论文标题
在ABS列表中搜索晶格方程的可集成的两组分版本
Search for integrable two-component versions of the lattice equations in the ABS-list
论文作者
论文摘要
在某些假设下，我们在ABS列表中搜索和分类四分之一方程的两个组件版本。自变量将称为$ y，z $，除了多线性和不可约性外，等式对还需要具有以下特定属性：（1）构成该对的两个方程式与$ y \ y \ leftrightArrow z $交易所相关。 （2）当$ z = y $时，两个方程式都会减少到ABS列表中的一个方程之一。 （3）在任何角方向上的进化是通过多线性方程对。构造这种两组分对的一种直接方法是使用ABS列表中的某些特定方程（以$ y $为单位），使用替换$ y \ y \ y \ y \ y \ leftrightArrow z $用于某些特定偏移，之后，该对的另一个方程式由属性获得（1）。这样，我们可以为每个启动方程式获得8对。我们的主要结果之一是，由于条件（3），实际上，对于H1，H3，Q1，Q3，这实际上是完整的。 （对于H2，我们还有一个案例，Q2，Q4我们没有检查。）至于CAC的集成性测试，对于每种选择底部方程式，我们原则上都可以具有$ 8^2 $可能的侧面方程。但是，我们发现只有用均匀数量的$ y \ leftrightArrow z $替换构建的方程式，并且对于每个此类方程式，都有两组“侧面”方程对产生（相同的）真正的bäcklund变换和lax对。
We search and classify two-component versions of the quad equations in the ABS list, under certain assumptions. The independent variables will be called $y,z$ and in addition to multilinearity and irreducibility the equation pair is required to have the following specific properties: (1) The two equations forming the pair are related by $y\leftrightarrow z$ exchange. (2) When $z=y$ both equations reduce to one of the equations in the ABS list. (3) Evolution in any corner direction is by a multilinear equation pair. One straightforward way to construct such two-component pairs is by taking some particular equation in the ABS list (in terms of $y$), using replacement $y \leftrightarrow z$ for some particular shifts, after which the other equation of the pair is obtained by property (1). This way we can get 8 pairs for each starting equation. One of our main results is that due to condition (3) this is in fact complete for H1, H3, Q1, Q3. (For H2 we have a further case, Q2, Q4 we did not check.) As for the CAC integrability test, for each choice of the bottom equations we could in principle have $8^2$ possible side-equations. However, we find that only equations constructed with an even number of $y \leftrightarrow z$ replacements are possible, and for each such equation there are two sets of "side" equation pairs that produce (the same) genuine Bäcklund transformation and Lax pair.
