Quaternionic Soliton Equations from Hamiltonian Curve Flows in HP^n

Abstract

A bi-Hamiltonian hierarchy of quaternion soliton equations is derived from geometric non-stretching flows of curves in the quaternionic projective space $H{P}^n$. The derivation adapts the method and results in recent work by one of us on the Hamiltonian structure of non-stretching curve flows in Riemannian symmetric spaces $M=G/H$ by viewing $H{P}^n\simeq \mathrm{U}(n+1,H)/\mathrm{U}(1,H)\times \mathrm{U}(n,H)\simeq \mathrm{Sp}(n+1)/\mathrm{Sp}(1)\times \mathrm{Sp}(n)$ as a symmetric space in terms of compact real symplectic groups and quaternion unitary groups. As main results, scalar-vector (multi-component) versions of the sine-Gordon (SG) equation and the modified Korteveg-de Vries (mKdV) equation are obtained along with their bi-Hamiltonian integrability structure consisting of a shared hierarchy of quaternionic symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in $H{P}^n$ are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schrodinger map.

25 pages; typos corrected

References in corpus (3)

• Classification of integrable polynomial vector evolution equations
• Bi-Hamiltonian operators, integrable flows of curves using moving frames, and geometric map equations
• Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces

Cited by in corpus (3)

• Hasimoto variables, generalized vortex filament equations, Heisenberg models and Schrodinger maps arising from group-invariant NLS systems
• 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)