Defects in pattern formation
Patterns of an almost periodic or multiperiodic nature turn up all over the place. They can be see as ripples on gently sloped beaches and desert dunes, as stripes on tigers and zebras, whorls in fingerprints, and spots on leopards. It turns out that these natural phonomena can be modeled by some mathematical systems which are governed by the minimization of some appropriate free energy. Examples of such systems include the Swift-Hohenberg and Cross-Newell energy of pattern formation, often leads to phase transitions, point and line defects (in two dimensions), and loop (in three dimensions). Our current work is kinds of relating to the bifurcation behaviour between line defects and point defects of the system. In the first half of the talk, I will recall some basic facts for SH and CN, and afterwards, I will show some figures that describe the condensation of Gauss curvature of solutions to the SH in the disk with fixed boundary condition. One of our current work is to better understand the structure of minimizers of the CN energy among the class of multi-valued functions on a semidisk. I will show some numerics for the SH solutions. By using the Hilbert transform, we are able to recover the phase solutions from the SH solutions, which bascially tell us what would be the right function space to put into CN. In the end, I will list some future directions. This is work is under the supervision of Prof. Alan Newell, Prof. Shankar Venkataramani, and joint with Dr. Rachel Neville.