Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces
Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces
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DOI：
10.1016/j.geomphys.2007.09.005
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发表时间：
2007-03
影响因子：
1.5
通讯作者：
S. Anco
中科院分区：
文献类型：
--
作者：
S. Anco
从黎曼对称空间M=G/H中的非伸展几何曲线流γ(t，x)出发，导出了群不变(多分量)孤子方程的普适双哈密顿族，包括G=K×K，H=图G时的紧致半单李群M=K.这些孤子族的推导使用了沿曲线流动的移动平行标架和连接1-形式，与Lie群G⊃H的克莱因几何有关，其中H是局部标架结构群。孤子方程显式地由主法向量N=∇xγx沿每条曲线的标架分量上的诱导流产生，并且在H中保持单位切线向量T=γx的等价子群下保持不变性。在曲线流的切线空间TγM中，它们的双哈密顿可积结构被几何地编码在关于平行标架及其连接1-形式的挠率和曲率的卡坦结构方程中。这些谱包括群不变版本的Sine-Gordon(SG)和Modify Korteweg-de Vries(MKdV)孤子方程，它们被发现是由描述G/H上非伸缩薛定谔映射的曲线流和mKdV类似的曲线流给出的。这些结果为许多已知的多分量孤子方程提供了几何解释和显式双哈密顿表述。此外，由本几何框架给出的群不变(多分量)孤子方程的所有例子都可以基于Cartan对对称空间的分类以显式方式构造。
Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows γ(t,x) in Riemannian symmetric spaces M=G/H, including compact semisimple Lie groups M=K for G=K×K, H=diagG. The derivation of these soliton hierarchies utilizes a moving parallel frame and connection 1-form along the curve flows, related to the Klein geometry of the Lie group G⊃H where H is the local frame structure group. The soliton equations arise in explicit form from the induced flow on the frame components of the principal normal vector N=∇xγxalong each curve, and display invariance under the equivalence subgroup in H that preserves the unit tangent vector T=γxin the framing at any point x on a curve. Their bi-Hamiltonian integrability structure is shown to be geometrically encoded in the Cartan structure equations for torsion and curvature of the parallel frame and its connection 1-form in the tangent space TγM of the curve flow. The hierarchies include group-invariant versions of sine–Gordon (SG) and modified Korteweg–de Vries (mKdV) soliton equations that are found to be universally given by curve flows describing non-stretching wave maps and mKdV analogs of non-stretching Schrödinger maps on G/H. These results provide a geometric interpretation and explicit bi-Hamiltonian formulation for many known multicomponent soliton equations. Moreover, all examples of group-invariant (multicomponent) soliton equations given by the present geometric framework can be constructed in an explicit fashion based on Cartan’s classification of symmetric spaces.