Integrable generalizations of Schrödinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces
The authors present a moving frame formulation of non-stretching geometric curve flows in Euclidean space to derive a \(1+1\)-dimensional hierarchy of integrable \(\text{SO}(3)\)-invariant vector models to describe the Heisenberg ferromagnetic spin model, and a spin vector version of the mKdV equation. Contents include: Introduction and summary; Vector models and space curve flows; Bi-Hamiltonian flows and operators; Geometric hierarchy of integrable vector models and dynamical maps; Surfaces and spatial Hamiltonian curve flows; Surface flows and \(2+1\) soliton equations; \(2+1\) vector models and dynamical maps; Geometric formulation; Concluding remarks; two appendices (dealing with technicalities of their discussion); and References (thirty items).
integrable \(\text{SO}(3)\) invariant vector models
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