Integrable motions of curves in projective geometries

Generated Surfaces via Inextensible Flows of Curves inR3

“…Consider the binormal motion of inextensible curves in R 3 , so = = 0. Then the evolution equation (9) takes the forṁ= = .…”

“…The connection between integrable systems and the differential geometry of curves has been studied extensively. Some integrable systems arise from invariant curve flows in certain geometries such as affine and centroaffine geometries [5][6][7] and similarity and projective geometries [8,9]. Motion of curves in Minkowski space 3 1 is studied in [10][11][12].…”

Evolution of Generalized Space Curve as a Function of Its Local Geometry

“…Nakayama, Segur and Wadati [14] set up a correspondence between the mKdV hierarchy and inextensible motions of plane curves in Euclidean geometry. Integrable systems satisfied by the curvatures of curves under inextensible motions in projective geometries are identified in [15]. Inextensible flows of curves in Galilean space are investigated in [16].…”

Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries

“…The parallel frames and other kinds of frames are also used to derive bi-Hamiltonian operators and associated hierarchies of multi-component soliton equations from non-stretching curve flows on Lie group manifolds [3,4,31,39]. The KdV equation, the modified KdV equation, the Sawada-Kotera equation and the Kaup-Kuperschmidt equation were shown to arise from the invariant curve flows respectively in centro-equiaffine geometry [7,9,48], Euclidean geometry [21], special affine geometry [11,35] and projective geometries [11,30,41]. The integrable systems with non-smooth solitary waves have drawn much attention in the last two decades because of their remarkable properties.…”