Exact analytical solution of a mechanical oscillator for phase transition involving spatially inhomogeneous distribution of the order parameter. (English) Zbl 1481.34004
Summary: 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.
MSC:
| 34A05 | Explicit solutions, first integrals of ordinary differential equations |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
| 33C75 | Elliptic integrals as hypergeometric functions |
Keywords:
coordinate-dependent mass; elliptic integral of the third kind; exact analytical solution; nonlinear oscillator; pattern formation; phase transitionReferences:
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