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Surfaces from deformation of parameters. (English) Zbl 1384.53004

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 17th international conference on geometry, integrability and quantization, Sts. Constantine and Elena (near Varna), Bulgaria, June 5–10, 2015. Sofia: Avangard Prima. Geometry, Integrability and Quantization, 318-343 (2016).
The aim of this paper is to introduce a new deformation, namely, deformation of parameters of solution of integrable equations to develop surfaces from integrable equations. Using this deformation, the authors construct surfaces from modified Korteweg-de Vries (mKdV) and sine-Gordon (SG) soliton solutions by the use of parametric deformations. For each integrable equation, there are two types of deformations arising from parameters of soliton solutions of the corresponding integrable equation. The first one gives surfaces on spheres and the second one gives highly complicated surfaces in \(\mathbb{R}^3\). The SG surfaces obtained are not the critical points of a functional where the Lagrange function is a polynomial function of the Gaussian (K) and mean (H) curvatures of the surfaces. In fact, the authors obtain quantities such as the first and second fundamental forms, the Gaussian and mean curvatures of the mKdV and SG surfaces. They find the position vector of mKdV and SG surfaces using the immersion function (F) and furthermore, they provide the graph of interesting mKdV and SG surfaces.
For the entire collection see [Zbl 1330.53003].

MSC:

53A05 Surfaces in Euclidean and related spaces
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
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Full Text: Euclid