Patterns, Nonlinear Dynamics and Applications - PANDA

Michael Proctor

Structures and dynamics in the Ginzburg-Landau equation

 
Abstract: The Ginzburg-Landau equation provides an asymptotic description of the spatially-extended dynamics of a pattern-forming instability where the pattern has an underlying periodicity but is modulated on a far longer length-scale. The simplest secondary instability that it captures is known as the Eckhaus instability, where the pattern wavelength becomes too far away from the `preferred' wavelength of the instability. By extending the description to include modulation in a second spatially-extended direction the Newell-Whitehead-Segel equation can be derived which models instabilities of a pattern to disturbances in this perpendicular direction. This lecture will discuss these modes of instability and examine recent work on other models of pattern-formation in spatially-extended systems to which the Ginsburg-Landau equation does not apply; for example when conservation laws are present.