ENG 
ESP Session S25 - New Methods and Emerging Applications in Dynamics, Networks, and Control
Monday, July 12, 17:25 ~ 17:45 UTC-3
Nematic liquid crystals: well posedness, optical solitons and control
Constanza Sánchez Fernández de la Vega
IMAS-CONICET y DM, Facultad de Ciencias Exactas y Naturales, UBA, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk we present results on well-posedness, decay, soliton solutions and control of the coupled nonlinear Schrödinger (NLS) equation \[\partial_{z}u= \frac{1}{2} \mathrm{i} \nabla^{2} u+ \mathrm{i} \gamma (\sin^2(\psi+\theta_0)-\sin^2(\theta_0)) u, \] \[ \nu \nabla^{2} \psi= \frac{1}{2}E_0^2\sin(2\theta_0)-\frac{1}{2}(E_0^2+|u|^{2})\sin(2(\psi+\theta_0)), \] where $u$ and $\psi$ depend on the ``optical axis'' coordinate $z \in \mathbb{R}$, and the ``transverse coordinates'' $(x, y) \in \mathbb{R}^2$. Also $\nabla^{2} = \partial_x^2 + \partial_y^2$ is the Laplacian in the transverse directions, $E_0$, $\nu$ and $\gamma $ are positive constants, and $\theta_0$ is a constant satisfying $\theta_0 \in (\pi/4, \pi/2)$. The model arises in the study of optical beam propagation in nematic liquid crystals, and in particular a set of experiments by Assanto and collaborators [3], [4], [5]. The complex field $u$ represents the electric field amplitude of a linearly polarized laser beam that propagates through a nematic liquid crystal along the optical axis $z$. The elliptic equation describes the effects of the beam electric field on the local orientation (director field) of the nematic liquid crystal and has an important regularizing effect, seen experimentally and understood theoretically in related models. The ``director field'' $\psi + \theta_0$ is a field of angles that describe the macroscopic orientation of the nematic liquid crystal molecules. The laser beam causes an additional deviation $\psi$ in the orientation of the liquid crystal molecules.
In [1] we show a ``saturation'' effect consistent with a bound $\theta_0 + \psi < \pi/2 $ on the total angle, implying that the molecular orientation can not be perpendicular to the optical axis. This seems to be a sharp bound on the saturation of the nonlinearity. In particular it is more precise than the bound obtained in [2] and follows from a more general model that has no small size assumptions for $\theta_0 - \pi/4$ and $\psi$.
Finally, we show some recent results concerning an optimal control problem where the external electric field varying in time is the control.
References:
[1] Borgna, Juan Pablo and Panayotaros, Panayotis and Rial, Diego and Sánchez de la Vega, Constanza. Optical solitons in nematic liquid crystals: Arbitrary deviation angle model. Physica D: Nonlinear Phenomena 408, 2020.
[2] Borgna, Juan Pablo and Panayotaros, Panayotis and Rial, Diego and Sánchez de la Vega, Constanza. Optical solitons in nematic liquid crystals: model with saturation effects. Nonlinearity 31 (4), 2018.
[3] Conti, C. and Peccianti, M. and Assanto, G. Route to nonlocality and observation of accessible solitons. Physical review letters 91 (7), 2003.
[4] Peccianti, M. and Assanto, G. Nematicons. Physics Reports 516 (4), 2012.
[5] Peccianti, M. and De Rossi, A. and Assanto, G. and De Luca, A. and Umeton, C. and Khoo, I. C. Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells. Applied Physics Letters 77 (1), 2000.
Joint work with Juan Pablo Borgna (CONICET - CEDEMA, Universidad Nacional de San Martin, Buenos Aires, Argentina), Panayotis Panayotaros (Departamento de Matemáticas y Mecánica, IIMAS, Universidad Nacional Autónoma de México, Cd. México, México) and Diego Rial (IMAS - CONICET and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, UBA, CABA, Argentina).