2014/11/11 by Chuu-Lian Terng, Zhiwei Wu, Terng, Chuu-Lian +1 · 3 citations
Mathematics · Physics and Astronomy · #37K10 #37K30 #37K35 #53C44 #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1411.2725
openalex publication_date 2014/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct a sequence of commuting central affine curve flows on Rn\backslash 0 invariant under the action of SL(n,R) and prove the following results: (a) The central affine curvatures of a solution of the j-th central affine curve flow is a solution of the j-th flow of Gelfand-Dickey (GDn) hierarchy on the space of n-th order differential operators. (b) We use the solution of the Cauchy problems of the GDn flow to solve the Cauchy problems for the central affine curve flows with periodic initial data and also with initial data whose central affine curvatures are rapidly decaying. (c) We obtain a bi-Hamiltonian structure for the central affine curve flow hierarchy and prove that it arises naturally from the Poisson structures of certain co-adjoint orbits. (d) We construct Backlund transformations, infinitely many families of explicit solutions and give a permutability formula for these curve flows.