二维外空间常参数KdV血管的分类
Classification of KdV vessels with constant parameters and two dimensional outer space
摘要：
在这篇文章中，我们对一些完全可积的偏微分方程的解的产生容器进行了分类，并给出了一个\textit{unified}方法。这一分类包括Korteweg-de Vries（KdV）方程和演化非线性Schr”odingier（ENLS）方程等重要例子。事实上，采用基本的矩阵代数技术，它表明，正好有两个规范形式的这种船只，使每个规范形式推广KdV或ENLS方程。特别是，狄拉克正则系统，其演变最近被插入到容器理论，被证明是等价的ENLS方程的意义上的船只。这项工作是重要的第一步，分类完全可积偏微分方程，这是可解的理论的船只。我们注意到，作者最近发表在《数学物理杂志》上的一篇论文表明，KdV方程具有解析初势的初值问题至少有一个“窄化”的时间解。所提出的分类，继承了这一思想和一个类似的定理可以很容易地证明所提出的偏微分方程。最后，本文的研究结果为以下问题的研究奠定了基础：1。广义KdV、ENLS方程的层次（通过推广血管方程），2.新的完全可积的偏微分方程（通过改变外层空间的维度），3。解决了给定任意偏微分方程的可积性问题（未来的分类将创建一个可由血管方程求解的列表，最终可能包括许多现有的偏微分方程类）。
英文摘要：
In this article we classify vessels producing solutions of some completely integrable PDEs, presenting a \textit{unified} approach for them. The classification includes such important examples as Korteweg-de Vries (KdV) and evolutionary Non Linear Schr\" odingier (ENLS) equations. In fact, employing basic matrix algebra techniques it is shown that there are exactly two canonical forms of such vessels, so that each canonical form generalize either KdV or ENLS equations. Particularly, Dirac canonical systems, whose evolution was recently inserted into the vessel theory, are shown to be equivalent to the ENLS equation in the sense of vessels. This work is important as a first step to classification of completely integrable PDEs, which are solvable by the theory of vessels. We note that a recent paper of the author, published in Journal of Mathematical Physics, showed that initial value problem with analytic initial potential for the KdV equation has at least a "narrowing" in time solution. The presented classification, inherits this idea and a similar theorem can be easily proved for the presented PDEs. Finally, the the resuts of the work serve as a basis for the investigation of the following problems: 1. hierarchy of the generalized KdV, ENLS equations (by generalizing the vessel equations), 2. new completely integrable PDEs (by changing the dimension of the outer space), 3. addressing the question of integrability of a given arbitrary PDE (the future classification will create a list of solvable by vessels equations, which may eventually include many existing classes of PDEs).
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