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.

在当前的工作中，我们探索了一些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）流动方法；（b）其变体，称为平方算子方法（SOM）；（c）以上两种方法中的每一种与所谓的放气方法的共同作用。还采用了更多的“传统”方法，例如牛顿法（及其带有通缩的变量）。我们的玩具示例从幼稚的一自由度（DOF）系统开始，以提供地面布局。随后，我们转向一个2-DOF系统，该系统受软物质结晶的无限维相场晶体（PFC）模型简化的推动。一旦阐明了2自由度系统的前景，