Theory of light-matter interaction in nematic liquid crystals and the second Painlevé equation

Abstract

Abstract. We study global minimizers of an energy functional arising as a thin sample limit in the theory of light-matter interaction in nematic liquid crystals. We show that depending on the parameters various defects are predicted by the model. In particular we show existence of a new type of topological defect which we call the shadow kink. Its local profile is described by the second Painlevé equation. As part of our analysis we find new solutions to this equation thus generalizing the well known result of Hastings and McLeod [23].

Total integrals of Ablowitz‐Segur solutions for the inhomogeneous Painlevé II equation

“…In this paper, we show that the explicit formula for the parametrix can be naturally established performing appropriate deformations and decompositions of the functionΦ (see the Appendix). This, in turn, will provide the total integral formulas (11) and (12). We remark that an alternative construction of the local parametrix for < 0 was also obtained in the recent paper Ref.…”

“…The integral formula (12) has an exponential from, which is different from the one from Theorem 1. Observe, however, that Theorem 2 says equivalently that the Cauchy principal value integral of the purely imaginary AS solution differs from the value arg (cos( ) + ) by an integer multiplicity of 2 .…”

“…Outline. In Section 2, we recall the RH problem for the inhomogeneous PII equation and analyze the related Flaschka-Newell Lax system to reduce the proof of formulas (11) and (12) to finding appropriate asymptotics of functions related with the solution Φ( , ). In Sections 3 and 4, we provide the representations of solutions for the RH problems obtained by steepest descent deformations of the graph of Φ( , ) and establish the asymptotic estimates of their component functions.…”

Singular asymptotics for solutions of the inhomogeneous Painlevé II equation

“…They satisfy the Painlevé property: the only movable singularities of a solution u are poles; see more details about the Painlevé equations and the historical developments in [21,24]. During the developments of the Painlevé equations, it has been realized that PII possesses a wild range of important applications in the modern theory of mathematics and physics, such as nonlinear wave motion [2,30,33], where PII arises as a similarity reduction of the KdV equation; liquid crystal [13,14,36], where PII plays a critical role in light-matter interaction experiments on nematic liquid crystal; random matrix theory [34,35], where PII appears in the celebrated Tracy-Widom distribution. It is worth mentioning that the Tracy-Widom distribution does not only describe the largest eigenvalue distribution in random matrix ensembles, but also appear in the distribution of the longest increasing subsequence of random permutations [3], totally asymmetric simple exclusion process [26].…”

Gradient theory of domain walls in thin, nematic liquid crystals films

“…For this purpose we will restrict our attention to the time independent solutions, the idea being that the system quickly relaxes to its stationary state. If one ignores the dependence on the transversal x 2 coordinate, the system exhibits two type of walls that separate domains that evanesce asymptotically [1,4]. One corresponds to the extension of Ising wall, standard kink, in this inhomogeneous system, which is a symmetric solution and centered in the region of the maximal illumination i.e.…”