arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1605.03472math-phmath.MPmath.RA
收藏

有理微分算子生成可积系统的一个充分条件

A sufficient condition for a Rational Differential Operator to generate an Integrable System

Sylvain Carpentier

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

对于一个有理微分算子$L=AB^{-1}$，Lenard-Magri可积方案是一个函数序列$F_n, n\geq 0$，使得（1）对所有$n \geq 0$都有$B(F_{n+1})=A(F_n)$，（2）函数$B(F_n)$两两交换。我们证明了，假设性质$(1)$成立，并且$B(F_n)$的微分阶的集合是无界的，性质$(2)$成立当且仅当$L$属于我们称之为可积的一类有理算子。此外，如果我们假设有理算子$L$是弱非局部的，并保持一定的分裂成偶数和奇数部分的函数代数，我们表明，人们总是可以找到这样一个序列$(F_n)$从任何功能Ker B。这一结果给出了一些洞察机制的递归算子，编码相应的可积方程的层次。

英文摘要：

For a rational differential operator $L=AB^{-1}$, the Lenard-Magri scheme of integrability is a sequence of functions $F_n, n\geq 0$, such that (1) $B(F_{n+1})=A(F_n)$ for all $n \geq 0$ and (2) the functions $B(F_n)$ pairwise commute. We show that, assuming that property $(1)$ holds and that the set of differential orders of $B(F_n)$ is unbounded, property $(2)$ holds if and only if $L$ belongs to a class of rational operators that we call integrable. If we assume moreover that the rational operator $L$ is weakly non-local and preserves a certain splitting of the algebra of functions into even and odd parts, we show that one can always find such a sequence $(F_n)$ starting from any function in Ker B. This result gives some insight in the mechanism of recursion operators, which encode the hierarchies of the corresponding integrable equations.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