| Abstract: |
| \begin{document}
\begin{abstract}
In this talk we present results
on well-posedness, decay and soliton solutions
of the coupled nonlinear
Schr\odinger (NLS) equation
\begin{align*}
& \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)),
\end{align*}
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 \cite{Peccianti2000,Conti2003,Peccianti2012}.
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.
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 \cite{Borgna2018} and follows from a more general model that has no small size assumptions
for $\theta_0 - \pi/4$ and $\psi$.
\end{abstract}
\begin{thebibliography}{99}
\bibitem{Borgna2018}
Borgna, Juan Pablo and Panayotaros, Panayotis and Rial, Diego and Sanchez de la Vega, Constanza. \textit{Optical solitons in nematic liquid crystals: model with saturation effects}.
Nonlinearity 31 (4), 2018.
\bibitem{Conti2003}
Conti, C. and Peccianti, M. and Assanto, G. \textit{Route to nonlocality and observation of accessible solitons}. Physical review letters 91 (7), 2003.
\bibitem{Peccianti2012} Peccianti, M. and Assanto, G. \textit{Nematicons}. Physics Reports 516 (4), 2012.
\bibitem{Peccianti2000}
Peccianti, M. and De Rossi, A. and Assanto, G. and De Luca, A. and Umeton, C. and Khoo, I. C. \textit{Electrically assisted self-confinement and waveguiding in planar
nematic liquid crystal cells}. Applied Physics Letters 77 (1), 2000.
\end{thebibliography}
\end{document} |
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