论文标题
部分可观测时空混沌系统的无模型预测
An insight into the $q$-difference two-dimensional Toda lattice equation, $q$-difference sine-Gordon equation and their integrability
论文作者
论文摘要
在我们以前的工作\ cite {lns}中，我们构建了准代表的解决方案，以实现非交通性$ q $ - 差异二维toda lattice（$ q $ -2DTL）方程（$ q $ -2DTL）方程，由Darboux转换，我们可以证明，我们可以证明，这可以证明，这可以证明BiLine $ -2 $ -2 $ -2DERTICE t hir hir hir hir hir hir hir hir hir hir hir hir hir hir hir hir hir hir。在交换环境中。实际上，不仅可以通过其darboux变换和二进制darboux变换来构建对孤子方程的解决方案，还可以解决其相应的b $ \ ddot {a} $ cklund变换的解决方案。更具体地说，通过迭代darboux变换和孤子方程的二进制darboux变换产生的本征函数除了对其b $ \ ddot {a} $ cklund变换的确定性解决方案外，其他任何东西。这揭示了Darboux转换与Hirota的双线性方法之间的深刻联系。在本文中，如果$ Q $ -2DTL等式，我们将阐述此观点。首先，我们得出了一个广义的双线性B $ \ ddot {a} $ cklund变换，因此是双线性$ q $ -2DTL方程的广义lax对。然后，我们成功地为$ Q $ -2DTL方程构建了二进制DARBOUX转换，基于该方程，以量子积分表示的Grammamian解决方案都是针对双线性$ Q $ -2DTL方程及其双线性b $ \ ddot {a} $ cklund变换的。最后，通过对$ q $ -2DTL方程的相应结果施加2个周期的减少，我们得出了$ q $ -Difference Sine-gordon方程，这是一种修改的$ q $ difference Sine-Gordon方程，并获得其相应的解决方案。
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, eigenfunctions produced by iterating Darboux transformations and binary Darboux transformations for soliton equations give nothing but determinant solutions to their B$\ddot{a}$cklund transformations, individually. This reveals the profound connections between Darboux transformations and Hirota's bilinear method. In this paper, we shall expound this viewpoint in the case of the $q$-2DTL equation. First, we derive a generalized bilinear B$\ddot{a}$cklund transformation and thus a generalized Lax pair for the bilinear $q$-2DTL equation. And then we successfully construct the binary Darboux transformation for the $q$-2DTL equation, based on which, Grammian solutions expressed in terms of quantum integrals are established for both the bilinear $q$-2DTL equation and its bilinear B$\ddot{a}$cklund transformation. In the end, by imposing the 2-periodic reductions on the corresponding results of the $q$-2DTL equation, we derive a $q$-difference sine-Gordon equation, a modified $q$-difference sine-Gordon equation and obtain their corresponding solutions.
