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

Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows gamma(t, x) in Riemannian symmetric spaces M = G/H, including compact semisimple Lie groups M = K for G = K x K, H = 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 superset of 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 = del(x)gamma(x) along each curve, and display invariance under the equivalence subgroup in H that preserves the unit tangent vector T = gamma(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 gamma 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. (C) 2007 Elsevier B.V. All rights reserved.