Abstract:

A G-strand is a map g :(t,s)\in RxR -> g(t,s)\in G into a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. G-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations. Some of these equations are completely integrable Hamiltonian systems that admit soliton solutions. For example, the SO(3)-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Various other examples will be discussed, including collisions of solutions with singular support (e.g. peakons) on Diff(R)-strands, in which Diff(R) is the group of diffeomorphisms of the real line R, for which the group product is composition of smooth invertible functions.