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Nonlinear evolution equations: B?cklund Transformations and B?cklund Charts (Conference Paper)

机译:非线性发展方程:B?cklund变换和B?cklund图表(会议论文)

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摘要

B?cklund Charts have been introduced to depict links, via B?cklund Transformations, which relate different Nonlinear Evolution Equations. Notably, the links established among different Nonlinear Evolution Equations can be extended to the whole Hierarchies of Nonlinear Evolution Equations. This approach proved to be very fruitful, as well known, when applied to scalar Nonlinear Evolution Equations since it induces very many interesting results. Some of them are here reviewed and further new research perspectives are shown. Specifically, when the non commutative analogue Nonlinear Evolution Equations are introduced new problems arise. Here, hierarchies of non-commutative Nonlinear Evolution Equations together with their links via B?cklund Transformations are considered. Specifically, a non-commutative analogues of Cole-Hopf and of Miura transformation are shown, respectively, to connect the operator versions of Burgers equation to heat equation and the operator version of KdV equation to the corresponding operator version of modified KdV equation. Again, the links are connecting not only the base member equations but all the corresponding equations in the hierarchy generated by the recursion operator it admits. Finally, it is pointed out how the method here presented allows to construct new non-commutative operator equation.
机译:引入了B?cklund图以通过B?cklund变换来描述链接,这些链接涉及不同的非线性发展方程。值得注意的是,在不同的非线性发展方程之间建立的联系可以扩展到非线性发展方程的整个层次。众所周知,这种方法在应用于标量非线性演化方程时非常有成果,因为它引起了很多有趣的结果。本文对其中一些进行了回顾,并展示了新的研究前景。具体地,当引入非可交换模拟非线性发展方程时,出现新问题。在这里,考虑了非交换非线性发展方程的层级及其通过B?cklund变换的联系。具体而言,分别示出了Cole-Hopf和Miura变换的非对易类似物,以将Burgers方程的算子版本连接到热方程,将KdV方程的算子版本连接到修改的KdV方程的对应算子版本。同样,链接不仅连接基本成员方程式,而且还连接它允许的递归运算符生成的层次结构中的所有相应方程式。最后,指出本文提出的方法如何构造新的非交换算子方程。

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