arXiv subjects

Guo-Fu Yu

Publications and source records attributed to Guo-Fu Yu.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic analysis of N-elliptic localized solutions for the Fokas--Lenells equation

This paper investigates the N-elliptic localized solutions of the Foka-Lenells equation. Based on the corresponding Lax pair, the Weierstrass elliptic functions are adopted to construct the elliptic function solutions and the fundamental solution matrix of the equation. The N-elliptic localized solutions are further derived via the N-fold Darboux-Backlund transformation. By virtue of the Cauchy determinant expressed with sigma functions, the asymptotic behaviors of the obtained solutions are systematically analyzed along and between their propagation directions, and the symmetry properties of these solutions are established.

nlin.SI↗

arXiv2025

Integrable variant Blaszak-Szum lattice equation

We derive a novel variant of the Blaszak-Szum lattice equation by introducing a new class of trigonometric-type bilinear operators. By employing Hirota's bilinear method, we obtain the Gram-type determinant solution of the variant Blaszak-Szum lattice equation. One-soliton and two-soliton solutions are constructed, with a detailed analysis of the asymptotic behaviors of the two-soliton solution. A B\"acklund transformation for the variant Blaszak-Szum lattice equation is established. By virtue of this B\"acklund transformation, multi-lump solutions of the equation are further constructed. Rational solutions are derived by introducing two differential operators applied to the determinant elements; in particular, lump solutions derived via these differential operators can be formulated in terms of Schur polynomials. Through parameter variation, three types of breather solutions are obtained, including the Akhmediev breather, Kuznetsov-Ma breather, and a general breather that propagates along arbitrary oblique trajectories. Finally, numerical three-periodic wave solutions to the variant Blaszak-Szum lattice equation are computed using the Gauss-Newton method.

nlin.SI↗

arXiv2025

Quasi-Pfaffians and applications

This paper presents a non-commutative generalization of the Pfaffian which we call a quasi-Pfaffian. This novel concept arises from solving linear systems with non-commutative skew-symmetric coefficients. A new non-commutative integrable system whose solutions are expressed in terms of these quasi-Pfaffians is presented. Derivative formulae and identities satisfied by these quasi-Pfaffians are presented.

math-ph↗

arXiv2025

A direct method in noncommutative integrable systems

We present a constructive framework for deriving noncommutative (NC) integrable equations directly from quasi-determinant solutions. Building upon the quasi-Wronskian structure, we extend the classical direct method to the NC setting, where standard determinant identities are replaced by algebraic relations intrinsic to quasi-determinants. By analyzing derivative identities satisfied by quasi-determinants, we recover the NC Kadomtsev-Petviashvili (ncKP) and Date-Jimbo-Kashiwara-Miwa (ncDJKM) equations by cancellations of nonlinear terms. Furthermore, by imposing flow constraints on the seed functions, we derive NC reductions such as the ncKdV and ncBoussinesq equations and obtain explicit matrix-valued soliton solutions. Our results highlight the quasi-determinant as a fundamental algebraic structure underpinning NC $\tau$-function structure.

nlin.SI↗

arXiv2025

Cauchy-Jacobi orthogonal polynomials and the discrete CKP equation

This paper intends to construct discrete spectral transformations for Cauchy-Jacobi orthogonal polynomials, and find its corresponding discrete integrable systems. It turns out that the normalization factor of Cauchy-Jacobi orthogonal polynomials acts as the $\tau$-function of the discrete CKP equation, which has applications in Yang-Baxter equation, integrable geometry, cluster algebra, and so on.

math-ph↗

arXiv2024

Integrable Semi-Discretization for a Modified Camassa-Holm Equation with Cubic Nonlinearity

In the present paper, an integrable semi-discretization of the modified Camassa-Holm (mCH) equation with cubic nonlinearity is presented. The key points of the construction are based on the discrete Kadomtsev-Petviashvili (KP) equation and appropriate definition of discrete reciprocal transformations. First, we demonstrate that these bilinear equations and their determinant solutions can be derived from the discrete KP equation through Miwa transformation and some reductions. Then, by scrutinizing the reduction process, we obtain a set of semi-discrete bilinear equations and their general soliton solutions in the Gram-type determinant form. Finally, we obtain an integrable semi-discrete analog of the mCH equation by introducing dependent variables and discrete reciprocal transformation. It is also shown that the semi-discrete mCH equation converges to the continuous one in the continuum limit.

