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Register a new userModulated structures in electroconvection in nematic liquid crystals
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Motivated by experiments in electroconvection in nematic liquid crystals with homeotropic alignment we study the coupled amplitude equations describing the formation of a stationary roll pattern in the presence of a weakly-damped mode that breaks isotropy. The equations can be generalized to describe the planarly aligned case if the orienting effect of the boundaries is small, which can be achieved by a destabilizing magnetic field. The slow mode represents the in-plane director at the center of the cell. The simplest uniform states are normal rolls which may undergo a pitchfork bifurcation to abnormal rolls with a misaligned in-plane director.We present a new class of defect-free solutions with spatial modulations perpendicular to the rolls. In a parameter range where the zig-zag instability is not relevant these solutions are stable attractors, as observed in experiments. We also present two-dimensionally modulated states with and without defects which result from the destabilization of the one-dimensionally modulated structures. Finally, for no (or very small) damping, and away from the rotationally symmetric case, we find static chevrons made up of a periodic arrangement of defect chains (or bands of defects) separating homogeneous regions of oblique rolls with very small amplitude. These states may provide a model for a class of poorly understood stationary structures observed in various highly-conducting materials (prechevrons or broad domains).
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We investigate a number of complex patterns driven by the electro-convection instability in a planarly aligned layer of a nematic liquid crystal. They are traced back to various secondary instabilities of the ideal roll patterns bifurcating at onset of convection, whereby the basic nemato-hydrodynamic equations are solved by common Galerkin expansion methods. Alternatively these equations are systematically approximated by a set of coupled amplitude equations. They describe slow modulations of the convection roll amplitudes, which are coupled to a flow field component with finite vorticity perpendicular to the layer and to a quasi-homogeneous in-plane rotation of the director. It is demonstrated that the Galerkin stability diagram of the convection rolls is well reproduced by the corresponding one based on the amplitude equations. The main purpose of the paper is, however, to demonstrate that their direct numerical simulations match surprisingly well new experiments, which serves as a convincing test of our theoretical approach.
We show experimentally that large matrixes of localized structures can be stored as elementary pixels in a nematic liquid crystal cell. Based on optical feedback with phase modulated input beam, our system allows to store, erase and actualize in parallel the localized structures in the matrix.
In uniaxial soft matter with a reorientational nonlinearity, such as nematic liquid crystals, a light beam in the extraordinary polarization walks off its wavevector due to birefringence, while it undergoes self-focusing via an increase in refractive index and eventually forms a spatial soliton. Hereby the trajectory evolution of solitons in nematic liquid crystals- nematicons- in the presence of a linearly varying transverse orientation of the optic axis is analysed. In this study we use and compare two approaches: i) a slowly varying (adiabatic) approximation based on momentum conservation of the soliton in a Hamiltonian sense; ii) the Frank-Oseen elastic theory coupled with a fully vectorial and nonlinear beam propagation method. The models provide comparable results in such a non-homogeneously oriented uniaxial medium and predict curved soliton paths with either monotonic or non-monotonic curvatures. The minimal power needed to excite a solitary wave via reorientation remains essentially the same in both uniform and modulated cases.
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.
The study of granular crystals, metamaterials that consist of closely packed arrays of particles that interact elastically, is a vibrant area of research that combines ideas from disciplines such as materials science, nonlinear dynamics, and condensed-matter physics. Granular crystals, a type of nonlinear metamaterial, exploit geometrical nonlinearities in their constitutive microstructure to produce properties (such as tunability and energy localization) that are not conventional to engineering materials and linear devices. In this topical review, we focus on recent experimental, computational, and theoretical results on nonlinear coherent structures in granular crystals. Such structures --- which include traveling solitary waves, dispersive shock waves, and discrete breathers --- have fascinating dynamics, including a diversity of both transient features and robust, long-lived patterns that emerge from broad classes of initial data. In our review, we primarily discuss phenomena in one-dimensional crystals, as most research has focused on such scenarios, but we also present some extensions to two-dimensional settings. Throughout the review, we highlight open problems and discuss a variety of potential engineering applications that arise from the rich dynamic response of granular crystals.
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