The Role Of Painleve II In Predicting New Liquid Crystal Self-Assembly Mechanisms

Abstract

We prove the existence of a new class of solutions, called shadow kinks, of the Painleve II equation ${{\frac {{\rm d}^{2} w}{{\rm d}z^{2}}}=2w^{3} +zw+\alpha,}$ where ${\alpha < 0}$ is a constant. Shadow kinks are sign changing solutions which satisfy ${ w(z) \sim -{\sqrt {-z/2}}\ {\rm as}\ z \to - \infty}$ and ${w(z) \sim -{\frac {\alpha}{z}} \ {\rm as}\ z \to \infty.}$ These solutions play a critical role in the prediction of a new class of topological defects, one dimensional shadow kinks and two dimensional shadow vortices, in light-matter interaction experiments on nematic liquid crystals. These new defects are physically important since it has recently been shown (Wang et al. in Nat Mater 15:106–112, 2016) that topological defects are a “template for molecular self-assembly” in liquid crystals. Connections with the modified KdV equation are also discussed.