Scattering of magnetic solitons in two dimensions

Solitons which have the form of a vortex-antivortex pair have recently been found in the Landau-Lifshitz equation which is the standard model for the ferromagnet. We simulate numerically head-on collisions of two ferromagnetic solitons and observe a right angle scattering pattern which is familiar from relativistic models. We offer a resolution of this highly nontrivial dynamical behavior by examining the Hamiltonian structure of the model, specifically the linear momentum of the two solitons. We also investigate the dynamics of vortices in a modified nonlinear σ model which arises in the description of antiferromagnets. We confirm numerically that a robust feature of the dynamics is the right angle scattering of two vortices which collide head-on. A nontrivial generalization of our theory is given for this relativistic model which leads to an understanding of the observed dynamical behavior of vortices.