非拉伸几何曲线流中的辛不变孤子方程
Symplectically-invariant soliton equations from non-stretching geometric curve flows
摘要：
利用黎曼对称空间$Sp(n+1)/Sp(1)\times Sp(n)$和$SU(2n)/Sp(n)$中曲线的几何非拉伸流的移动标架形式，导出了两个双Hamilton族的辛不变孤子方程.作为主要结果，得到了sine-Gordon（SG）方程和修正的Korteweg-de Vries（mKdV）方程的多分量形式的$Sp(1)\times Sp(n-1)$不变性，以及它们的双Hamilton可积性结构，该结构由一个遗传递归算子产生的对称性和守恒律的共享族组成.相应的几何曲线流在$Sp(n+1)/Sp(1)\times Sp(n)$和$SU(2n)/Sp(n)$示出的非拉伸波映射和mKdV模拟的非拉伸薛定谔映射描述。
英文摘要：
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.
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