A noncommutative semi-discrete Toda equation and its quasideterminant solutions
C. X. Li, J. J. C. Nimmo
考虑半离散Toda方程的非对易形式。找到了一个拉克斯对及其达布变换和二元达布变换，并用它们构造了两族拟行列式解.
A noncommutative version of the semi-discrete Toda equation is considered. A Lax pair and its Darboux transformations and binary Darboux transformations are found and they are used to construct two families of quasideterminant solutions.
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On a direct approach to quasideterminant solutions of a noncommutative modified KP equation
本文讨论了修正的KP方程的非对易形式及其拟行列式解族。利用达布变换解释了这些解的来源，并直接验证了这些解的正确性.我们还直接验证了非交换mKP方程和非交换KP方程的三浦变换所产生的拟行列式的解决方案之间的显式连接。
On a direct approach to quasideterminant solutions of a noncommutative KP equation
本文讨论了KP方程的一个非对易解及其两类拟行列式解。这些解决方案的起源解释达布和二元达布变换的装置。此外，它表明，这些解决方案也可以直接验证。这种方法让人想起wronskian技术用于广田双线性形式的定期，交换KP方程，但在非交换的情况下，没有双线性变换。
In [1], a generalized type of Darboux transformations defined in terms of a twisted derivation was constructed in a unified form. Such twisted derivations include regular derivations, difference operators, superderivatives and $q$-difference operators as special cases. The formulae for the iteration of Darboux transformations are expressed in terms of quasideterminants. This approach not only enables one to recover the known Darboux transformations and quasideterminant solutions to the noncommutative KP equation, the non-Abelian two-dimensional Toda lattice equation, the non-Abelian Hirota-Miw…
We construct solutions of an infinite Toda system and an analogue of the Painlev'e II equation over noncommutative differential division rings in terms of quasideterminants of Hankel matrices.
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Quasideterminant solutions of a non-Abelian Toda lattice and kink solutions of a matrix sine-Gordon equation
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We discuss extension of soliton theories and integrable systems into noncommutative spaces. In the framework of noncommutative integrable hierarchy, we give infinite conserved quantities and exact soliton solutions for many noncommutative integrable equations, which are represented in terms of Strachan's products and quasi-determinants, respectively. We also present a relation to an noncommutative anti-self-dual Yang-Mills equation, and make comments on how "integrability" should be considered in noncommutative spaces.
Factorization of differential operators, quasideterminants, and nonabelian Toda field equations
我们集成非交换户田场方程的类型A，B，C，根系统的功能，在任何结合代数的值。解用拟行列式表示。在附录中，我们回顾了有关非交换孤子方程的一些结果。
On solutions to the non-Abelian Hirota-Miwa equation and its continuum limits
本文构造了一类非阿贝尔Hirota-Miwa方程的类文法拟行列式解。通过这个非阿贝尔Hirota-Miwa方程的连续极限和它的拟行列式解，我们构造了一个以非对易KP方程为结尾的非对易微分差分方程的级联.对于每一个这些系统的拟行列式的解决方案，以及构造。
In this paper, I construct the Darboux transformations for the non-commutative Toda solutions at $ n=1 $ with the help of linear systems whose compatibility condition yields zero curvature representation of associated systems of non-linear differential equations. I also derive the quasideterminant solutions of the non-commutative Painlevé II equation by taking the Toda solutions at $ n=1 $ as a seed solution in its Darboux transformations. Further by iteration, I generalize the Darboux transformations of the seed solutions to $ N$-th form. At the end I describe the zero curvature representatio…
The extended versions of the noncommutative KP and mKP equations and Miura transformation
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Matrix solutions of a noncommutative KP equation and a noncommutative mKP equation
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Nonabelian Integrable Systems, Quasideterminants, and Marchenko Lemma
我们发现显式（多孤子）解决方案的nonabelian可积系统，如周期Toda场方程，朗缪尔方程，薛定谔方程的功能与值在任何关联代数。作者在前一篇文章（q-alg/9701008）中用前两位作者引入的拟行列式表示了A、B、C型根系的非交换Toda场方程的解。为了找到多孤子解的周期户田方程和其他nonabelian系统，我们使用这些想法相结合的重要引理，这是由于马尔琴科。
The non-Abelian two-dimensional Toda lattice and matrix sine-Gordon equations with self-consistent sources
本文建立并求解了带自相容源的非阿贝尔二维Toda晶格，给出两族拟行列式解，并首次通过周期与拟周期约化构造带自相容源的矩阵sine-Gordon方程及其拟行列式精确解。
With a Cole-Hopf transformation to solutions of the noncommutative KP hierarchy in terms of Wronski matrices
在KP层次的情况下，其中因变量取值于（任意）结合代数A中，已知存在可以用求解线性热层次的Wronski矩阵的拟行列式来表示的解。我们得到这些解决方案，而无需准行列式的帮助下，在一个简单的方式通过矩阵KP族（在A上）的解决方案，并通过使用Cole-Hopf变换。对于这类精确解，我们与“弱非结合”代数对应。
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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.
Bäcklund Transformations for Noncommutative Anti-Self-Dual Yang-Mills Equations
本文对规范群为G=GL（2）的非对易反自对偶Yang-Mills方程提出了Backlund变换，并利用它从一个简单的种子解生成了一系列精确解.由这种方法产生的解决方案表示在quasideterminaters和属于非交换版本的Atiyah-Ward anomaly。在交换极限下，我们的结果与Corrigan，Fairlie，Yates和Goddard的结果一致.
