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Contour dynamics models for Hamiltonian and dissipative pattern formation
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Contour dynamics models for Hamiltonian and dissipative pattern formation
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Dissertation
Contour dynamics models for Hamiltonian and dissipative pattern formation
1994
Overview
In this thesis, we investigate the nonlinear dynamics of three systems effectively defined by contour dynamics. Following an introductory chapter, in chapter 2 we investigate a class of curve dynamics in the plane equivalent to the modified Korteweg-de Vries (mKdV) equation. The mKdV equation is integrable, and we show that some of the the standard results of integrable systems theory have geometric interpretations. Finally, we show the relation between the abstract curve dynamics and vortex patch dynamics. In chapter 3, we investigate a class of curve dynamics in three dimensions, equivalent to the integrable nonlinear Schrodinger (NLS) equation. Again, many of the features of NLS have geometric interpretations. We rederive a known result about the relation between NLS and vortex line dynamics in two different ways. Finally, in chapter 4, we investigate the pattern formation of a reaction-diffusion system defined in two spatial dimensions. The patterns are effectively defined by interfaces, so we develop a contour dynamics to describe the time evolution of the patterns. We show that these patterns are not equivalent to Turing patterns and that wildly branched and complex patterns are possible.
