非等谱正则系统的Bäcklund-达布变换与Riemann-Hilbert问题

多参数同谱 Fokker-Planck 方程

Multi-parameter isospectral Fokker-Planck equations

本文通过多步和逆向 darboux-Crum 变换，构造了给定 Fokker-Planck 方程的多参数形变伴生同谱方程，恢复了被删除的本征能级，并简要讨论了向分数阶方程的推广及黑洞热势方法的应用。

分数阶达布变换

Fractional Darboux Transformations

本文利用Ricatti方程关于线性分式变换的协方差定义了一类共形等价的二阶微分方程。这促使引入分数达布变换，它也可以被认为是广义科尔-霍普夫变换。我们应用这些变换找到薛定谔方程的等谱势和线性化的一些新的类的非线性偏微分方程。

Essentially isospectral transformations and their applications

We define and study the properties of Darboux-type transformations between Sturm--Liouville problems with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary conditions). Using these transformations, we obtain various direct and inverse spectral results for these problems in a unified manner, such as asymptotics of eigenvalues and norming constants, oscillation of eigenfunctions, regularized trace formulas, and inverse uniqueness and existence theorems.

Exceptional Legendre Polynomials and Confluent Darboux Transformations

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of "exceptional" degrees. In this paper we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explic…

KdV-ghost系统的对称性、Lax对及幂零Darboux变换

Symmetries, Lax pair, and nilpotent Darboux transformation for the KdV-ghost system

本文研究源于BF型拓扑场论BRST形式化的KdV-ghost系统，分析其对称性、Lax对等，构造幂零Darboux变换，为ghost部分提供Darboux理论解释且区别于标准超对称KdV谱系。

一维量子力学中非线性超对称性的因式分解。I：约化的一般分类与三阶代数的分析

Factorization of nonlinear supersymmetry in one-dimensional Quantum Mechanics. I: general classification of reducibility and analysis of the third-order algebra

研究了一维量子力学中超对称变换分解为非奇异系数的初等达布变换链的可能性。给出了不可约（几乎）等谱变换及其相关的SUSY代数的分类。对三阶情形的超对称代数和等谱算子进行了详细的分析。

Lax单域算子的协变形式：从阿贝尔到非交换

Covariant forms of Lax one-field operators: from Abelian to non-commutative

在量子理论的背景下，因子分解理论，超对称性和达布变换等谱变形的链接被认为是。导出了一类谱问题中分解算子的无限链方程。链的闭包定义了系统的对称性。分析了Pauli型和Dirac型矩阵微分算子的例子。

'Riemann Equations' in Bidifferential Calculus

We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, …

向量/矩阵非线性 Schrödinger 方程的可积离散化及相关 Yang-Baxter 映射

Integrable discretization of the vector/matrix nonlinear Schrödinger equation and the associated Yang-Baxter map

本文将 NLS 层级的二元 Bäcklund-Darboux 变换重构为矩阵值离散谱问题，建立向量/矩阵 NLS 与 mKdV 的可积空间离散化，并由二维格一致性导出 NLS 型 Yang-Baxter 映射。