Symplectically invariant soliton equations from non-stretching geometric curve flows

Abstract

Bi-Hamiltonian hierarchies of symplectically invariant soliton equations are derived from geometric non-stretching flows of curves in the Riemannian symmetric spaces Sp(n + 1)/Sp(1) × Sp(n) and SU(2n)/Sp(n). The derivation uses Hasimoto variables defined by a moving parallel frame along the curves. As main results, two new multi-component versions of the sine–Gordon equation and the modified Korteweg–de Vries (mKdV) equation exhibiting Sp(1) × Sp(n − 1) invariance are obtained along with their bi-Hamiltonian integrability structure consisting of a hierarchy of symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in both Sp(n + 1)/Sp(1) × Sp(n) and SU(2n)/Sp(n) are shown to be described by a non-stretching wave map and a mKdV analogue of a non-stretching Schrödinger map.