非局部变换对非线性广义修正Emden型方程线性化和精确可解性的影响
Effect of nonlocal transformations on the linearizability and exact solvability of the nonlinear generalized modified Emden type equations
摘要：
非线性广义修正Emden型方程（Gestival）可以通过非局域变换线性化为简谐振子（HO）或阻尼谐振子（DHO）。据此，我们表明，非局部变换的结构和线性化到HO或DHO决定的性质/结构的动力学力量参与（因此，确定结构的动力学方程）。然而，逆向工程的策略是使用，使新兴的Gestival的精确解进行非局部变换，找到HO和DHO动力学方程的精确解。因此，虽然HO的精确解仍然是教科书上的精确解，但DHO的精确解（据我们所知，从未在其他地方报道过）显然是最明确和最通用的解，其为相关的欠阻尼、临界阻尼和过阻尼情况（即，没有用于坐标和/或速度的复杂设置是显著的/可行的）。此外，对于所有新兴的动力系统，我们报告说明性的数字，为每个解决方案以及相应的相空间轨迹，因为它们随着时间的推移。
英文摘要：
The nonlinear generalized modified Emden type equations (GMEE) are known to be linearizable into simple harmonic oscillator (HO) or damped harmonic oscillators (DHO) via some nonlocal transformations. Hereby, we show that the structure of the nonlocal transformation and the linearizability into HO or DHO determine the nature/structure of the dynamical forces involved (hence, determine the structure of the dynamical equation). Yet, a reverse engineering strategy is used so that the exact solutions of the emerging GMEE are nonlocally transformed to find the exact solutions of the HO and DHO dynamical equations. Consequently, whilst the exact solution for the HO remains a textbook one, the exact solution for the DHO (never reported elsewhere, to the best of our knowledge) turns out to be manifestly the most explicit and general solution that offers consistency and comprehensive coverage for the associated under-damping, critical-damping, and over-damping cases (i.e., no complex settings for the coordinates and/or the velocities are eminent/feasible). Moreover, for all emerging dynamical system, we report illustrative figures for each solution as well as the corresponding phase-space trajectories as they evolve in time.
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