Nonvariational mechanism of front propagation: Theory and experiments

Multistable systems exhibit a rich front dynamics between equilibria. In one-dimensional scalar gradient systems, the spread of the fronts is proportional to the energy difference between equilibria. Fronts spreading proportionally to the energetic difference between equilibria is a characteristic of one-dimensional scalar gradient systems. Based on a simple nonvariational bistable model, we show analytically and numerically that the direction and speed of front propagation is led by nonvariational dynamics. We provide experimental evidence of nonvariational front propagation between different molecular orientations in a quasi-one-dimensional liquid-crystal light valve subjected to optical feedback. Free diffraction length allows us to control the variational or nonvariational nature of this system. Numerical simulations of the phenomenological model have quite good agreement with experimental observations.

Figure 1

Front propagation in the bistable variational model Eq. (1) with $\epsilon =0$. The upper panels represent the potential $V\left(u\right)$ for different values of $\eta$, (a) $\eta =0.2$, (b) $\eta =0$, and (c) $\eta =-0.2$, with $\mu =1.0$. The middle and lower panels illustrate the front profiles and their respective spatiotemporal evolutions for (d), (f) $\mu =1.0, \eta =0.3$, and (e), (g) $\eta =-0.3$.

Figure 2

Nonvariational front propagation model Eq. (1) at Maxwell's point ($\eta =0, \epsilon =1$, and $\mu =1$). (a) Potential $V\left(u\right)$. Front profiles at a given instant for (b) positive $c=3$, (c) negative $c=-3$, and $b=0$. (d) and (e) represent the spatiotemporal evolutions of front solutions with positive and negative parameters $c$ respectively and $b=0$. (f) Dimensionless front speed as a function of parameter $c$. Points account for the numerical front speed obtained from Eq. (5) with $b=0, \eta =0$, and $\epsilon =1$, the solid straight line is obtained from the analytical formula $v_{NV}\approx (2c-b)\epsilon \mu \sqrt{2}/5$, and the curve is obtained using formula (4) with a numerical front profile $u_F$.

Figure 3

Front propagation model Eq. (1) with $\eta =0.3$ and $\mu =1$. (a) Potential $V\left(u\right)$. (b) Front profiles for zero (dashed line) and positive (solid line) $c$ and $b=0$. (c) and (d) represent the spatiotemporal evolutions of the front solution with zero and positive parameter $c$. (e) Dimensionless front speed as a function of parameter $c$. Points account for the numerical front speed and continuous curve $v=v_V+v_{NV}$.

Figure 4

Nonvariational front propagation in a LCLV with optical feedback. (a) Schematic representation of the experimental setup. $\{L1,L2\}$ stands for two lenses with a focal distance $f=25\text{cm}$, M is a mirror, FB is an optical fiber bundle, PBS is a polarizing beam splitter cube, BS represents a beam splitter, and SLM is a spatial light modulator driven by a computer. $V_0$ external voltage applied across the LCLV. (b) Temporal sequence of snapshots of front propagation from top to bottom. (c) Front speeds as a function of voltage $V_0$ for different free propagation lengths $L$.

Figure 5

Front propagation of the numerical simulation of the phenomenological model of LCLV with optical feedback Eq. (5). (a) Bifurcation diagram of molecular average orientations ${\theta }_0$ as a function of voltage $V_0$. $V_0^c$ accounts for a critical value of voltage for which the system exhibits nascent bistability at ${\theta }_0={\theta }_c$. (b) Front speeds as functions of voltage $V_0$ for different free propagation lengths.