Multi-Turing instabilities & spontaneous patterns in discrete nonlinear systems : simplicity and complexity, cavities and counterpropagation

Alan Turing's profound insight into morphogenesis, published in 1952, has provided the cornerstone for understanding the origin of pattern and form in Nature. When the uniform states of a nonlinear reaction-diffusion system are sufficiently stressed, arbitrarily-small disturbances can drive spontaneous self-organization into simple patterns with finite amplitude. Emergent structures have a universal quality (including hexagons, honeycombs, squares, stripes, rings, spirals, vortices), and they are characterized by a single dominant scalelength that is associated with the most-unstable Fourier component. In this paper, we extend Turing's ideas to three wave-based discrete nonlinear optical models with a wide range of boundary conditions. In each case, the susceptibility of the uniform states to symmetry-breaking fluctuations is addressed and we predict a threshold instability spectrum for static patterns that comprises a multiple-minimum structure. These Turing systems are also studied numerically, and we uncover examples of simple and complex (i.e., fractal, or multi-scale) pattern formation.