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 γ ( t , x ) in Riemannian symmetric spaces M = G / H , including compact semisimple Lie groups M = K for G = K × 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 ⊃ 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 = ∇ x γ x along each curve, and display invariance under the equivalence subgroup in H that preserves the unit tangent vector T = γ 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 γ 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 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.