论文标题
一种代数方法来分散时间可集成性
An algebraic approach to discrete time integrability
论文作者
论文摘要
我们提出了主要基于代数考虑的经典和量子二维时空晶格的系统构建，即，关于相关的R-Matrices以及基本的空间和时间上的古典和量子代数。这是一种新颖的结构，可导致完全离散的集成系统的推导，该系统由一致的集成非线性时空差异方程组成。为了说明所提出的方法，我们得出了两个完全离散的非线性Schrodinger类型系统的版本。第一个是基于有理R-matrix的存在，而第二个是完全离散的Ablowitz-Ladik模型，并且与三角r-matrix相关联。还针对第一个离散化方案（主要是一致性检查），并且在完全离散的热方程的解决方案方面，Darboux涂抹方法也用于第一个离散化方案。在这种情况下，完全离散系统的量化是很自然的，因此还检查了二维量子晶格。
We propose the systematic construction of classical and quantum two dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical and quantum algebras. This is a novel construction that leads to the derivation of fully discrete integrable systems governed by sets of consistent integrable non-linear space-time difference equations. To illustrate the proposed methodology, we derive two versions of the fully discrete non-linear Schrodinger type system. The first one is based on the existence of a rational r-matrix, whereas the second one is the fully discrete Ablowitz-Ladik model and is associated to a trigonometric r-matrix. The Darboux-dressing method is also applied for the first discretization scheme, mostly as a consistency check, and solitonic as well as general solutions, in terms of solutions of the fully discrete heat equation, are also derived. The quantization of the fully discrete systems is then quite natural in this context and the two dimensional quantum lattice is thus also examined.
