Symplectically-invariant soliton equations from non-stretching geometric curve flows

Abstract

A moving frame formulation of geometric non-stretching flows of curves in the Riemannian symmetric spaces $Sp(n+1)/Sp(1)\times Sp(n)$ and $SU(2n)/Sp(n)$ is used to derive two bi-Hamiltonian hierarchies of symplectically-invariant soliton equations. As main results, multi-component versions of the sine-Gordon (SG) equation and the modified Korteweg-de Vries (mKdV) equation exhibiting $Sp(1)\times Sp(n-1)$ invariance are obtained along with their bi-Hamiltonian integrability structure consisting of a shared hierarchy of symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in $Sp(n+1)/Sp(1)\times Sp(n)$ and $SU(2n)/Sp(n)$ are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schrödinger map.

39 pages; remarks added on algebraic aspects of the moving frame used in the construction

References in corpus (4)

• Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces
• Classification of integrable polynomial vector evolution equations
• Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces
• New reductions of integrable matrix PDEs: $Sp(m)$-invariant systems

Cited by in corpus (4)

• Hasimoto variables, generalized vortex filament equations, Heisenberg models and Schrodinger maps arising from group-invariant NLS systems
• Multi-Hamiltonian Structures on Spaces of Closed Equicentroaffine Plane Curves Associated to Higher KdV Flows
• Unitarily-invariant integrable systems and geometric curve flows in $SU(n+1)/U(n)$ and $SO(2n)/U(n)$
• Integrable Systems with Unitary Invariance from Non-stretching Geometric Curve Flows in the Hermitian Symmetric Space Sp(n)/U(n)