Unitarily-invariant integrable systems and geometric curve flows in and
arXiv:1408.5290 · doi:10.1088/1751-8121/aaa193
Abstract
Bi-Hamiltonian hierarchies of soliton equations are derived from geometric non-stretching (inelastic) curve flows in the Hermitian symmetric spaces and . The derivation uses Hasimoto variables defined by a moving parallel frame along the curves. As main results, new integrable multi-component versions of the Sine-Gordon (SG) equation and the modified Korteveg-de Vries (mKdV) equation, as well as a novel nonlocal multi-component version of the nonlinear Schrödinger (NLS) equation are obtained, along with their bi-Hamiltonian structures and recursion operators. These integrable systems are unitarily invariant and correspond to geometric curve flows given by a non-stretching wave map and a mKdV analog of a non-stretching Schrödinger map in the case of the SG and mKdV systems, and a generalization of the vortex filament bi-normal equation in the case of the NLS systems.
32 pages. Typos fixed. To appear in J. Phys. A
References in corpus (8)
- Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces
- Bi-Hamiltonian operators, integrable flows of curves using moving frames, and geometric map equations
- Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces
- Hamiltonian Flows of Curves in symmetric spaces G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type
- Systematic method of generating new integrable systems via inverse Miura maps
- Symplectically-invariant soliton equations from non-stretching geometric curve flows
- Quaternionic Soliton Equations from Hamiltonian Curve Flows in HP^n
- Integrable Systems with Unitary Invariance from Non-stretching Geometric Curve Flows in the Hermitian Symmetric Space Sp(n)/U(n)