可计算可积性。第5章：LPDO的因子分解
Computable Integrability. Chapter 5: Factorization of LPDOs
摘要：
可积性的不同定义通常使用初始方程的线性化和/或一些基本函数的展开，这些基本函数本身就是一些线性微分方程的解。这里重要的事实是，某些微分方程的线性化是它的简化，但尚未解决。例如，在线性薛定谔方程的情况下，我们不能明确地找到它的解，而只能将它们命名为约斯特函数，并利用它们的有用性质（见前几章）。另一方面，众所周知的事实是，对于常系数LODE算子本身总是可以分解成一阶因子，因此问题归结为求解几个一阶LODE可解的求积。在变系数微分算子的情况下，因子分解并不总是可能的，但对于大量的算子，BK-因子分解明确给出了因子分解条件，我们将在本章中证明。利用BK分解构造了任意阶LPDO的不变量集，并讨论了这些新不变量与二阶双曲LPDO的经典Laplace不变量之间的相互关系.
英文摘要：
Different definitions of integrability, as a rule, use linearization of initial equation and/or expansion on some basic functions which are themselves solutions of some linear differential equation. Important fact here is that linearization of some differential equation is its simplification but not solving yet. For instance, in case of linear Schroedinger equation, we are not able to find its solutions explicitly but only to name them Jost functions and to exploit their useful properties (see previous Chapters). On the other hand, well-known fact is that for LODE with constant coefficients operator itself can always be factorized into first-order factors and thus the problem is reduced to the solving of a few first-order LODEs which are solvable in quadratures. In case of differential operators with variable coefficients factorization is not always possible but for the great number of operators BK-factorization gives factorization conditions explicitly which we are going to demonstrate in this Chapter. BK factorization is used to construct set of invariants for LPDO of arbitrary order; interconnections of these new invariants with classic Laplace invariants for second order hyperbolic LPDOs are discussed.
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。
