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

We prove the existence of a new class of solutions, called shadow kinks, of the Painleve II equation d 2w dz2 = 2w3 + zw + α, where α < 0 is a constant. Shadow kinks are sign changing solutions which satisfyw(z) ∼ −√−z/2 as z → −∞ and w(z) ∼ −z as z → ∞.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.