An r-Matrix Approach to Nonstandard Classes of Integrable Equations
An r-Matrix Approach to Nonstandard Classes of Integrable Equations
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DOI：
10.2977/prims/1195166743
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发表时间：
1993-08
影响因子：
1.2
通讯作者：
B. Konopelchenko;W. Oevel
中科院分区：
文献类型：
--
作者：
B. Konopelchenko;W. Oevel
考虑了伪微分算子代数和相应的r-矩阵的三种不同的分解。在广义Lax方程和Sato方法的框架下讨论了3类相关的1 + 1和2 + 1维非线性可积方程. 2 + 1维方程族分别与Kadomtsev-Petviashvili（KP）方程、修正的KP方程和Dym方程相关联.一般方程族的约化导出了其它已知的2 + 1维可积方程以及各种1 + 1维可积方程。它示出，如何多哈密顿结构的1 + 1维方程可以得到从底层的r-矩阵。此外，密切的关系与三个不同的r-矩阵的方程之间的揭示。这三个类相关的达布定理起源于规范变换和倒数链接的拉克斯运营商。这些连接进行了讨论的一般水平上，导致统一的图片（倒数）Backlund和自动Backlund变换所涵盖的KP，修改后的KP和Dym层次的大类可积方程。§
Three different decompositions of the algebra of pseudo-differential operators and the corresponding r-matrices are considered. Three associated classes of nonlinear integrable equations in 1 +1 and 2 + 1 dimensions are discussed within the framework of generalized Lax equations and Sato's approach. The 2 +1-dimensional hierarchies are associated with the Kadomtsev-Petviashvili (KP) equation, the modified KP equation and a Dym equation, respectively. Reductions of the general hierarchies lead to other known integrable 2 + 1dimensional equations as well as to a variety of integrable equations in 1 +1 dimensions. It is shown, how the multi-Hamiltonian structure of the 1 + 1-dimensional equations can be obtained from the underlying r-matrices. Further, intimate relations between the equations associated with the three different r-matrices are revealed. The three classes are related by Darboux theorems originating from gauge transformations and reciprocal links of the Lax operators. These connections are discussed on a general level, leading to a unified picture on (reciprocal) Backlund and auto-Backlund transformations for large classes of integrable equations covered by the KP, the modified KP, and the Dym hierarchies. §