Integrable equations arising from motions of plane curves

On the blow-up of solutions to the integrable modified Camassa–Holm equation

“…Second is the existence of peakons, which are nonanalytic solitary waves that are global weak solutions, and, moreover, interact cleanly as do solitons. Third is the variety of interesting geometric formulations of the CH equation [2,7,16,19]. Well-posedness and wave breaking of the CH equation were studied in a number of papers.…”

SIMILAR AND SELF-SIMILAR CURVES IN MINKOWSKI n-SPACE

“…K.-S. Chou and C. Qu [9] showed that the motions of curves in two-, three-and n-dimensional (n > 3) similarity geometries correspond to the Burgers hierarchy, Burgers-mKdV hierarchy and a multi-component generalization of these hierarchies by using the similarity invariants of curves in comparison with its invariants under the Euclidean motion. Also, they [8] found that many 1+1-dimensional integrable equations like KdV, Burgers, Sawada-Kotera, Harry-Dym hierarchies and Camassa-Holm equations arise from motions of plane curves in centro-affine, similarity, affine and fully affine geometries. The motion of curves on two-dimensional surfaces in E 3 1 was considered by Gürses [18].…”

“…for the curve given by (8) . Letκ 1 denote the function − 1 κ1 dκ1 dσ andκ i denote the function κi κ1 for i = 2, 3, .…”

Algebraic and Differential Invariants

“…Invariants of these group actions typically arise to reduce a problem or to decide if two objects, geometric or abstract, are obtained from one another by the action of a group element. [8,9,10,11,13,15,17,39,40,42,43,45,46,52,59] are a few recent references of applications. Both algebraic and differential invariant theories have become in recent years the subject of computational mathematics [13,14,17,40,60].…”