Exploring Critical Points of Energy Landscapes: From Low-Dimensional Examples to Phase Field Crystal PDEs

Abstract

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 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 crystallization.