Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 030, 20 pages      arXiv:1205.5329      https://doi.org/10.3842/SIGMA.2012.030
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Motions of Curves in the Projective Plane Inducing the Kaup-Kupershmidt Hierarchy

Emilio Musso
Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy

Received February 08, 2012, in final form May 11, 2012; Published online May 24, 2012

Abstract
The equation of a motion of curves in the projective plane is deduced. Local flows are defined in terms of polynomial differential functions. A family of local flows inducing the Kaup-Kupershmidt hierarchy is constructed. The integration of the congruence curves is discussed. Local motions defined by the traveling wave cnoidal solutions of the fifth-order Kaup-Kupershmidt equation are described.

Key words: local motion of curves; integrable evolution equations; Kaup-Kupershmidt hierarchy; geometric variational problems; projective differential geometry.

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