nlin.SI↗

arXiv2023

Integrable discretizations for a generalized sine-Gordon equation and the reductions to the sine-Gordon equation and the short pulse equation

In this paper, we propose fully discrete analogues of a generalized sine-Gordon (gsG) equation $u_{t x}=\left(1+\nu \partial_x^2\right) \sin u$. The bilinear equations of the discrete KP hierarchy and the proper definition of discrete hodograph transformations are the keys to the construction. Then we derive semi-discrete analogues of the gsG equation from the fully discrete gsG equation by taking the temporal parameter $b\rightarrow0$. Especially, one full-discrete gsG equation is reduced to a semi-discrete gsG equation in the case of $\nu=-1$ (Feng {\it et al. Numer. Algorithms} 2023). Furthermore, $N$-soliton solutions to the semi- and fully discrete analogues of the gsG equation in the determinant form are constructed. Dynamics of one- and two-soliton solutions for the discrete gsG equations are discussed with plots. We also investigate the reductions to the sine-Gordon (sG) equation and the short pulse (SP) equation. By introducing an important parameter $c$, we demonstrate that the gsG equation reduces to the sG equation and the SP equation, and the discrete gsG equation reduces to the discrete sG equation and the discrete SP equation, respectively, in the appropriate scaling limit. The limiting forms of the $N$-soliton solutions to the gsG equation also correspond to those of the sG equation and the SP equation.

nlin.SI↗

arXiv2023

Multiple skew orthogonal polynomials and 2-component Pfaff lattice hierarchy

In this paper, we introduce multiple skew-orthogonal polynomials and investigate their connections with classical integrable systems. By using Pfaffian techniques, we show that multiple skew-orthogonal polynomials can be expressed by multi-component Pfaffian tau-functions upon appropriate deformations. Moreover, a two-component Pfaff lattice hierarchy, which is equivalent to the Pfaff-Toda hierarchy studied by Takasaki, is obtained by considering the recurrence relations and Cauchy transforms of multiple skew-orthogonal polynomials.

math-ph↗

arXiv2022

Matrix-valued Cauchy bi-orthogonal polynomials and a novel noncommutative integrable lattice

Matrix-valued Cauchy bi-orthogonal polynomials were proposed in this paper, together with its quasideterminant expression. It is shown that the coefficients in four-term recurrence relation for matrix-valued Cauchy bi-orthogonal polynomials should satisfy a novel noncommutative integrable system, whose Lax pair is given by fractional differential operators with non-abelian variables.

math-ph↗

arXiv2022

Discrete orthogonal ensemble on the exponential lattices

Inspired by Aomoto's $q$-Selberg integral, the orthogonal ensemble in the exponential lattice is considered in this paper. By introducing a skew symmetric kernel, the configuration space of this ensemble is constructed to be symmetric and thus, corresponding skew inner product, skew orthogonal polynomials as well as correlation functions are explicitly formulated. Examples including Al-Salam & Carlitz, $q$-Laguerre, little $q$-Jacobi and big $q$-Jacobi cases are considered.

math-ph↗

arXiv2021

An integrable semi-discretization of the modified Camassa-Holm equation with linear dispersion term

In the present paper, we are concerned with integrable discretization of a modified Camassa-Holm equation with linear dispersion term. The key of the construction is the semi-discrete analogue for a set of bilinear equations of the modified Camassa-Holm equation. Firstly, we show that these bilinear equations and their determinant solutions either in Gram-type or Casorati-type can be reduced from the discrete KP equation through Miwa transformation. Then, by scrutinizing the reduction process, we obtain a set of semi-discrete bilinear equations and their general soliton solution in Gram-type or Casorati-type determinant form. Finally, by defining dependent variables and discrete hodograph transformations, we are able to derive an integrable semi-discrete analogue of the modified Camassa-Holm equation. It is also shown that the semi-discrete modified Camassa-Holm equation converges to the continuous one in the continuum limit.

nlin.SI↗

arXiv2021

$q$-Pearson pair and moments in $q$-deformed ensembles

The generalisation of continuous orthogonal polynomial ensembles from random matrix theory to the $q$-lattice setting is considered. We take up the task of initiating a systematic study of the corresponding moments of the density from two complementary viewpoints. The first requires knowledge of the ensemble average with respect to a general Schur polynomial, from which the spectral moments follow as a corollary. In the case of little $q$-Laguerre weight, a particular ${}_3 \phi_2$ basic hypergeometric polynomial is used to express density moments. The second approach is to study the $q$-Laplace transform of the un-normalised measure. Using integrability properties associated with the $q$-Pearson equation for the $q$-classical weights, a fourth order $q$-difference equation is obtained, generalising a result of Ledoux in the continuous classical cases.

