## Nonlinear Dynamics and Pattern Formation

### Parametrically excited spin waves

This was the topic of my PhD thesis. I have studied theoretically the formation and the dynamics of dissipative patterns caused by parametric excitation of spin waves. Parametric excitation of waves is caused by parametric resonance which is an instability phenomenon. For the first time analytical methods (i.e., multiple-scale perturbation theory and amplitude equations) well-known from hydrodynamic pattern formation has been applied to parametrically excited spin waves. A simple toy-model is discussed in Ref. 2, 3, and 5 of the publication list. A comparison between pattern formation in hydrodynamics and high-power ferromagnetic resonance has been published in Ref. 10. During my stay at the University of California at San Diego (UCSD) I started to do more realistic calculations with experimental predictions. The results of this work appeared in Ref.  13, 16, 20, and 26. The influence of hybridization on the type and the threshold of the main instability has been studied in Ref.  27.
 These are examples of stationary patterns of a ferromagnetic insulating film which is driven by out-of-plane parallel pump (i.e., the static magnetic field is perpendicular to the film plane and the magnetic field of a microwave is parallel to the static field). The images are synthetic photographs generated on the computer. They simulate real photographs where Faraday rotation is used to visualize the patterns. Fig. (a) is a square pattern built by two standing waves. Fig. (b) and (c) are built up by three standing waves forming a periodic pattern and a quasi-periodic one, respectively. These patterns are in general not stationary. The dynamics switches between them, a process which is very sensitive on noise. For more details see Ref.  26. There is also movie (MPEG, 1327Kb) which shows the pattern switching.

### Nonlocal pattern formation

In a series of papers I have discussed on a general level the influence of nonlocal or global terms in the equation of motion on the formation of patterns (see Ref. 4, 6-8). As an application of this theory it turned out that under certain conditions a new kind of oscillatory behavior in the electrothermal instability of the ballast resistor is possible (see Ref. 9). Also parallels in the kind of nonlocality between hydrodynamics and micromagnetism has been uncovered (see Ref. 10).

### Front propagation into an unstable state

During my stay at USCD I discussed with Harry Suhl and a student of him a simple micromagnetic model. We were interested into the problem of propagation of a stable state into a linearly unstable one. We found a new dynamical behavior namely that the initial front can split into two fronts propagating with different velocities leaving an unstable state in-between. Together with Jean-Pierre Eckmann and a student of him we were able to show that this a generic behavior. For more details see Ref. 14 and 17.

### Dynamical behavior of atomic force microscopy

An atomic force microscope is a linear oscillator which becomes nonlinear due to the interaction with the sample. Some preliminary results appeared in Ref. 15. Recently I have calculated the renormalization of the eigen frequencies due to induced motion in a medium (see Ref. 28).