Solitons as sections in non trivial bundles

Abstract

We interpret soliton solutions to classical wave equations as sections in some vector bundles; the peculiarity of soliton behaviour comes from the non-trivial character of the fibration, which also includes the “broken symmetry” aspect. We review in this light the most well-known cases: Kink, sine-Gordon soliton, vortex, monopole and instanton, and make contact with the former homotopy classification through the construction of bundles over spheres.