Singularities and Defects in Patterns far from Threshold

                                    by

                               Nick Ercolani
              University of Arizona, Department of Mathematics

              INSCC 110, 3:30pm  Monday, September 14, 1998


                                Abstract


Patterns are frequently observed in spatially extended systems 
driven far from equilibrium. Convection rolls in 
ordinary fluids, electro-hydrodynamic instabilities in liquid
crystals and transverse structures in Raman and Maxwell-Bloch laser
systems are just some of the striking examples of pattern formation.  

In the spirit of modulation theory, models of pattern 
formation far from threshold have been derived in which it is the
phase of an envelope which organizes the structure of the pattern. 
These models have been quite successful in capturing the qualitative 
nature of these patterns as well as their defects.

This talk will describe recent analytical investigations of one
of these models, the Cross-Newell phase diffusion equation. By use
of Legendre transforms, explicit multivalued solutions of this
equation can be constructed and their singularities studied. 
These can be related to physical solutions by introducing a natural 
variational regularization of the equations. In the singular limit, 
as the regularization is removed,
one recovers weak solutions of the Cross-Newell equation which,
in an interesting class of cases, can be shown to be asymptotic
minimizers of the free energy. 

Our regularized phase diffusion equation can be viewed as a
generalization of the vector Ginzburg-Landau equation when the vector 
fields are constrained to be irrotational. Remarkably, this equation 
has independently arisen in contexts other than pattern formation 
such as the modelling of thin film blistering and of crumbled paper.


Request for preprints and reprints to ercolani@math.arizona.edu
This information can be found at http://www.math.utah.edu/research/