G-strands and peakon collisions on diff(ℝ)

A G-strand is a map g: ℝ × ℝ! G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. Some G-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations that admit soliton solutions as completely integrable Hamiltonian systems. For example, the SO(3)-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Previous work has shown that G-strands for diffeomorphisms on the real line possess solutions with singular support (e.g. peakons). This paper studies collisions of such singular solutions of G-strands when G = Diff(ℝ) is the group of diffeomorphisms of the real line ℝ, for which the group product is composition of smooth invertible functions. In the case of peakon-antipeakon collisions, the solution reduces to solving either Laplace's equation or the wave equation (depending on a sign in the Lagrangian) and is written in terms of their solutions. We also consider the complexified systems of G-strand equations for G = Diff(ℝ) corresponding to a harmonic map g: ℂ → Diff(ℝ) and find explicit expressions for its peakon-antipeakon solutions, as well.