arXiv 0905.42152009-05math-phmath.MP

HP ^n中Hamilton曲线流的四元数孤子方程

Quaternionic Soliton Equations from Hamiltonian Curve Flows in HP^n

Stephen C. Anco, Esmaeel Asadi

备注：: 25 pages; typos corrected
发表信息：: J. Phys. A: Math. Theor. 42 (2009), 485201
DOI：: 10.1088/1751-8113/42/48/485201

摘要：

在四元数射影空间$HP^n$中，利用曲线的几何非拉伸流导出了一类双Hamilton四元数孤子方程族。推导适应的方法和结果在最近的工作中，我们之一的非拉伸曲线流的哈密顿结构的黎曼对称空间$M=G/H$通过查看$HP^n \simeq {\rm U}(n+1,H)/{\rm U}(1,H) \times {\rm U}(n,H)\simeq {\rm Sp}(n+1)/{\rm Sp}(1)\times {\rm Sp}(n)$作为一个对称空间的紧实辛群和四元数酉群。作为主要结果，得到了sine-Gordon（SG）方程和修正的Korteveg-de Vries（mKdV）方程的标量-矢量（多分量）形式及其双Hamilton可积性结构，该结构由四元数对称性和守恒律的共享族组成，由遗传递归算子生成.相应的几何曲线流在$HP^n$中示出由非拉伸波映射和非拉伸薛定谔映射的mKdV模拟来描述。

英文摘要：

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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