Exploring critical points of energy landscapes: From low-dimensional examples to phase field crystal PDEs
Exploring critical points of energy landscapes: From low-dimensional examples to phase field crystal PDEs
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DOI：
10.1016/j.cnsns.2020.105679
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发表时间：
2020-08
期刊：
影响因子：
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通讯作者：
P. Subramanian;I. Kevrekidis;P. Kevrekidis
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文献类型：
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作者：
P. Subramanian;I. Kevrekidis;P. Kevrekidis
在本工作中，我们探讨了几种寻根方法在一系列原型实例中的应用。我们考虑的方法包括：(a)所谓的连续时间Nesterov （CTN）流法；(b)称为平方算子法（SOM）的一种变体；(c)以上两种方法的共同作用，即所谓的通货紧缩方法。更“传统”的方法，如牛顿的方法（及其与通货紧缩的变体）也被引入。我们的玩具示例从一个简单的单自由度（DOF）系统开始，以提供陆地的位置。随后，我们转向2-DOF系统，该系统是由无限维软物质结晶相场晶体（PFC）模型的减少所驱动的。一旦2-DOF系统的景观被阐明，我们转向完整的PDE模型，并说明低维例子的见解如何导致PDE水平的新解决方案，这些解决方案与软物质结晶的完整框架相关且感兴趣。
In the present work we explore the application of a few root-finding methods to a series of prototypical examples. The methods we consider include: (a) the so-called continuous-time Nesterov (CTN) flow method; (b) a variant thereof referred to as the squared-operator method (SOM); and (c) the joint action of each of the above two methods with the so-called deflation method. More “traditional” methods such as Newton’s method (and its variant with deflation) are also brought to bear. Our toy examples start with a naive one degree-of-freedom (DOF) system to provide the lay of the land. Subsequently, we turn to a 2-DOF system that is motivated by the reduction of an infinite-dimensional, phase field crystal (PFC) model of soft matter crystallisation. Once the landscape of the 2-DOF system has been elucidated, we turn to the full PDE model and illustrate how the insights of the low-dimensional examples lead to novel solutions at the PDE level that are of relevance and interest to the full framework of soft matter crystallisation.