math-ph↗

arXiv2021

Evaluations of certain Catalan-Hankel Pfaffians via classical skew orthogonal polynomials

This paper is to evaluate certain Catalan-Hankel Pfaffians by the theory of skew orthogonal polynomials. Due to different kinds of hypergeometric orthogonal polynomials underlying the Askey scheme, we explicitly construct the classical skew orthogonal polynomials and then give different examples of Catalan-Hankel Pfaffians with continuous and $q$-moment sequences.

math.CA↗

arXiv2020

Christoffel transformations for (partial-)skew-orthogonal polynomials and applications

In this article, we consider the Christoffel transformations for skew-orthogonal polynomials and partial-skew-orthogonal polynomials. We demonstrate that the Christoffel transformations can act as spectral problems for discrete integrable hierarchies, and therefore we derive certain integrable hierarchies from these transformations. Some reductional cases are also considered.

math-ph↗

arXiv2020

Two-parameter generalisations of Cauchy bi-orthogonal polynomials and integrable lattices

In this article, we consider the generalised two-parameter Cauchy two-matrix model and corresponding integrable lattice equation. It is shown that with parameters chosen as $1/k_i$ when $k_i\in\mathbb{Z}_{>0}$ ($i=1,\,2$), the average characteristic polynomials admit $(k_1+k_2+2)$-term recurrence relations, which provide us spectral problems for integrable lattices. The tau function is then given by the partition function of the generalised Cauchy two-matrix model as well as Gram determinant. The simplest example with exact solvability is demonstrated.

math-ph↗

arXiv2019

Matrix integral solutions to the related Leznov lattice equations

Matrix integrals used in random matrix theory for the study of eigenvalues of matrix ensembles have been shown to provide $ \tau $-functions for several hierarchies of integrable equations. In this paper, we construct the matrix integral solutions to the Leznov lattice equation, semi-discrete and full-discrete version and the Pfaffianized Leznov lattice systems, respectively. We demonstrate that the partition function of Jacobi unitary ensemble is a solution to the semi-discrete Leznov lattice and the partition function of Jacobi orthogonal/symplectic ensemble gives solutions of the Pfaffianized Leznov lattice.

nlin.SI↗

arXiv2019

Rank shift conditions and reductions of 2d-Toda theory

This paper focuses on different reductions of 2-dimensional (2d-)Toda hierarchy. Symmetric and skew symmetric moment matrices are firstly considered, resulting in the differential relations between symmetric/skew symmetric tau functions and 2d-Toda's tau functions, respectively. Furthermore, motivated by the Cauchy two-matrix model and Bures ensemble from random matrix theory, we study the rank one shift condition in symmetric case and rank two shift condition in skew symmetric case, from which the C-Toda hierarchy and B-Toda hierarchy are found respectively, together with their special Lax matrices and integrable structures.

nlin.SI↗

arXiv2019

Invariant subspaces of biconfluent Heun operators and special solutions of Painlev\'e IV

We show that there is a full correspondence between the parameters space of the degenerate biconfluent Heun connection (BHC) and that of Painlev\'{e} IV that admits special solutions. The BHC degenerates when either the Stokes' data for the irregular singularity at $\infty$ degenerates or the regular singular point at the origin becomes an apparent singularity. We show that if the BHC is written as isomonodromy family of biconfluent Heun equations (BHE), then the BHE degenerates precisely when it admits eigen-solutions of the biconfluent Heun operators, after choosing appropriate accessory parameter, of specially constructed invariant subspaces of finite dimensional solution spaces spanned by parabolic cylinder functions. We have found all eigen-solutions over this parameter space apart from three exceptional cases after choosing the right accessory parameters. These eigen-solutions are expressed as certain finite sum of parabolic cylinder functions. We extend the above sum to new convergent series expansion in terms of parabolic cylinder functions to the BHE. The infinite sum solutions of the BHE terminates precisely when the parameters of the BHE assumes the same values as those of the degenerate biconfluent Heun connection except at three instances after choosing the right accessory parameter.

math.CA↗