  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.
