Jo Tae-Chang, Armbruster Dieter
Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Jul;68(1 Pt 2):016213. doi: 10.1103/PhysRevE.68.016213. Epub 2003 Jul 16.
The Mathieu partial differential equation (PDE) is analyzed as a prototypical model for pattern formation due to parametric resonance. After averaging and scaling, it is shown to be a perturbed nonlinear Schrödinger equation (NLS). Adiabatic perturbation theory for solitons is applied to determine which solitons of the NLS survive the perturbation due to damping and parametric forcing. Numerical simulations compare the perturbation results to the dynamics of the Mathieu PDE. Stable and weakly unstable soliton solutions are identified. They are shown to be closely related to oscillons found in parametrically driven sand experiments.
马修偏微分方程（PDE）被作为由参数共振导致的模式形成的一个典型模型进行分析。经过平均化和尺度变换后，它被证明是一个受扰非线性薛定谔方程（NLS）。应用孤子的绝热微扰理论来确定NLS中的哪些孤子能在因阻尼和参数强迫引起的微扰中幸存下来。数值模拟将微扰结果与马修PDE的动力学进行了比较。识别出了稳定和弱不稳定的孤子解。结果表明它们与在参数驱动的沙子实验中发现的振荡子密切相关。
