From the point of view of mathematics,
micromagnetics is an ideal playground for a pattern forming system in materials
science: There are abundant experiments on a wealth of visually
attractive phenomena and there is a well--accepted continuum model.
In this talk, I will focus on a specific
experimental observation for thin film ferromagnetic elements: Elements
with elongated rectangular cross--section are saturated along the longer
axis by a strong external field. Then the external field is slowly
reduced. At a certain field strength, the uniform magnetization buckles
into a quasiperiodic domain pattern which resembles a concertina.
Our hypothesis is that the period of this pattern is the
frozen--in period of the unstable mode at critical field.
Starting point for the analysis is the micromagnetic model which has three
length scales. We rigorously identify four scaling regimes for the
critical field. One of the regimes has been
overseen by the physics literature. It displays an
oscillatory unstable mode, which we identify asymptotically.
In this parameter regime, we identify
a scaling limit for the bifurcation.
The analysis amounts to the combination of an asymptotic
limit with a bifurcation argument. This is carried out by a suitable
``blow--up'' of the energy landscape in form of Gamma--convergence.
Numerical simulation of the normal form visualizes a homotopy
from a turning point to
the strongly nonlinear concertina pattern.
This is joint work with Ruben Cantero--Alvarez and Jutta Steiner.
