# APMA 2210: Topics in Nonlinear Dynamical Systems (Autumn 2009)

## Class information

Instructor | Björn Sandstede |

Phone | 863-2815 |

Email | bjorn_sandstede@brown.edu |

Class meetings | MW 2:30-3:20pm and F 2:00-2:50pm, in Room 102a in 180 George Street |

## Course material

I plan to talk about the dynamics of nonlinear waves and patterns using "spatial dynamics", covering the motivation behind this approach, the techniques needed for carrying it out, and several applications.

Nonlinear waves organize the dynamics of many physical systems. In one-dimensional media, they correspond to travelling-wave solutions *u(x,t)=v(x-ct)* of, say, reaction-diffusion systems *u*_{t}=*u*_{xx}+*f*(*u*). Travelling waves *v(x-ct)* satisfy the ordinary differential equation *-cv*_{y}=*v*_{yy}+*f*(*v*) which can be analysed using dynamical-systems methods. Spatial dynamics refers to the idea of studying more complicated patterns, such as travelling waves on multidimensional domains, time-periodic structures, and spiral waves, from exactly the same view point, namely by considering the spatial variable *x*, and not the time *t*, as the evolution variable. These patterns cannot be captured by ordinary differential equations but satisfy an elliptic PDE: nevertheless, it turns out that many dynamical-systems ideas carry over. Similar ideas can also be used to investigate systems on discrete lattices.

I will develop the underlying theory for studying patterns in this fashion and show how concepts such as group velocities and transport, which are commonly used in physics and provide a great deal of intuition about how patterns and waves should behave, are reflected by spatial dynamics. The key mathematical techniques we will learn about are exponential dichotomies, spectral theory of travelling waves, Fredholm theory, and Lyapunov-Schmidt reduction for homoclinic orbits. I will also discuss applications to waves in nonlinear optics and fluids.