applicable to gauge groups of all ranks and clarifies several long-standing puzzles in the field.

hep-th↗

arXiv2025

Non-crossing permutations for the KP solitons under the Gel'fand-Dickey reductions and the vertex operators

We give a classification of the $regular$ soliton solutions of the KP hierarchy, referred to as the $KP solitons$, under the Gel'fand-Dickey $\ell$-reductions in terms of the permutation of the symmetric group. As an example, we show that the regular soliton solutions of the (good) Boussinesq equation as the 3-reduction can have $at ~most$ one resonant soliton in addition to two sets of solitons propagating in opposite directions. We also give a systematic construction of these soliton solutions for the $\ell$-reductions using the vertex operators. In particular, we show that the $non-crossing$ permutation gives the regularity condition for the soliton solutions.

nlin.SI↗

arXiv2025

Tau functions of the UC hierarchy as partition functions of matrix models

We present a family of matrix models such that their partition functions are tau functions of the universal character (UC) hierarchy. This develops one of the topics of our previous paper arXiv:2410.14823. We found new matrix models associated with the product of two spheres with embedded graphs via a gluing matrix. We also generalize these studies to multi-matrix models case, which corresponds to the multi-component UC hierarchy.

hep-th↗

arXiv2024

Orthogonal polynomials: from Heun equations to Painlevé equations

In this paper, we {\color{black}study four kinds of polynomials orthogonal with the singularly perturbed Gaussian weight $w_{\rm SPG}(x)$, the deformed Freud weight $w_{\rm DF}(x)$, the jumpy Gaussian weight $w_{\rm JG}(x)$, and the Jacobi-type weight $w_{\rm {\color{black}JC}}(x)$. The second order linear differential equations satisfied by these orthogonal polynomials and the associated Heun equations are presented. Utilizing the method of isomonodromic deformations from [J. Dereziński, A. Ishkhanyan, A. Latosiński, SIGMA 17 (2021), 056], we transform these Heun equations into Painlevé equations. It is interesting that the Painlevé equations obtained by the way in this work are same as the results satisfied by the related three term recurrence coefficients or the auxiliaries studied by other authors. In addition, we discuss the asymptotic behaviors of the Hankel determinant generated by the first weight, $w_{\rm SPG}(x)$, under a suitable double scalings for large $s$ and small $s$, where the Dyson's constant is recovered.}

math.CA↗

arXiv2024

Hopf link invariants and integrable hierarchies

The goal of this note is to study integrable properties of a generating function of the HOMFLY-PT invariants of the Hopf link colored with different representations. We demonstrate that such a generating function is a $τ$-function of the KP hierarchy. Furthermore, this Hopf generating function in the case of composite representations, which is a generating function of the 4-point functions in topological string (corresponding to the resolved conifold with branes on the four external legs), is a $τ$-function of the universal character(UC) hierarchy put on the topological locus. We also briefly discuss a simple matrix model associated with the UC hierarchy.

hep-th↗

arXiv2024

Inner Product in Highest-Weight Representation

In this paper, we study the inner product of states corresponding to weights of finite-dimensional highest-weight representations of classical groups. We prove that the action of the raising operators would reduce a state of hight-weight representation to a linear combination of states of highest-weight representation, with the level decreased by one. Then we propose an iterative algorithm for calculating the inner products of sates efficiently, revealing the intricate structure of the representation. As applications, we discuss the unitarity of the highest-weight representation and propose a conjecture. We determine the norm of a special class of states. And we completely determine the inner products of states of the minuscule representations. The algorithm proposed is applicable to the highest-weight representation of affine Lie algebra without modifications. These findings can be used to study the construction of solutions to Kapustin-Witten equations which are based on the fundamental solutions of Toda systems.

math-ph↗

arXiv2024

Invariants of rigid surface operators

Lusztig used the symbol invariant to describe the Springer correspondence for classical groups. Similarly, the fingerprint invariant can describe the Kazhdan-Lusztig map. Both invariants pertain to rigid semisimple operators labeled by pairs of partitions $(λ', λ'')$. It is conjectured that the symbol invariant is equivalent to the fingerprint invariant for rigid surface operators. In this study, we provide a proof of this conjecture. We classify the maps that preserve the fingerprint invariant and demonstrate that they also preserve the symbol invariant. Conversely, we classify the maps that preserve the symbol invariant and show that they also preserve the fingerprint invariant. The constructions of the symbol and fingerprint invariants in prior works are crucial to the proof. Additionally, we found that one condition in the definition of the fingerprint invariant is redundant for rigid surface operators. In the appendix, we present an alternative strategy to prove the equivalence of these invariants.

math.RT↗

arXiv2024

Generation of multi-component super integrable hierarchy associated with higher-dimensional Lie superalgebra spo(2N,2) and osp(2N,2)

A new type of high-dimensional Lie superalgebras is constructed, including Lie superalgebras spo(4,2) and osp(4,2). Based on it, two different coupled nonisospectral super AKNS hierarchies and their bi-Hamiltonian structures are obtained. Then, the Lie superalgebras spo(4,2) and osp(4,2) are expanded to infinite-dimensional Lie superalgebras spo(2N,2) and osp(2N,2). It follows that we present the generation of multi-component super integrable hierarchy, and deduce two different multi-component super integrable AKNS hierarchies and their Hamiltonian structures.

nlin.SI↗

arXiv2023

Generation of higher-dimensional isospectral-nonisospectral integrable hierarchies associated with a new class of higher-dimensional column-vector loop algebras

