Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces

Abstract

A moving frame formulation of non-stretching geometric curve flows in Euclidean space is used to derive a 1+1 dimensional hierarchy of integrable SO(3)-invariant vector models containing the Heisenberg ferromagnetic spin model as well as a model given by a spin-vector version of the mKdV equation. These models describe a geometric realization of the NLS hierarchy of soliton equations whose bi-Hamiltonian structure is shown to be encoded in the Frenet equations of the moving frame. This derivation yields an explicit bi-Hamiltonian structure, recursion operator, and constants of motion for each model in the hierarchy. A generalization of these results to geometric surface flows is presented, where the surfaces are non-stretching in one direction while stretching in all transverse directions. Through the Frenet equations of a moving frame, such surface flows are shown to encode a hierarchy of 2+1 dimensional integrable SO(3)-invariant vector models, along with their bi-Hamiltonian structure, recursion operator, and constants of motion, describing a geometric realization of 2+1 dimensional bi-Hamiltonian NLS and mKdV soliton equations. Based on the well-known equivalence between the Heisenberg model and the Schrodinger map equation in 1+1 dimensions, a geometrical formulation of these hierarchies of 1+1 and 2+1 vector models is given in terms of dynamical maps into the 2-sphere. In particular, this formulation yields a new integrable generalization of the Schrodinger map equation in 2+1 dimensions as well as a mKdV analog of this map equation corresponding to the mKdV spin model in 1+1 and 2+1 dimensions.

Published version with typos corrected. Significantly expanded version of a talk given by the first author at the 2008 BIRS workshop on "Geometric Flows in Mathematics and Physics"

References in corpus (1)

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

Cited by in corpus (14)

• Integrable (2+1)-Dimensional Spin Models with Self-Consistent Potentials
• Integrable Kuralay equations: geometry, solutions and generalizations
• Darboux Transformation and Exact Solutions of the integrable Heisenberg ferromagnetic equation with self-consistent potentials
• Integrable motion of two interacting curves, spin systems and the Manakov system
• Hasimoto variables, generalized vortex filament equations, Heisenberg models and Schrodinger maps arising from group-invariant NLS systems
• Multi-Hamiltonian Structures on Spaces of Closed Equicentroaffine Plane Curves Associated to Higher KdV Flows
• Symplectically-invariant soliton equations from non-stretching geometric curve flows
• From Euler's elastica to the mKdV hierarchy, through the Faber polynomials
• Modified non-linear Schrödinger models, $\mathcal{C}{\mathcal{P}}_s{\mathcal{T}}_d$ invariant $N-$bright solitons and infinite towers of anomalous charges
• Unitarily-invariant integrable systems and geometric curve flows in $SU(n+1)/U(n)$ and $SO(2n)/U(n)$
• Integrable U(1)-invariant peakon equations from the NLS hierarchy
• Elastic null curve flows, nonlinear $C$-integrable systems, and geometric realization of Cole-Hopf transformations
• A fourth-order dispersive flow equation for closed curves on compact Riemann surfaces
• Integrable spinor/quaternion generalizations of the nonlinear Schrodinger equation