Abstract

Certain geometric flows of curves in homogeneous plane and space geometries are well-known to encode scalar soliton equations through the induced evolution of geometrical invariants of the curve. A broad generalization of these results has been obtained on the Hamiltonian structure of non-stretching flows of curves in homogeneous Riemannian geometries (i.e. symmetric spaces) For any such geometry, this gives a general geometric derivation of group-invariant (multi-component) mKdV(modified Korteweg and de Vries) and SG(Sine-Gordon) equations along with their bi-Hamiltonian integrability structure consisting of a shared hierarchy of symmetries and conservation laws generated by a group-invariant recursion operator. Adaptation of this derivation to non-stretching curve flows in the quaternion projective geometry viewed as a symmetric space yield a bi- Hamiltonian hierarchy of quaternion soliton equations.



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