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

Bostock, C; Christian, JM; Leite, AB; McDonald, GS; Huang, JG

Abstract

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.

Citation

Bostock, C., Christian, J., Leite, A., McDonald, G., & Huang, J. (2015, June). Multi-Turing instabilities & spontaneous patterns in discrete nonlinear systems : simplicity and complexity, cavities and counterpropagation. Presented at 12th International Conference on the Mathematical and Numerical Aspects of Wave Propagation, Karlsruhe, Germany

• Presentation Conference Type: Other
• Conference Name: 12th International Conference on the Mathematical and Numerical Aspects of Wave Propagation
• Conference Location: Karlsruhe, Germany
• Start Date: Jun 21, 2015
• End Date: Jun 25, 2015
• Acceptance Date: Apr 1, 2015
• Publication Date: Jun 1, 2015
• Deposit Date: Jul 7, 2015
• Publicly Available Date: Apr 5, 2016
• Related Public URLs: http://waves2015.math.kit.edu/
• Additional Information: Event Type : Conference