Nonisospectral deformations of noncommutative Laurent biorthogonal polynomials and matrix discrete Painlevé-type equations
本文通过非交换洛朗双正交多项式的非等谱变形建立其与矩阵离散潘勒韦方程的联系，并导出矩阵离散潘勒韦型方程。
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Factorization methods for Noncommutative KP and Toda hierarchy
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Quantitative analysis and simulation of Noncommutative Langmuir Oscillator solutions
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Noncommutative nonisospectral Toda and Lotka-Volterra lattices, and matrix discrete Painlevé equations
本文提出非交换非等谱Toda与Lotka-Volterra格，通过非等谱变形导出矩阵离散Painlevé I方程，并给出拟行列式解以证明平稳约化的合理性。
Non-Abelian discrete Toda chains and related lattices
该研究推导了具有拟行列式解与多形式Lax对的二维离散Toda格非阿贝尔类似物，探讨了其与多类低维格点非阿贝尔模型的关联，并研究了离散Toda方程到Somos-N序列再到离散Painlevé方程体系的非交换类比。
We present exact soliton solutions of anti-self-dual Yang-Mills equations for G=GL(N) on noncommutative Euclidean spaces in four-dimension by using the Darboux transformations. Generated solutions are represented by quasideterminants of Wronski matrices in compact forms. We give special one-soliton solutions for G=GL(2) whose energy density can be real-valued. We find that the soliton solutions are the same as the commutative ones and can be interpreted as one-domain walls in four-dimension. Scattering processes of the multi-soliton solutions are also discussed.
We discuss trivial deformations of the canonical Poisson brackets associated with the Toda lattices, relativistic Toda lattices, Henon-Heiles, rational Calogero-Moser and Ruijsenaars-Schneider systems and apply one of these deformations to construct a new trivial family of noncommutative integrable systems.
We introduce and solve the non-commutative version of the Hermite-Padé type I approximation problem. Its solution, expressed by quasideterminants, leads in a natural way to a subclass of solutions of the non-commutative Hirota (discrete Kadomtsev--Petviashvili) system and of its linear problem. We also prove integrability of the constrained system, which in the simplest case is the non-commutative discrete-time Toda lattice equation known from the theory of non-commutative Padé approximants and matrix orthogonal polynomials.
Some aspects of noncommutative geometry and physics
介绍了非对易几何的一些选定方面。有限集和格提供了这方面的简单例子。作为一个应用，它是如何解释的非线性户田格和离散时间版本，它可以被理解为广义西格玛模型的基础上非交换几何。特别是，通过这种方式，人们可以简单地理解户田格的完全可积性。此外，还简要讨论了有限集和格上的广义度量结构。
A fully noncommutative analog of the Painlevé IV equation and a structure of its solutions
本研究提出第四Painlevé方程的完全非交换推广形式，证明其解可由带导子的含单位元结合除环上的无穷Toda系统构造，拓展了非交换可积系统的研究范畴。
Integrable discretisations of the noncommutative NLS equation
本文利用 Darboux 变换的兼容性条件，导出了非交换非线性 Schrödinger 方程的可积离散化，并构建了非交换 Adler--Yamilov 型系统和离散 Toda 方程及其 Bäcklund 变换。
Darboux transformations and related non-Abelian integrable differential-difference systems of the derivative nonlinear Schrödinger type
本研究针对7种具Lax表示的非阿贝尔导数非线性Schrödinger方程，构建相容的线性与二次Darboux矩阵，推广了非阿贝尔可积模型，并借助拟行列式得到高次Darboux矩阵的分解条件。
Noncommutative Ward's Conjecture and Integrable Systems
非对易Ward猜想是原Ward猜想的一个非对易版本，原Ward猜想认为几乎所有的可积方程都可以由反自对偶Yang-Mills方程通过约化得到。本文证明了（2+1）维和（1+1）维非对易可积方程实际上是具有有限规范群的非对易反自对偶Yang-Mills方程的约化，其中包括Calogero-Bogoyavlenskii-Schiff方程的非对易形式，Zakharov系统，Ward的手征和拓扑手征模型，（修改的）Korteweg-de Vries，非线性Schroedinger，Boussinesq，N波，（仿射）Toda，sine-Gordon，Liouville，Tzitzeica，（Ward的）调和映射方程，这将保证它们的扭量描述的存在性和N=2弦理论中相应的物理情形，并导致非对易可积系统和弦论的富有成效的应用。还讨论了它们的一些可积性。
Integrability and geometry of the Wynn recurrence
本文将Wynn递推视为离散Schwarzian KP方程的约化，揭示其射影几何意义，并建立非交换Padé理论、非交换Toda方程及其在正则语言和离散解析函数中的应用。
In our previous work \cite{LNS}, we constructed quasi-Casoratian solutions to the noncommutative $q$-difference two-dimensional Toda lattice ($q$-2DTL) equation by Darboux transformation, which we can prove produces the existing Casoratian solutions to the bilinear $q$-2DTL equation obtained by Hirota's bilinear method in commutative setting. It is actually true that one can not only construct solutions to soliton equations but also solutions to their corresponding B$\ddot{a}$cklund transformations by their Darboux transformations and binary Darboux transformations. To be more specific, eigenf…
Some properties of the correspondence between the non-commutative versions of the (generalized) sine-Gordon (NCGSG$_{1,2}$) and the massive Thirring (NCGMT$_{1,2}$) models are studied. Our method relies on the master Lagrangian approach to deal with dual theories. The master Lagrangians turn out to be the NC versions of the so-called affine Toda model coupled to matter fields (NCATM$_{1,2}$), in which the Toda field $g$ belongs to certain subgroups of $ GL(3)$, and the matter fields lie in the higher grading directions of an affine Lie algebra. Depending on the form of $g$ one arrives at two d…
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