AI 中文总结
本文从标准Burgers层级推导出二阶正Burgers方程，通过多种方法确立其可积性，推导变换并分类行波等解，证明Hirota微扰级数有限截断，为相关领域理论理解提供支撑。
AI 中文摘要
本文从标准Burgers层级中推导出二阶正Burgers方程，以探究其完全可积性与精确解析解。我们通过将递归算子系统应用于经典Burgers方程，构造出该高阶非线性演化方程。在此结构框架基础上，我们进一步推导出显式三阶方程，并利用完全Bell多项式对n阶层级进行推广。通过Cole-Hopf变换，该二阶非线性方程可严格映射为线性三阶色散方程。我们通过明确构造标量Lax对（零曲率表示）确立了其完全可积性，这使得我们能够直接推导相关的微分与代数Darboux变换。为系统分类显式定态、随时间变化的行波及自相似剖面，我们在线性域中应用了多种技术：分离变量法、Hirota微扰法、基于多项式完全判别系统方法（CDSPM）的行波约化，以及李相似约化。关键的是，我们的分析证明了Hirota微扰级数的精确有限截断。最后，我们概述了该方程对流体动力学、非线性输运现象及高阶波传播的理论理解产生的影响。
英文摘要
This paper derives the second-order positive Burgers' equation from the standard Burgers' hierarchy to explore its complete integrability and exact analytical solutions. We construct this higher-order nonlinear evolution equation by systematically applying the recursion operator to the classical Burgers' equation. Expanding on this structural framework, we then derive the explicit third-order equation and use Complete Bell Polynomials to generalize the $n$-th order hierarchy. Through the Cole-Hopf transformation, the second-order nonlinear equation rigorously maps to the linear third-order dispersion equation. We establish complete integrability by explicitly formulating the scalar Lax pair (zero-curvature representation), which allows us to directly derive the associated differential and algebraic Darboux transformations. To systematically classify explicit stationary, time-dependent traveling wave, and self-similar profiles, we apply several techniques to the linear domain: separation of variables, the Hirota perturbation method, traveling wave reduction via the Complete Discrimination System for Polynomial Method (CDSPM), and Lie similarity reduction. Crucially, our analysis demonstrates the exact finite truncation of the Hirota perturbation series. We conclude by outlining how this equation impacts the theoretical understanding of fluid dynamics, nonlinear transport phenomena, and higher-order wave propagation.
