Travelling Waves with a Singularity in Magnetic Nanowires

Abstract

We model the evolution of the magnetization in an infinite cylinder by harmonic map heat flow with an additional external field. Using variational methods, we prove the existence of corotationally symmetric travelling wave solutions with a moving vortex. We moreover show that for weak and strong fields the travelling waves connect the original state anti-parallel to the external magnetic field with the totally reversed state in direction of the external field. Our results match numeric simulations. For thicker wires several groups have found a reversal mode where a domain wall with a corotational symmetry and a vortex is propagating through the wire.

Keywords:

• Harmonic map
• Magnetism
• Travelling waves
• Variational methods

Mathematics Subject Classification:

• 82D40
• 47J30

Acknowledgments

The present results are part of my PhD thesis [19,, Theorem 3.3]. Many thanks are due to my supervisor Prof. S. Müller for suggesting the topic and for many helpful discussions. This work was supported by the DFG Priority Programme SPP 1095.