# Helix-Free Ferroelectric Liquid Crystals: Electro Optics and Possible Applications
## 5 Theoretical Model of the Ferroelectric Liquid Crystal Deformation and Director Reorientation
### Ferroelectric Liquid Crystal Director Reorientation  
The bulk density of the free energy for the electrostatic interaction of the electric field and spontaneous polarization can be written as  
\[\frac{1}{2}P_{S}E\cos\varphi_{0}\bigg{(}\frac{l-l_{0}}{l_{0}}\bigg{)}^{2}, \tag{7}\]  
where \(\varphi_{0}\) is the initial azimuth angle of the director orientation. Now we minimize the expression in Equation (3) and record the equation of a balance of moments, describing the change \(\psi\) in the electric field \(E\):  
\[-\gamma_{\psi}\frac{\partial\psi}{\partial t}=K\frac{\partial^{2}\psi}{ \partial y^{2}}+\frac{(P_{S}E\cos\varphi_{0}+M)}{2}\psi(\Theta_{0}^{2}-\psi^{ 2}) \tag{8}\]where \(K\partial^{2}\psi/\partial y^{2}\) and \(\gamma_{\psi}\partial\psi/\partial t\) are the elastic and viscous moments defining the director reorientation by the angle \(\psi\); \(K\) and \(\gamma_{\psi}\) are elasticity and viscosity coefficients; \(t\) is the time; and the electric field switches on at \(t=0\). If we introduce variables  
\[\alpha=\frac{d^{2}(P_{S}E\cos\varphi_{0}+M)}{2K},y^{\prime}=\frac{y}{d},t^{ \prime}=\frac{t}{t_{0}},\text{ and }t_{0}=\frac{\gamma_{\psi}d^{2}}{K}, \tag{9}\]  
then Equation (8) will be transformed to  
\[\frac{\partial\psi}{\partial t^{\prime}}+\frac{\partial^{2}\psi}{\partial y^{ \prime 2}}+\alpha\Theta_{0}{}^{2}\psi-\alpha\psi^{3}=0. \tag{10}\]  
Since \(\psi\) in a general case can depend not only on the coordinates but also on the time, we have \(\psi=\psi\) (\(y^{\prime}\),\(t^{\prime}\)) = \(\psi_{0}\) exp (\(-i\xi t^{\prime}\)), where \(\xi>0\) is a constant, and \(\psi_{0}\) is the amplitude, which is a slow function of time [24]. Thus, Equation (8) is converted to the following:  
\[\frac{\partial\psi_{0}}{\partial t^{\prime}}+\frac{\partial^{2}\psi_{0}}{ \partial y^{\prime 2}}+\psi_{0}\Big{(}\alpha\Theta_{0}{}^{2}+i\xi\Big{)}- \alpha\psi_{0}{}^{3}=0. \tag{11}\]  
After substitution of \(\psi_{0}=\partial\)exp(\(-i\eta\)), where \(\Phi=\Phi(y^{\prime})\) and \(\eta=\eta(t^{\prime})\), Equation (11) is transformed to a system of two equations:  
\[\left\{ \begin{array}{l}\partial^{2}\Phi/\partial y^{\prime 2}+\Phi\alpha \Theta_{0}{}^{2}-\alpha\,\Phi^{3}=0,\\ \Phi(\partial\eta/\partial t^{\prime})-\Phi^{2}_{5}=0.\end{array} \right. \tag{12}\]  
Using the hyperbolic function \(sn\), one can describe \(\Phi\) as follows:  
\[\Phi=\frac{\sqrt{2}\Theta_{0}k}{\sqrt{1+k^{2}}}sn\bigg{[}\frac{\Theta_{0} \sqrt{\alpha}(y^{\prime}+C_{1})}{\sqrt{1+k^{2}}},k\bigg{]},\text{ and }\eta=\xi t^{\prime}+C_{2}. \tag{13}\]  
Now the function \(\psi=\psi_{0}\exp(-i\xi t^{\prime})\) using \(sn\) can be written as follows:  
\[\psi=\frac{\sqrt{2}\Theta_{0}k}{\sqrt{1+k^{2}}}sn\bigg{[}\frac{\Theta_{0} \sqrt{\alpha}(y^{\prime}+C_{1})}{\sqrt{1+k^{2}}},k\bigg{]}\exp\bigl{(}-2i\xi t^ {\prime}-iC_{2}\bigr{)}. \tag{14}\]  
For the extremely nonlinear situation at \(k\)\(\rightarrow\)1 when the wave period tends to infinity, this relation (using the hyperbolic function \(th\)) results in the spatially localized waves of a stationary profile--solitons. \[\psi=\Theta_{0}\,th\bigg{(}\frac{\Theta_{0}\sqrt{\alpha}\,(y^{\prime}+S_{1})} {2}\bigg{)}\,\exp(-2i\xi t^{\prime}-iC_{2}) \tag{15}\]  
From the boundary and initial conditions, we take \(C_{1}=C_{2}=0\) and \(\eta=\text{const}\). Then,  
\[\psi=\Theta_{0}\,th\bigg{(}\frac{\Theta_{0}\sqrt{\alpha}y^{\prime}}{2}\bigg{)} \,\exp(-2i\xi t^{\prime}). \tag{16}\]  
This equation defines the width of the soliton localization region and its amplitude. However, it does not describe the soliton movement along the coordinate \(y\), namely, its velocity \(V\).