Mathematics faculty and graduate students are invited to attend the PhD
Defense of Brenden Balch.

You may also attend on Zoom

Advisor:  Dr. Patrick Shipman

Committee:  Dr. Chris Peterson, Dr. Clayton Shonkwiler, Dr. Richard
Bradley, Dr. Hua Chen

Title:  Notes on pattern formation

Abstract:   We discuss two topics from pattern formation. The first part
investigates the dispersive Swift-Hohenberg equation (DSHE), formed by
adding a linear dispersive term to the classic Swift-Hohenberg model for
pattern formation.  The DSHE generates stripe patterns with extended
defects called seams, characterized as dislocations smeared along
oblique line segments.  We compare these defects with those found in
previous work on the dispersive Kuramoto-Sivashinsky equation (DKSE).
Unlike the DKSE, the DSHE exhibits a narrow band of unstable wavelengths
near an instability threshold, enabling analytical advancements. We
establish a connection between an amplitude equation derived from the
DSHE and the anisotropic complex Ginzburg-Landau equation (ACGLE),
demonstrating that seams in the DSHE correspond to spiral waves in the
ACGLE. These spiral waves tend to organize into chains, and we derive
formulas for their core velocity and spacing. Additionally, under strong
dispersion, we establish a perturbative relationship between the stripe
pattern’s amplitude, wavelength, and propagation velocity. Analytical
results are confirmed through numerical integrations of the ACGLE and
DSHE. The second part of this talk will explore the measurement and
characterization of order in non-equilibrium pattern forming systems.
The study focuses on the use of topological measures of order, via
persistent homology and the Wasserstein metric. We investigate the
quantification of order with respect to ideal lattice patterns and
demonstrate the effectiveness of the introduced measures of order in
analyzing imperfect three-dimensional patterns and their time evolution.

Join Zoom Meeting:
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Meeting ID: 974 8405 1784
Passcode: 244911