Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces

Abstract

Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows $\map(t,x)$ in Riemannian symmetric spaces $M=G/H$, including compact semisimple Lie groups $M=K$ for $G=K\times K$, $H={\rm diag} G$. 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 $G\supset H$ 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 $N=\covder{x}\mapder{x}$ along each curve, and display invariance under the equivalence subgroup in $H$ that preserves the unit tangent vector $T=\mapder{x}$ in 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 $T_\map M$ 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 Schrodinger 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.