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Energy minimizers in a periodic phase transition model of light-matter interaction in nematic liquid crystals

Panayotis Smyrnelis, Marcel G. Clerc, Manuel Diaz-Zuniga, Michał Kowalczyk

TL;DR

The paper analyzes global minimizers of a forced, non-autonomous, one-dimensional phase-transition model arising from a thin-sample reduction of nematic liquid crystal energy under a two-period forcing. By deriving and exploiting renormalized-energy bounds, it shows that the minimizers can display up to three zeros and that their structure is governed by two thresholds, with detailed regimes depending on the parameter $\alpha$ and the two-bump structure of $\mu$. The results connect to the physical nematic-light interaction model and reproduce the vortex configurations observed in donut-beam experiments, including shadow and standard kink patterns and their transitions as forcing strength varies. The analysis covers both nonperiodic and periodic $\mu$, providing explicit threshold expressions, zero-location convergences, and asymptotic kink profiles (shadow, standard, and giant kinks) in the small-$\varepsilon$ limit. Overall, the work consolidates the link between the 1D variational model and the qualitative vortex dynamics seen in the 2D physical setting, with precise energy-based characterizations of the minimizers across regimes.

Abstract

In this paper we complete the study of global minimizers of a forced, non autonomous, one dimensional, phase transition model, initiated in [8]. Motivated by the recent findings in [9], revealing new configurations of topological structures in light, we consider a forcing term having two periods. We show that depending on the strength of the forcing, at most two thresholds that determine the structure of the minimizers (kinks) are attained. These kinks are now a combination of the previous types encountered in [8], and they may have at most three zeros. The existence of these complex types of phase transition follows from a periodic one dimensional model of matter-light interaction in nematic liquid crystal based on a thin sample limit of the Oseen-Frank energy. We show that the qualitative behaviour of global minimizers is consistent with the original model.

Energy minimizers in a periodic phase transition model of light-matter interaction in nematic liquid crystals

TL;DR

The paper analyzes global minimizers of a forced, non-autonomous, one-dimensional phase-transition model arising from a thin-sample reduction of nematic liquid crystal energy under a two-period forcing. By deriving and exploiting renormalized-energy bounds, it shows that the minimizers can display up to three zeros and that their structure is governed by two thresholds, with detailed regimes depending on the parameter and the two-bump structure of . The results connect to the physical nematic-light interaction model and reproduce the vortex configurations observed in donut-beam experiments, including shadow and standard kink patterns and their transitions as forcing strength varies. The analysis covers both nonperiodic and periodic , providing explicit threshold expressions, zero-location convergences, and asymptotic kink profiles (shadow, standard, and giant kinks) in the small- limit. Overall, the work consolidates the link between the 1D variational model and the qualitative vortex dynamics seen in the 2D physical setting, with precise energy-based characterizations of the minimizers across regimes.

Abstract

In this paper we complete the study of global minimizers of a forced, non autonomous, one dimensional, phase transition model, initiated in [8]. Motivated by the recent findings in [9], revealing new configurations of topological structures in light, we consider a forcing term having two periods. We show that depending on the strength of the forcing, at most two thresholds that determine the structure of the minimizers (kinks) are attained. These kinks are now a combination of the previous types encountered in [8], and they may have at most three zeros. The existence of these complex types of phase transition follows from a periodic one dimensional model of matter-light interaction in nematic liquid crystal based on a thin sample limit of the Oseen-Frank energy. We show that the qualitative behaviour of global minimizers is consistent with the original model.
Paper Structure (8 sections, 26 theorems, 85 equations)

This paper contains 8 sections, 26 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Physical motivation
  3. The mathematical model
  4. Main results
  5. Basic results for minimizers when the function $\mu$ has two bumps
  6. Renormalized energy
  7. The structure of the kinks for the different regimes
  8. The case of a periodic function $\mu$

Key Result

Theorem 2.1

Assume hyp1 and hyp2. Then there exists $v \in H^1(\mathbb{R})$ such that $E(v)=\min_{H^1(\mathbb{R})} E$. In addition, if $\mu$ and $f$ are continuous, then $v\in C^2(\mathbb{R})$ is a classical solution of the O.D.E.

Theorems & Definitions (38)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • Proposition 2.8
  • proof : Proof of Proposition \ref{['p3']}
  • ...and 28 more