The onset of layer undulations in smectic A liquid crystals due to a strong magnetic field

TL;DR

This paper studies how a strong magnetic field causes layer undulations in three-dimensional smectic A liquid crystals, identifying critical points and bifurcation behaviors, and relates the findings to micromagnetic models.

Contribution

It extends bifurcation analysis of layer undulations from 2D to 3D by handling eigenvalue multiplicity and symmetry, revealing multiple bifurcation branches and stability properties.

Findings

Identified a critical magnetic field for instability

Described multiple bifurcation branches including a stable one

Linked the 3D problem to micromagnetic models to analyze minimizers

Abstract

We investigate the effect of a strong magnetic field on a three dimensional smectic A liquid crystal. We identify a critical field above which the uniform layered state loses stability; this is associated to the onset of layer undulations. In a previous work, Garc\'ia-Cervera and Joo considered the two dimensional case and analyzed the transition to the undulated state via a simple bifurcation. In dimension n=3 the situation is more delicate because the first eigenvalue of the corresponding linearized problem is not simple. We overcome the difficulties inherent to this higher dimensional setting by identifying the irreducible representations for natural actions on the functional that take into account the invariances of the problem thus allowing for reducing the bifurcation analysis to a subspace with symmetries. We are able to describe at least two bifurcation branches, one of which is stable, highlighting the richer landscape of energy critical states in the three dimensional setting. Finally, we analyze a reduced two dimensional problem, assuming the magnetic field is very strong, and are able to relate this to a model in micromagnetics studied, from where we deduce the periodicity property of minimizers.