We construct a new class of higher-dimensional column-vector loop algebras. Based on it, a method for generating higher-dimensional isospectral-nonisospectral integrable hierarchies is proposed. As an application, we derive a generalized nonisospectral integrable Schrödinger hierarchy which can be reduced to the famous derivative nonlinear Schrödinger equation. By using the higher-dimensional column-vector loop algebras, we obtain an expanded isospectral-nonisospectral integrable Schrödinger hierarchy which can be reduced to many classical and new equations, such as the expanded nonisospectral derivative nonlinear Schrödinger system, the heat equation, the Fokker-Plank equation which has a wide range of applications in stochastic dynamic systems. Furthermore, we deduce a ZN nonisospectral integrable Schrödinger hierarchy, which means that the coupling results are extended to an arbitrary number of components. Additionally, the Hamiltonian structures of these hierarchies are discussed by using the quadratic form trace identity.

nlin.SI↗

arXiv2021

Rigid Surface Operator and Symbol Invariant of Partitions

The symbol is used to describe the Springer correspondence for the classical groups by Lusztig. We refine the explanation that the $S$-duality maps of the rigid surface operators are symbol preserving maps. And we find that the maps $X_S$ and $Y_S$ used in the construction of $S$-duality maps are essentially the same. We clear up cause of the mismatch problem of the total number of the rigid surface operators between the $B_n$ and $C_n$ theories. And we construct all the $B_n/C_n$ rigid surface operators which can not have a dual. A classification of the problematic surface operators is made.

math-ph↗

arXiv2021

Painlevé V for a Jacobi unitary ensemble with random singularities

In this paper, we focus on the relationship between the fifth Painlevé equation and a Jacobi weight perturbed with random singularities, \begin{equation*} w(z)=\left(1-z^2\right)^α{\rm e}^{-\frac{t}{z^2-k^2}},~~~z,k\in[-1,1],~α,t>0. \end{equation*} By using the ladder operator approach, we obtain that an auxiliary quantity $R_n(t)$, which is closely related to the recurrence coefficients of monic polynomials orthogonal with $w(z)$, satisfies a particular Painlevé V equation.

math-ph↗

arXiv2021

Darboux transformations and solutions of nonlocal Hirota and Maxwell-Bloch equations

In this paper, based on the Hirota and Maxwell-Bloch (H-MB) system and its application in the theory of the femtosecond pulse propagation through an erbium doped fiber, we define two kinds of nonlocal Hirota and Maxwell-Bloch (NH-MB) systems, namely, $PT$-symmetric NH-MB system and reverse space-time NH-MB system. Then we construct the Darboux transformations of these NH-MB systems. Meanwhile, we derive the explicit solutions by the Darboux transformations.

nlin.SI↗

arXiv2020

The smallest eigenvalue of large Hankel matrices generated by a singularly perturbed Laguerre weight

An asymptotic expression of the orthonormal polynomials $\mathcal{P}_{N}(z)$ as $N\rightarrow\infty$, associated with the singularly perturbed Laguerre weight $w_α(x;t)=x^α{\rm e}^{-x-\frac{t}{x}},~x\in[0,\infty),~α>-1,~t\geq0$ is derived. Based on this, we establish the asymptotic behavior of the smallest eigenvalue, $λ_{N}$, of the Hankel matrix generated by the weight $w_α(x;t)$.

math-ph↗

arXiv2020

Virasoro symmetries of Multi-Component Gelfand-Dickey systems

In this paper, we mainly study the additional symmetry and $τ$ functions of a multi-component Gelfand-Dickey hierarchy which includes many classical integrable systems, such as the multi-component KdV hierarchy and the multi-component Boussinesq hierarchy. With other kinds of reductions, we can derive a B type multi-component Gelfand-Dickey hierarchy and a C type multi-component Gelfand-Dickey hierarchy. In our research, the additional flows of the additional symmetries can not all survive. By calculating, we find that the generator of the additional symmetry of the C type multi-component Gelfand-Dickey hierarchy is different from that of the B type multi-component Gelfand-Dickey hierarchy, while the forms of their surviving additional flows are the same.

nlin.SI↗

arXiv2020

Extensions of the finite nonperiodic Toda lattices with indefinite metrics

In this paper, we firstly construct a weakly coupled Toda lattices with indefinite metrics which consist of $2N$ different coupled Hamiltonian systems. Afterwards, we consider the iso-spectral manifolds of extended tridiagonal Hessenberg matrix with indefinite metrics what is an extension of a strict tridiagonal matrix with indefinite metrics. For the initial value problem of the extended symmetric Toda hierarchy with indefinite metrics, we introduce the inverse scattering procedure in terms of eigenvalues by using the Kodama's method. In this article, according to the orthogonalization procedure of Szegö, the relationship between the $τ$-function and the given Lax matrix is also discussed. We can verify the results derived from the orthogonalization procedure with a simple example. After that, we construct a strongly coupled Toda lattices with indefinite metrics and derive its tau structures. At last, we generalize the weakly coupled Toda lattices with indefinite metrics to the $Z_{n}$-Toda lattices with indefinite metrics.

nlin.SI↗

arXiv2019

$π$-type Fermions and $π$-type KP hierarchy

In this paper, we firstly construct $π$-type Fermions. According to these, we define $π$-type Boson-Fermion correspondence which is a generalization of the classical Boson-Fermion correspondence. We can obtain $π$-type symmetric functions $S_λ^π$ from the $π$-type Boson-Fermion correspondence, analogously to the way we get the Schur functions $S_λ$ from the classical Boson-Fermion correspondence (which is the same thing as the Jacobi-Trudi formula). Then as a generalization of KP hierarchy, we construct the $π$-type KP hierarchy and obtain its tau functions.

nlin.SI↗