# Light-Induced Ring Pattern in a Dye-Doped Nematic Liquid Crystal
## 3 Mathematical Modeling for Photo-Isomerization in Dye-Doped Liquid Crystals
### Homogeneous Illumination and Bifurcation Diagram  
Considering a spatially homogeneous illumination--that is, \(C_{0}\) is a constant--the parameters that characterize the nematic and isotropic liquid transition are renormalized and independent of the space. The effective model has the form  
\[\partial_{t}S = -\tilde{A}S+\tilde{B}S^{2}-S^{3}-\
u\
abla^{2}S-\
abla^{4}S+bS \
abla^{2}S. \tag{15}\]  
The homogeneous phases of this model have the form \(S_{0}=0\) and  
\[S_{\pm}=\frac{\tilde{B}\pm\sqrt{\tilde{B}^{2}-4\tilde{A}}}{2}, \tag{16}\]  
where \(S_{0}\) and \(S_{\pm}\) account for the liquid isotropic and nematic phase. For high temperatures--i.e., a large \(\tilde{A}\)--it is expected that the only stable state is the isotropic liquid phase \(S_{0}\). When decreasing \(\tilde{A}\), the system presents a coexistence between the isotropic and nematic phase for \(\tilde{A}=A_{sn}\equiv\tilde{B}^{2}/4\). This bifurcation occurs due to the emergence of two new equilibria:the saddle-node bifurcation causes the emergence of a stable nematic state (\(S_{+}\)) and an unstable state (\(S_{-}\)). Figure 5 shows the bifurcation diagram of the model in Equation (15). As \(\tilde{A}\) continues to decrease, the isotropic liquid state \(S_{0}\) is as favorable as the nematic phase \(S_{+}\) for \(A=A_{M}\equiv 2\tilde{B}^{2}/9\)--the Maxwell point [46]. Then, a flat wall between these two phase states at this critical point is characterized by being motionless. When further decreasing \(\tilde{A}\), the isotropic liquid phase presents a spatial instability. We can study this instability by linearizing Equation (15) around the isotropic liquid state \(S_{0}=0\), and considering the ansatz \(S(\vec{r},t)=S^{\prime}e^{i\vec{k}\vec{r}+i\sigma t}\), we obtain the growth rate equation:  
\[\sigma=-\tilde{A}+v\vec{k}^{2}-\vec{k}^{4}, \tag{17}\]  
The instability condition is \(d\sigma(k_{c})/dk=0\) and \(\sigma(k_{c})=0\). The first condition determines the critical length \(k_{c}=\sqrt{\vec{k}^{2}}\equiv\sqrt{\
u/2}\), and the second defines the critical relation of the parameters for the spatial instability, which has the form \(\tilde{A}=A_{sp}\equiv\
u^{2}/4\). Weakly nonlinear analysis shows that this instability is of a supercritical nature for a small \(\tilde{B}\). Thus, despite the fact that the linear term is positive, \(0<\tilde{A}<A_{sp}\), the isotropic liquid state is unstable. For \(\tilde{A}=A_{T}\equiv 0\), the system presents a transcritical bifurcation between unstable states. To study the stability of the nematic phase, we use a similar strategy as in the study of the spatial stability of the isotropic liquid phase. Let us consider the linear perturbation \(S=S_{\pm}+\chi\), where \(\chi\) is a small variable that satisfies the equation  
\[\partial_{t}\chi=(-\tilde{A}+2\tilde{B}-3S_{\pm}^{2})\chi-(\
u+bS_{\pm})\
abla ^{2}\chi-\
abla^{4}\chi. \tag{18}\]  
Introducing the ansatz \(\chi(\vec{r},t)=\chi^{\prime}e^{i\vec{k}\vec{r}+i\sigma t}\) in the above equation, we obtain  
\[\sigma=-\tilde{A}+2\tilde{B}-3S_{\pm}^{2}+(\
u+bS_{\pm})\vec{k}^{2}-\vec{k}^{4}. \tag{19}\]  
Imposing the spatial instability conditions, we obtain \(k_{c}=\sqrt{\vec{k}^{2}}=\sqrt{(\
u+bS_{\pm})/2}\) and  
\[2\tilde{A}+\tilde{B}S_{\pm}=-\frac{(\
u+bS_{\pm})^{2}}{4}. \tag{20}\]  
Figure 5: Schematic representation of the bifurcation diagram of the effective model in Equation (15) with constant coefficients. The order parameter \(S\) as a function of the bifurcation parameter \(\tilde{A}\). \(S_{0}\), \(S_{+}\), and \(S_{-}\) account for the isotropic liquid and nematic phases, respectively. The continuous and dashed lines account for stable and unstable states respectively. \(A_{sn}\), \(A_{M}\), and \(A_{T}\) are the critical points that account for the emergence of the nematic phase; both phases are equally favored, with a transcritical bifurcation of the isotropic liquid phase. \(A_{sp}\), \(A_{sp}^{+}\), and \(A_{sp}^{-}\) account for the spatial instabilities of the homogeneous phases.