In this paper, we derive the exact analytical solution for the periodic oscillations of a nonlinear mechanical oscillator that is capable of describing phase transition phenomenon with spatially inhomogeneous distribution of the order parameter. The exact analytical solution was derived in terms of the elliptic integral of the third kind and covers the cases where the physical variable influencing the order parameter is negative or positive. The conditions for obtaining periodic oscillations for the two cases were discussed, and results were simulated for small- and large-amplitude nonlinear oscillations. The results also cover periodic responses with bistable oscillations, which indicate the existence of bifurcation in phase transition. Furthermore, the error of the numerical solution and other published approximate analytical solutions were analyzed and discussed. The present study can be viewed as a benchmark solution for the mechanical oscillator for phase transition and can be used to verify the accuracy of existing and future approximate solutions.

中文翻译：

---

机械振荡器的精确解析解，用于涉及阶参数空间不均匀分布的相变

在本文中，我们推导出了非线性机械振荡器周期性振荡的精确解析解，该振荡器能够描述阶参数在空间上不均匀分布的相变现象。精确解析解是根据第三类椭圆积分推导出来的，涵盖了影响阶参数的物理变量为负或正的情况。讨论了两种情况下获得周期振荡的条件，并对小幅度和大幅度非线性振荡的结果进行了模拟。结果还涵盖了双稳态振荡的周期性响应，这表明相变中存在分叉。此外，对数值解和其他已发表的近似解析解的误差进行了分析和讨论。本研究可被视为相变机械振荡器的基准解决方案，可用于验证现有和未来近似解决方案的准确性。