Matthews P C, Susanto H
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Dec;84(6 Pt 2):066207. doi: 10.1103/PhysRevE.84.066207. Epub 2011 Dec 19.
Localized structures appear in a wide variety of systems, arising from a pinning mechanism due to the presence of a small-scale pattern or an imposed grid. When there is a separation of length scales, the width of the pinning region is exponentially small and beyond the reach of standard asymptotic methods. We show how this behavior can be obtained using a variational method, for two systems. In the case of the quadratic-cubic Swift-Hohenberg equation, this gives results that are in agreement with recent work using exponential asymptotics. In addition, the method is applied to a discrete system with cubic-quintic nonlinearity, giving results that agree well with numerical simulations.
局域结构出现在各种各样的系统中，这是由于存在小尺度图案或外加网格而产生的钉扎机制所致。当存在长度尺度分离时，钉扎区域的宽度呈指数级减小，超出了标准渐近方法的适用范围。我们展示了如何使用变分方法来获得两个系统的这种行为。对于二次 - 三次Swift - Hohenberg方程的情况，这给出的结果与最近使用指数渐近法的工作一致。此外，该方法应用于具有三次 - 五次非线性的离散系统，得到的结果与数值模拟结果吻合良好。
