Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2006-10-31
IMA Volumes in Mathematics and its Applications, Vol. 144, Symmetries and Overdetermined Systems of Partial Differential Equat
Nonlinear Sciences
Exactly Solvable and Integrable Systems
Final version with improved derivation of vector SG equations. To appear in Proceedings of IMA workshop on Symmetries and Over
Scientific paper
A bi-Hamiltonian hierarchy of complex vector soliton equations is derived from geometric flows of non-stretching curves in the Lie groups $G=SO(N+1),SU(N)\subset U(N)$, generalizing previous work on integrable curve flows in Riemannian symmetric spaces $G/SO(N)$. The derivation uses a parallel frame and connection along the curves, involving the Klein geometry of the group $G$. This is shown to yield the two known $U(N-1)$-invariant vector generalizations of the nonlinear Schrodinger (NLS) equation and the complex modified Korteweg-de Vries (mKdV) equation, as well as $U(N-1)$-invariant vector generalizations of the sine-Gordon (SG) equation found in recent symmetry-integrability classifications of hyperbolic vector equations. The curve flows themselves are described in explicit form by chiral wave maps, chiral variants of Schrodinger maps, and mKdV analogs.
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