Topological Solitons Seminar: Inspecting Baby Skyrmions with Effective Geometries
22 January 2014 13:00 in CM219
It has been known for some time that the propagation of the excitations of nonlinear field theories in a given background is governed by an effective metric that depends on the background field configuration and on the details of the non-linear dynamics. The perturbations in systems such as moving fluids, Bose-Einstein condensates, superfluids, and nonlinear electromagnetism, to mention a few, evolve in a curved effective spacetime different from the background geometry (which can even be Euclidean). Among other developments, the effective metric led to the construction of analog models of gravity, which imitate the kinematical properties of gravitational fields, and to insights on the evolutionary properties of hyperbolic nonlinear PDE's: regularity of solutions, emergence of
instabilities, confinement of waves and others.
In this talk we discuss effective geometries in the context of the Baby Skyrme wave map, where fields take values in the unit sphere. In particular, we derive a fourth order Fresnel-like equation for the high frequency waves and show that it factorizes into two quadratic characteristic polynomials. It follows that the causal structure of the theory is governed by a duplicity of effective geometries which depend explicitly on the pulled-back metric (strain) for the map. As a consequence, "large" background solutions of the Baby Skyrme equations behave as a birefringent medium for the linearized waves. In optics, birefringence means the emergence of two rays (ordinary and extraordinary) propagating with different velocities inside the medium. It appears in materials having a refractive index that depends on the polarization such as crystals with asymmetric structures, as well as plastics under mechanical stresses. In physical terms, there will be a double refraction process whereby a ray, when incident upon a soliton, is split into two rays taking slightly different paths. We discuss several features of the propagation in the context of Sine-Gordon-like solutions and Hedgehogs in 2+1. We conclude with some possible extensions and further applications of effective metrics to solitonic physics.
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