In the present work, a new method for solving a strong nonlinear oscillator equation of the form $\ddot{x}$ + F(x) = 0, where F(−x) = −F(x), is carried out. This method consists of approximating function F(x) by means of a suitable Chebyshev polynomial: F(x) ≈ P(x) = px + qx³ + rx⁵, and then, the original oscillator is replaced by the cubic–quintic Duffing equation $\ddot{x}$ + px + qx³ + rx⁵ = 0 with arbitrary initial conditions, which admits the exact solution in terms of elliptic functions. The efficacy of the present method is demonstrated through the fluid multi-ion plasma equations and a generalized pendulum problem. For the generalized pendulum problem, the governing motion is directly reduced to the cubic–quintic Duffing oscillator with the help of the Chebyshev polynomial, and the approximate analytical and exact solutions are obtained. In addition, the comparison between our solutions and the Runge–Kutta numerical solution is examined. Moreover, the periodic time formula of the oscillations for both the approximate analytical solution and the exact solution is deduced, and the comparison between them is implemented. With respect to the plasma application, the fluid plasma equations of its particles are reduced to the Extended Korteweg–de Vries (EKdV) equation utilizing a reductive perturbation method. Then, we proved for the first time that any undamped polynomial oscillator of the nth degree can be reduced to a (2n − 1)th odd parity Duffing. Accordingly and after applying the previous theory to the EKdV equation, it was converted to the cubic–quintic Duffing equation. Finally, we can deduce that our new solutions and theory help us to understand and investigate many nonlinear phenomena in various branches of science.

中文翻译：

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解决强保守奇奇校验非线性振荡器的新方法：在等离子体物理学和刚性转子中的应用

在目前的工作中，一种新的求解形式为强非线性振荡器方程的方法 $\ddot{X}$+ F（x）= 0，其中F（-x）=- F（x）被执行。该方法包括近似函数的˚F（X借助于）合适的切比雪夫多项式：˚F（X）≈ P（X）= PX + QX ³ + RX ⁵，然后，原始振荡器是由立方五次杜芬取代方程$\ddot{X}$+ px + qx ³ + rx ⁵在任意初始条件下= 0，这允许椭圆函数的精确解。通过流体多离子等离子体方程和广义摆问题证明了本方法的有效性。对于广义摆问题，借助Chebyshev多项式，将控制运动直接简化为三次五次Duffing振荡器，并获得了近似的解析和精确解。此外，还检验了我们的解决方案与Runge–Kutta数值解决方案之间的比较。此外，推导了近似解析解和精确解的振荡周期时间公式，并进行了比较。关于等离子应用，使用还原摄动法将其粒子的流体等离子体方程简化为扩展的Korteweg-de Vries（EKdV）方程。然后，我们首次证明了任何无阻尼的多项式振荡器第n个度可以降低到第（2 n -1）个奇偶校验Duffing。因此，在将先前的理论应用于EKdV方程后，将其转换为立方五次Duffing方程。最后，我们可以推断出我们的新解决方案和理论有助于我们理解和研究科学各个领域的许多非线性现象。