Bi-Hamiltonian operators, integrable flows of curves using moving frames, and geometric map equations

Abstract

Moving frames of various kinds are used to derive bi-Hamiltonian operators and associated hierarchies of multi-component soliton equations from group-invariant flows of non-stretching curves in constant curvature manifolds and Lie group manifolds. The hierarchy in the constant-curvature case consists of a vector mKdV equation coming from a parallel frame, a vector potential mKdV equation coming from a covariantly-constant frame, and higher order counterparts generated by an underlying vector mKdV recursion operator. In the Lie-group case the hierarchy comprises a group-invariant analog of the vector NLS equation coming from a left-invariant frame, along with higher order counterparts generated by a recursion operator that is like a square-root of the mKdV one. The corresponding respective curve flows are found to be given by geometric nonlinear PDEs, specifically mKdV and group-invariant analogs of Schrodinger maps. In all cases the hierarchies also contain variants of vector sine-Gordon equations arising from the kernel of the respective recursion operators. The geometric PDEs that describe the corresponding curve flows are shown to be wave maps. Full details of these results are presented for two main cases: $S^2,S^3\simeq SU(2)$.