Elastic null curve flows, nonlinear C-integrable systems, and geometric realization of Cole-Hopf transformations. Issue 3 (2nd July 2020)

• Record Type: Journal Article
• Authors: Küçükarslan Yüzbaşı, Zühal; Anco, Stephen C.
• Is Part Of: Journal of nonlinear mathematical physics. Volume 27:Issue 3(2020)
• Page Start: 357
• Page End: 392
• Publication Date: 2020-07-02
• Subjects: null curve -- Minkowski space -- elastic flow -- Cole-Hopf transformation -- integrable system; 14H70 -- 37K10 -- 37K05; Nonlinear theories -- Periodicals; Mathematical physics -- Periodicals; 515.252
• DOI: 10.1080/14029251.2020.1757223
• Languages: English
• ISSNs: 1402-9251

Abstract

Abstract : Elastic (stretching) flows of null curves are studied in three-dimensional Minkowski space. As a main tool, a natural type of moving frame for null curves is introduced, without use of the pseudo-arclength. This new frame is related to a Frenet null frame by a gauge transformation that belongs to the little group contained in the Lorentz group SO (2, 1) and provides an analog of the Hasimoto transformation (relating a parallel frame to a Frenet frame for curves in Euclidean space). The Cartan structure equations of the transformed frame are shown to encode a hereditary recursion operator giving a two-component generalization of the recursion operator of Burgers equation, as well as a generalization of the Cole-Hopf transformation. Three different hierarchies of integrable systems are obtained from the various symmetries of this recursion operator. The first hierarchy contains two-component Burgers-type and nonlinear Airy-type systems; the second hierarchy contains novel quasilinear Schrödinger-type (NLS) systems; and the third hierarchy contains semilinear wave equations (in two-component system form). Each of these integrable systems is shown to correspond to a geometrical flow of a family of elastic null curves in three-dimensional Minkowski space.