Matrix solutions of a noncommutative KP equation and a noncommutative mKP equation
C. R. Gilson, J. J. C. Nimmo, C. M. Sooman
讨论了一类非交换KP方程和一类非交换mKP方程的矩阵解，并给出了它们的拟行列式表示。特别地，我们研究了两孤子解的相互作用性质。
Matrix solutions of a noncommutative KP and a noncommutative mKP equation which can be expressed as quasideterminants are discussed. In particular, we investigate interaction properties of two-soliton solutions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The extended versions of the noncommutative KP and mKP equations and Miura transformation
该研究统一构造了非交换KP与mKP方程的扩展版本，给出两类拟行列式解，明确其在交换情形下对应带自相容源的双线性方程的Wronskian与Grammian解，并建立了两类扩展非交换方程的Miura变换。
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 the structure of $(2+1)$--dimensional commutative and noncommutative integrable equations
我们发展了符号表示方法，从标量Lax算子导出了$(2+1)$维可积方程族，并研究了它们的整体性质。该方法适用于交换和非交换的情况下，在这个意义上，因变量采取其值在${\mathbb C}$或非交换结合代数。我们证明了这些层次确实是准局部的交换的情况下，由Mikhailov和Yamilov在1998年。我们提出了一个基于符号表示的环扩展。作为例子，我们给出了KP，mKP和Boussinesq方程的非交换版本。
In this paper we conjecture that the generating function of the intersection numbers on the moduli spaces of Riemann surfaces with boundary, constructed recently by R. Pandharipande, J. Solomon and R. Tessler and extended by A. Buryak, is a tau-function of the KP integrable hierarchy. Moreover, it is given by a simple modification of the Kontsevich matrix integral so that the generating functions of open and closed intersection numbers are described by the MKP integrable hierarchy. Virasoro constraints for the open intersection numbers naturally follow from the matrix integral representation.
Functional representations of integrable hierarchies
我们考虑一个一般的框架，可积阶层的Lax形式，并得出某些普遍的方程，从其中的“功能表示”的特定层次（如KP，离散KP，mKP，AKNS），即配方方面的功能方程，是系统的，很容易得到。这种形式主义真正适用于因变量位于非交换（通常是矩阵）代数中的层次结构。得到的功能表示可以理解为“非交换”类似物的“费伊身份”的KP层次。
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。
Dispersionless limit of the noncommutative potential KP hierarchy and solutions of the pseudodual chiral model in 2+1 dimensions
通常的分散性限制的KP层次不工作的情况下，因变量的值在一个非交换（如矩阵）代数。传递到潜在的KP层次，有一个相应的缩放限制在非交换的情况下，这原来是一个“pseudodual手征模型”在2+1维的层次（“pseudodual”的层次扩展沃德的（修改）可积手征模型）。将标度过程应用于从矩阵线性热族的解生成矩阵（势）KP族的精确解的方法，得到了相应的生成矩阵无色散势KP族的精确解的方法，即伪对偶手征模型族.我们利用这个结果构造了2+1维su（m）伪对偶手征模型的精确解，包括各种多块组态。
R-matrix approach to integrable systems on time scales
介绍了时标上可积类孤子系统的一般统一框架。将$R$ -矩阵形式应用于$δ$ -微分算子的代数中，利用它可以构造无穷族的交换向量场.该理论说明了两个无穷域可积族的时间尺度上，这是不同的对应物的KP和mKP。作为相关的有限场限制，构造了AKNS和Kaup-Broer孤子系统的差分对应。
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。
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.
I revisit the so called "bispectral problem" introduced in a joint paper with Hans Duistermaat a long time ago, allowing now for the differential operators to have matrix coefficients and for the eigenfunctions, and one of the eigenvalues, to be matrix valued too. In the last example we go beyond this and allow both eigenvalues to be matrix valued.
We consider Fredholm determinants of matrix convolution operators associated to matrix versions of the $n - $th Airy functions. Using the theory of integrable operators, we relate them to a fully noncommutative Painlevé II hierarchy, defined through a matrix valued version of the Lenard operators. In particular, the Riemann-Hilbert technique used to study these integrable operators allows to find a Lax pair for each member of the hierarchy. Finally, the coefficients of the Lax matrices are explicitely written in terms of these matrix valued Lenard operators and some solution of the hierarchy a…
It is known that the local Yang--Baxter equation is a generator of potential solutions to Zamolodchikov's tetrahedron equation. In this paper, we show under which additional conditions the solutions to the local Yang--Baxter equation are tetrahedron maps, namely solutions to the set-theoretical tetrahedron equation. This is exceptionally useful when one wants to prove that noncommutative maps satisfy the Zamolodchikov's tetrahedron equation. We construct new noncommutative maps and we prove that they possess the tetrahedron property. Moreover, by employing Darboux transformations with noncommu…
We extend the formalism of integrable operators a' la Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a semi-infinite interval and to matrix integral operators with a kernel of the form E_1^T(x) E_2(y)/(x+y) thus proving that their resolvent operators can be expressed in terms of solutions of some specific Riemann-Hilbert problems. We also describe some applications, mainly to a noncommutative version of Painleve' II (recently introduced by Retakh and Rubtsov), a related noncommutative equation of Painleve' type. We construct a particular family of solutions of the noncom…
In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on $R$-matrix description which provides Lax pairs in terms of quantum and classical $R$-matrices. First, we prove that for relativistic (and non-relativistic) tops such Lax pairs with spectral parameter follow from the associative Yang-Baxter equation and its degenerations. Then we proceed to matrix extensions of the models and find out that some additional constraints are required for their construction. We describe a matrix version of ${\mathbb Z}_2$ reduced elliptic t…
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。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。