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Register a new userTheory of light-matter interaction in nematic liquid crystals and the second Painleve equation
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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 {it shadow kink}. Its local profile is described by the second Painleve equation. As part of our analysis we find new solutions to this equation thus generalizing the well known result of Hastings and McLeod.
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We consider the simplified Ericksen-Leslie model in three dimensional bounded Lipschitz domains. Applying a semilinear approach, we prove local and global well-posedness (assuming a smallness condition on the initial data) in critical spaces for initial data in $L^3_{sigma}$ for the fluid and $W^{1,3}$ for the director field. The analysis of such models, so far, has been restricted to domains with smooth boundaries.
In the first part of this paper, we will consider minimizing configurations of the Oseen-Frank energy functional $E(n, m)$ for a biaxial nematics $(n, m):Omegato mathbb S^2times mathbb S^2$ with $ncdot m=0$ in dimension three, and establish that it is smooth off a closed set of $1$-dimension Hausdorff measure zero. In the second part, we will consider a simplified Ericksen-Leslie system for biaxial nematics $(n, m)$ in a two dimensional domain and establish the existence of a unique global weak solution $(u, n, m)$ that is smooth off at most finitely many singular times for any initial and boundary data of finite energy. They extend to biaxial nematics of earlier results corresponding to minimizing uniaxial nematics by Hardt-Kindelerherer-Lin cite{HKL} and a simplified hydrodynamics of uniaxial liquid crystal by Lin-Lin-Wang cite{LLW10} respectively.
In this paper we study qualitative properties of global minimizers of the Ginzburg-Landau energy which describes light-matter interaction in the theory of nematic liquid crystals near the Friedrichs transition. This model is depends on two parameters: $epsilon>0$ which is small and represents the coherence scale of the system and $ageq 0$ which represents the intensity of the applied laser light. In particular we are interested in the phenomenon of symmetry breaking as $a$ and $epsilon$ vary. We show that when $a=0$ the global minimizer is radially symmetric and unique and that its symmetry is instantly broken as $a>0$ and then restored for sufficiently large values of $a$. Symmetry breaking is associated with the presence of a new type of topological defect which we named the shadow vortex. The symmetry breaking scenario is a rigorous confirmation of experimental and numerical results obtained in our earlier work.
In this paper, we study the Cauchy problem of the Poiseuille flow of full Ericksen-Leslie model for nematic liquid crystals. The model is a coupled system of a parabolic equation for the velocity and a quasilinear wave equation for the director. For a particular choice of several physical parameter values, we construct solutions with smooth initial data and finite energy that produce, in finite time, cusp singularities - blowups of gradients. The formation of cusp singularity is due to local interactions of wave-like characteristics of solutions, which is different from the mechanism of finite time singularity formations for the parabolic Ericksen-Leslie system. The finite time singularity formation for the physical model might raise some concerns for purposes of applications. This is, however, resolved satisfactorily; more precisely, we are able to establish the global existence of weak solutions that are Holder continuous and have bounded energy. One major contribution of this paper is our identification of the effect of the flux density of the velocity on the director and the reveal of a singularity cancellation - the flux density remains uniformly bounded while its two components approach infinity at formations of cusp singularities.
We study the solutions of the second Painleve equation in the space of initial conditions first constructed by Okamoto, in the limit as the independent variable, x, goes to infinity. Simultaneously, we study solutions of the related equation known as the thirty-fourth Painleve equation. By considering degenerate cases of the autonomous flow, we recover the known special solutions, which are either rational functions or expressible in terms of Airy functions. We show that the solutions that do not vanish at infinity possess an infinite number of poles. An essential element of our construction is the proof that the union of exceptional lines is a repellor for the dynamics in Okamotos space. Moreover, we show that the limit set of the solutions exists and is compact and connected.
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