Integrable motions of space curves in affine geometry
Relations between motions of curves and soliton equations are interesting to not only mathematicians but also physicists. In the affine geometry direction, we have as samples the work of Chern and Terng in 1980 giving an affine analog of the Bäcklund transformation, and that of Pinkall in 1995 realizing the KdV equation as a Hamiltonian curve flow which keeps the total centro-affine curvature invariant. The authors of this articie under review connect motions of curves in affine and centro-affine three-dimensional spaces to the Korteweg-de Vries, Harry-Dym, and Sawada-Kotera hierarchies, as well as the Boussinesq, Kaup-Kupershmidt, Tzitzeica, and Hirota-Satsuma equations. Moreover, they explicitly construct the motions of curves corresponding to the soliton solutions of the KdV and Sawada-Kotera equations.
scientific article; zbMATH DE number 1783200
scientific article