Elastic null curve flows, nonlinear C-integrable systems, and geometric realization of Cole-Hopf transformations

DOI

10.1080/14029251.2020.1757223How to use a DOI?

Keywords

null curve; Minkowski space; elastic flow; Cole-Hopf transformation; integrable system

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.

Copyright

© 2020 The Authors. Published by Atlantis and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Zühal Küçükarslan Yüzbaşı
AU  - Stephen C. Anco
PY  - 2020
DA  - 2020/05/04
TI  - Elastic null curve flows, nonlinear C-integrable systems, and geometric realization of Cole-Hopf transformations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 357
EP  - 392
VL  - 27
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2020.1757223
DO  - 10.1080/14029251.2020.1757223
ID  - Yüzbaşı2020
ER  -