When
Where
Student: Ed McDugald, Program in Applied Mathematics
Title: From knees to dislocations: order parameters from data for the phase grain boundaries of stripe patterns
Advisor: Dr. Shankar Venkataramani, Mathematics & Dr. Alan Newell, Mathematics
Location: MATH 402
Abstract: A phase grain boundary of the Swift–Hohenberg equation, where two mirror-image patches of stripes meet, is a smooth knee at shallow far-field angles and a sequence of dislocations at steep ones. The phase theory of Cross and Newell describes each end state, but not the boundaries in between, nor the passage from one to the other in time. This dissertation reads these boundaries through order parameters extracted directly from the pattern: the wavevector, phase and amplitude, estimated with the Riesz transform and the monogenic signal and validated on fields whose order parameters are known exactly. Classified by where the defect charge lives, the relaxed boundaries fall into three regimes: knees, with no charge; an intermediate regime, in which the pattern carries half-turn defects of its stripe direction that vanish when the amplitude is divided out; and dislocations, in which the phase itself carries one pair of defects per spacing. The same regimes are reached whether the boundary starts as a knee or as dislocations. Followed in time, nucleation runs the same way at every angle tested: the first change of topology appears in the field that carries the amplitude, and the phase acquires its defects afterwards. Where the phase theory should hold, sparse regression recovers it from the extracted fields; where defects form, the amplitude is needed to describe what the pattern does.