Pattern Size in Gaussian Fields from Spinodal Decomposition

We study the two-dimensional snake-like pattern that arises in phase separation of alloys described by spinodal decomposition in the Cahn-Hilliard model. These are somewhat universal pattern due to an overlay of eigenfunctions of the Laplacian with a similar wave-number. Similar structures appear in other models like reaction-diffusion systems describing animal coats' patterns or vegetation patterns in desertification. Our main result studies random functions given by cosine Fourier series with independent Gaussian coefficients, which are taken over domains in Fourier space that grow and scale with aparameter of order 1/ε. Using a theorem by Edelman and Kostlan and ergodic theory, we show that on any straight line through the spatial domain the average distance of zeros of the series is asymptotically of order ε with a precisely given constant

Noisy patterns: Bridging the gap between stochastics and dynamics

In this thesis, we study travelling waves in stochastic reaction-diffusion equations. We extend techniques from the deterministic theory for travelling waves to apply to the stochastic version, which allows us to compute the stochastic wave speed and shape, and draw conclusions on the stability of the wave.Analysis and Stochastic