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Competing patterns in generic models

Working Group 8.43

Pattern formation far from equilibrium

Many systems in living and inanimate nature are far from thermodynamic equilibrium. Thus, the observed structures and patterns can not simply be rationalized by free energy minimization. Structure formation is rather a dynamic, nonlinear process driven by an energy source. For instance, the activation of action potentials, resulting in the spatiotemporal excitation patterns of cardiac tissue, is driven by energy provided by a concentration difference of ions in the intra- and extracellular medium. Time-invariant structures may also be sustained by energy input. Rayleigh-Bénard convection cells, for instance, require a certain temperature difference between bottom and ceiling. The Swift-Hohenberg equation (equation 1, where $\alpha=0$) describes structure formation at a length scale $1/k_{1}$. It is used to model pattern phenomena from diverse areas such as hydrodynamics and biophysics. For $\epsilon > 0$, the uniform solution $u_{1} = 0$ becomes unstable and stationary Turing patterns develop.

 

Coupled Swift-Hohenberg equations

$\begin{align}\partial_t \, u_1 &= \epsilon \, u_1 - \left( k_1^{-2} \, \nabla^2 \, + \, 1 \right)^2 u_1 - u_1^3 - \alpha \, u_2 \label{gekoppelteSwiftHohenbergGl_1} \\\partial_t \, u_2 &= \epsilon \, u_2 - \left( k_2^{-2} \, \nabla^2 \, + \, 1 \right)^2 u_2 - u_2^3 + \alpha \, u_1\label{gekoppelteSwiftHohenbergGl_2}\end{align}$

Coupling two Swift-Hohenberg equations asymmetrically, as in equations (1) and (2), leads to another instability [Schüler et.al., Chaos 24(4), 2014]: For $\alpha>\alpha_{c}$ and $\epsilon>\epsilon_{c}$, traveling waves (see Fig. 2) may arise in addition to Turing patterns. Using both numerical and analytical methods and in cooperation with project group A5 ("Pattern formation in systems with multiple states") of the collaborative research center 910, we investigate solutions of equations (1) and (2). In particular, we derive and analyze amplitude equations which describe spatiotemporal modulations of Turing patterns and traveling waves in the vicinity of the (supercritical) bifurcation. For sufficiently large coupling $\alpha$ and one spatial dimension, for instance, solutions of these amplitude equations reveal patches where one wave direction is dominating. These patches are separated by sources and sinks. They are observable in direct numerical simulations of equations (1) and (2), see Fig. 2. In the vicinity of the bifurcation, amplitude equations may also provide quantitative predictions regarding linear growth of perturbations, nonlinear saturation, and possible modulations at large scales in space and time. We will also address control of the patterns. A possible control scheme is delayed feedback, where a term is introduced which couples to the delayed function value $u_{1}(t-\tau)$.

 

Fig. 2: Wave instability of the equations (1) and (2): The homogenous solution u1 = u2 = 0 is unstable and small perturbations lead to traveling waves in u1 (blue) and u2 (green). From t ≈ 50 s, sources and sinks appear at x ≈ 60 and x ≈ 125, respectively.