矩阵NLS系统的解及其离散化：统一处理
Solutions of matrix NLS systems and their discretisations: A unified treatment
摘要：
利用可积偏微分或差分方程的双微分分次代数方法，给出了连续、半离散（Ablowitz-Ladik）和全离散矩阵NLS系统的统一处理.这些方程起源于一个普遍的方程在这个框架内，通过指定一个表示的双微分分次代数和实施减少。通过应用的一般结果，相应的家庭的精确解得到，特别是包括矩阵孤子的解决方案在聚焦NLS的情况下。的解决方案是参数化的恒定矩阵数据的西尔维斯特方程（以前出现在可积系统文献中的秩条件）。这些数据表现出一定的冗余，我们在很大程度上减少了它。更确切地说，我们首先考虑更一般的AKNS型系统，从两个不同的矩阵NLS系统出现通过减少。在连续情况下，熟悉的厄米共轭约化导致连续矩阵（包括向量）NLS方程，但众所周知，这在离散情况下不起作用。另一方面，有一个复杂的共轭减少，这显然是以前没有研究过。它导致方阵NLS系统，但在所有三种情况下（连续，半和全离散）。这项工作的很大一部分是致力于探索相应的解决方案，特别是规则性和渐近行为的矩阵孤子解决方案。
英文摘要：
Using a bidifferential graded algebra approach to integrable partial differential or difference equations, a unified treatment of continuous, semi-discrete (Ablowitz-Ladik) and fully discrete matrix NLS systems is presented. These equations originate from a universal equation within this framework, by specifying a representation of the bidifferential graded algebra and imposing a reduction. By application of a general result, corresponding families of exact solutions are obtained that in particular comprise the matrix soliton solutions in the focusing NLS case. The solutions are parametrised in terms of constant matrix data subject to a Sylvester equation (which previously appeared as a rank condition in the integrable systems literature). These data exhibit a certain redundancy, which we diminish to a large extent. More precisely, we first consider more general AKNS-type systems from which two different matrix NLS systems emerge via reductions. In the continuous case, the familiar Hermitian conjugation reduction leads to a continuous matrix (including vector) NLS equation, but it is well-known that this does not work as well in the discrete cases. On the other hand there is a complex conjugation reduction, which apparently has not been studied previously. It leads to square matrix NLS systems, but works in all three cases (continuous, semi- and fully-discrete). A large part of this work is devoted to an exploration of the corresponding solutions, in particular regularity and asymptotic behaviour of matrix 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。
