Nonlinear PDEs and pattern formation (MVSpec)

Motivation:
After discussing dynamical systems on the level of ODEs in the last semester, we here will discuss how to transfer the knowledge to PDEs with a focus on pattern formation and solitons. Pattern formation is one of the most fascinating and intriguing phenomena in nature: it takes place in a wide variety of physical, chemical and biological systems and on very different spatial and temporal scales: examples are convection phenomena in geosciences and meteorology, but also patterns occurring in chemical reactions and bacterial colonies. In some circumstances, pattern formation is undesired, for instance the formation of spiral waves leading to cardiac arrhythmias in the heart muscle. In other contexts, pattern formation is even essential for the functioning of a system as in cell division and embryo development.

Contents:
We will study spatio-temporal behavior, i.e. partial differential equations (PDEs) and discuss the main questions in pattern formation: when will a homogeneous state become structured, i.e. unstable towards a pattern? What are the generic scenarios/types of patterns? When are patterns stable and are they unique? What determines the wavelength / period in time / amplitude of a pattern? A universal description of pattern dynamics exists, i.e. amplitude equations can be derived that are related to the famous Ginzburg-Landau equation (Nobel Prize in Physics 2003, originally derived for superconductivity). Nonlinear waves and solitons (localized waves) will also be discussed. They again occur in many systems, from coupled nonlinear springs to hydrodynamic surface waves and nonlinear optics.

Prerequisites: The course „Dynamical Systems“ from last year or a good knowledge of dynamical systems on the level of the book by SH Strogatz, Nonlinear dynamics and chaos, Westview 1994.

Literature: Cross M C and Greenside H, Pattern formation and dynamics in nonequilibrium systems (Cambridge, Cambridge Univ. Press, 2009).