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.