Quaternionic Soliton Equations from Hamiltonian Curve Flows in HP^n

Details Quaternionic Soliton Equations from Hamiltonian Curve Flows in HP^n Quaternionic Soliton Equations from Hamiltonian Curve Flows in HP^n

2009-05-26

arxiv.org/abs/0905.4215v3

J. Phys. A: Math. Theor. 42 (2009), 485201

Physics

Mathematical Physics

25 pages; typos corrected

Scientific paper

A bi-Hamiltonian hierarchy of quaternion soliton equations is derived from geometric non-stretching flows of curves in the quaternionic projective space $HP^n$. 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 $HP^n \simeq {\rm U}(n+1,H)/{\rm U}(1,H) \times {\rm U}(n,H)\simeq {\rm Sp}(n+1)/{\rm Sp}(1)\times {\rm Sp}(n)$ 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 $HP^n$ are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schrodinger map.

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