Non-equilibrium patterns in polarizable active layers

• Formulation of the problem of pattern formation in a polarizable active layer.
• Long-wave monotonic and short-wave oscillatory instabilities.
• Spontaneous emergence of deformation, polarization, and chemical activity.
• Lagrangian numerical computation with nonlinear elasticity.
• Establishment of permanent polarity in spherical geometry.

We formulate and explore a generic continuum model of a polarizable active layer with nonlinear elasticity and chemo-mechanical interactions. Homogeneous solutions of the model equations exhibit a monotonic long-wave instability when the medium is activated by expansion, and an oscillatory short-wave instability in the case of compressive activation. Both regimes are investigated analytically and numerically. The long-wave instability initiates a coarsening process, which provides a possible mechanism for the establishment of permanent polarization in spherical geometry.