通过精确的 proper 操作符指数对 Magnus 系列进行替代表示
Alternative representation of Magnus series by exact proper operator exponent
AI总结：
本文提出通过精确的 proper 操作符指数对非线性微分方程进行替代表示，从而得到解析解的泰勒展开。
摘要：
本文重点在于通过 proper 操作符（矩阵）指数解微分方程（系统）来替代 Magnus 系列。主要思想在于对非线性微分方程进行精确的线性表示。我们从 Dyson 的时间有序解出发，仅利用对 well-known 的 Baker-Campbell-Hausdorff（BCH）和 Zassenhaus 公式在 t 依赖操作符上的一般化，直接将它们转换为简单的 proper 操作符指数。该方法在操作符和形式解作为普通指数表达方面都是显式的，这使得计算以变量 t 的 Taylor 函数级数形式的解的解析表达式变得非常容易。如果引入一个可以互逆的变量 t 变换到原始方程中，然后找到这个新方程的普通指数形式的解，就可以得到所需的函数的完全不同的泰勒展开。该方法的本质在于重新求和级数。
英文摘要：
In this report the emphasis is on an alternative representation of the Magnus series by proper operator (matrix) exponential solutions to differential equations (systems), both linear and nonlinear ODEs and PDEs. The main idea here is in \emph{exact} \emph{linear} representations of the \emph{nonlinear} DEs. We proceeded from Dyson's time-ordered solutions, and using only generalizations of the well-known Baker-Campbell- Hausdorff (BCH) and Zassenhaus formulae for $t$-dependent operators directly converted them to simple proper operator exponents. The method being explicit both in terms of the operator and in terms of expressing the formal solution as an ordinary exponential, makes it quite easy to calculate analytical expressions to solutions in the form of a Taylor function series in one variable $t$. If introduce a mutually invertible change of variable $t$ into the original equations and then find a solution to this new equation in the form with ordinary exponential, one can obtain a completely different Taylor expansion of the desired function. The essence of this method comes down to resuming the series.
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。
