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 γ(t,x)γ(t,x) in Riemannian symmetric spaces M=G/HM=G/H, including compact semisimple Lie groups M=KM=K for G=K×KG=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⊃HG⊃H where HH 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=∇xγxN=∇xγx along each curve, and display invariance under the equivalence subgroup in HH that preserves the unit tangent vector T=γxT=γx in the framing at any point xx 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 TγMTγM 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/HG/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.