Journal
SIAM JOURNAL ON APPLIED MATHEMATICS
Volume 82, Issue 5, Pages 1808-1828Publisher
SIAM PUBLICATIONS
DOI: 10.1137/21M1447404
Keywords
bifurcation; Landau-de Gennes model; nematic liquid crystals; solution landscape; saddle point
Categories
Funding
- National Key Research and Development Program of China [2021YFF1200500]
- National Natural Science Foundation of China [12225102, 12050002]
- Royal Society Newton International Fellowship [NIF\R1\201143]
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By studying the solution landscape and bifurcation diagrams of confined nematic liquid crystals, novel solution states are discovered, and the effects of geometrical anisotropy on confined defect patterns are revealed.
We study the solution landscape and bifurcation diagrams of nematic liquid crystals confined on a rectangle, using a reduced two-dimensional Landau-de Gennes framework in terms of two geometry-dependent variables: half short edge length lambda and aspect ratio b. First, we analytically prove that, for any b with a small enough lambda or for a large enough b with a fixed domain size, there is a unique stable solution that has two line defects on the opposite short edges. Second, we numerically construct solution landscapes by varying lambda and b, and report a novel X state, which emerges from saddle-node bifurcation and serves as the parent state in such a solution landscape. Various new classes are then found among these solution landscapes, including the X class, the S class, and the L class. By tracking the Morse indices of individual solutions, we present bifurcation diagrams for nematic equilibria, thus illustrating the emergence mechanism of critical points and several effects of geometrical anisotropy on confined defect patterns.
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