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Integrable curves and surfaces. (English) Zbl 1351.53009

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. 13-71 (2016).
Authors’ abstract: The surfaces in three dimensional Euclidean space \(\mathbb{R}^3\) obtained through the use of the soliton techniques are called integrable surfaces. Integrable equations and their Lax equations possess certain symmetries. Infinitesimal versions of these symmetries are deformations which are responsible in constructing the integrable surfaces. There are four different types of deformations. The spectral parameter, the gauge, the generalized symmetries and integration parameters deformations. We shall present here how these deformations generate surfaces in \(\mathbb{R}^3\) and also in three-dimensional Minkowski space \(\mathbb{M}^3\). The key point here is to start with an integrable equation and its Lax representation.
In this work, we assume that the Lax equations of integrable equations are given in terms of a group \(G\), its algebra \(\mathfrak{g}\), and \(\mathfrak{g}\)-valued functions. The surfaces in \(\mathbb{R}^3\) are also represented via \(\mathfrak{g}\)-valued functions. While constructing integrable surfaces we need the solutions of both the integrable equations and their corresponding Lax equations. In this work, we use the one-soliton solutions of the integrable equations. We solve the Lax equations for one-soliton solutions of the integrable equations. Then, choosing a deformation one can construct several types of surfaces. After obtaining these surfaces the next step is to search for their properties. Most of these surfaces are Weingarten surfaces, Willmore-like surfaces, and surfaces which are derivable from a variational principle. We give sketches of the interesting surfaces of Korteweg-de Vries (KdV), modified Korteweg-de Vries (mKdV), nonlinear Schrödinger (NLS), and sine Gordon (SG) equations.
For the entire collection see [Zbl 1330.53003].

MSC:

53A05 Surfaces in Euclidean and related spaces
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q51 Soliton equations
35Q35 PDEs in connection with fluid mechanics
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Full Text: Euclid