Equilibria of a discrete Landau–de Gennes theory for nematic liquid crystals

Abstract

We study equilibria of a discrete version of the Landau-de Gennes energy functional for nematic liquid crystals. We consider the regime of small intersite coupling and present necessary and sufficient conditions for the continuation of equilibria of the decoupled Landau-de Gennes system. We also identify a class of small coupling Landau-de Gennes equilibria that correspond to equilibria of a discretized Oseen-Frank energy. The theory is presented for the case of 2×2 Q-tensors and for the one-elastic-constant energy functionals in finite lattices and graphs. We also present some immediate consequences of the continuation theory to Landau-de Gennes and Oseen-Frank gradient dynamics in the small intersite coupling limit. The results rely on continuation and symmetry arguments and are also related to the construction of breather solutions of discrete nonlinear Schrödinger equations near the anticontinuous limit, a problem arising in photonics. 0123456789().: V,-vol