Binormal motion of curves of constant curvature and torsion. Generation of soliton surfaces. (English) Zbl 0981.53061
Summary: The purely binormal motion of curves of constant curvature or torsion, respectively, is shown to lead to integrable extensions of the Dym and classical sine-Gordon equations. In the case of the extended Dym equation, a reciprocal invariance is used to establish the existence of novel dual-soliton surfaces associated with a given soliton surface. A cc-ideal formulation is adduced to obtain a matrix Darboux invariance for the extended Dym and reciprocally linked \(m^2\)KdV equations. A Bäcklund transformation with a classical-length property is thereby constructed, which allows the generation of associated soliton surfaces. Analogues of both Bäcklund’s and Bianchi’s classical transformations are derived for the extended sine-Gordon system.
MSC:
53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
35Q53 | KdV equations (Korteweg-de Vries equations) |
53A05 | Surfaces in Euclidean and related spaces |
53A04 | Curves in Euclidean and related spaces |