\(G\)-strands and peakon collisions on \(\text{Diff}(\mathbb R)\) (Q352495)

The authors uses the Euler-Poincaré (EP) framework to study the \(G\)-strand equations making up the PDE system obtained from the EP variational equations for a \(G\)-invariant Lagrangian and an auxiliary zero-curvature equation, often arising in integrable chiral models, where it sets up the Lax-pair formulation of the system. When the \(G\)-invariant Lagrangian has been choosed, the combined dynamics and zero-curvature system of \(G\)-strand equations follows in the EP framework.NEWLINENEWLINEThe authors study collisions of solutions with singular support (peakons) of \(G\)-strands arising for diffeomorphisms on the real line when \(G=\mathrm{diff}(\mathbb R)\), for which the group product is the composition of smooth integrable functions. The peakon-peakon and peakon-antipeakon collision interactions are found to admit elementary solution methods reducing to solving either the Laplace or the wave equations depending on the sign in the Lagrangian. Also two complexified version of the \(\mathrm{Diff}(\mathbb R)\)-strand equations are introduced together with a formulation of their peakon solutions, solved then by elementary means, where the peakon-peakon and peakon-antipeakon collision interactions involve the solution of the Laplace or wave operator.