Patterns Selected by Spatial Inhomogeneity

Abstract

This thesis studies patterns that form in environments with sharp spatial variation. In a uniform environment, spots or stripes typically form with a self-consistent width. This width is taken from an interval around a characteristic value determined by the system. With a dramatic spatial variation, our environments only allow patterns in half the spatial region. This sets up a region of patterns directly adjacent to an area where patterns are suppressed. We show that this environmental inhomogeneity significantly restricts the widths of patterns that may occur in a given system. That is, the length of the interval around the characteristic value is significantly reduced. We examine this phenomenon using a universal partial differential equation model. Reduction techniques from dynamical systems simplify our study to the behavior in a normal form equation (amplitude equation). A difficulty arrises at the location of the discontinuous inhomogeneity; results in the normal form equations on the left and right cannot be directly compared. We construct a transformation of variables that bridges this jump and allows a heteroclinic gluing argument. The explicit form of this transformation determines the widths of patterns that can occur in the inhomogeneous environment.