Classification of integrable Weingarten surfaces possessing an \mathfrak{sl} (2)-valued zero curvature representation

@article{Baran2010ClassificationOI,
  title={Classification of integrable Weingarten surfaces possessing an \mathfrak\{sl\} (2)-valued zero curvature representation},
  author={H. Baran and M. Marvan},
  journal={Nonlinearity},
  year={2010},
  volume={23},
  pages={2577-2597}
}
In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an (2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary differential equation to govern the functional dependence of the principal curvatures. Employing the scaling and translation (offsetting) symmetry, we give a general solution of the… Expand

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References

SHOWING 1-10 OF 47 REFERENCES
A generalized formula for integrable classes of surfaces in Lie algebras
We discuss relations between the approach of Fokas and Gelfand to immersions on Lie algebras and the theory of soliton surfaces of Sym. We show that many results concerning immersions on Lie algebrasExpand
On integrability of Weingarten surfaces: a forgotten class
Rediscovered by a systematic search, a forgotten class of integrable surfaces is shown to disprove the Finkel–Wu conjecture. The associated integrable nonlinear partial differential equationExpand
Integrable systems, harmonic maps and the classical theory of surfaces
Many geometers in the 19th and early 20th century studied surfaces in R 3 with particular conditions on the curvature. Examples include minimal surfaces, surfaces of constant mean curvature andExpand
A formula for constructing infinitely many surfaces on Lie algebras and integrable equations
Abstract. Surfaces immersed in Lie algebras can be characterized by the so called fundamental forms. The coefficients of these forms satisfy a system of nonlinear partial differential equationsExpand
Surfaces on Lie groups, on Lie algebras, and their integrability
It is shown that the problem of the immersion of a 2-dimensional surface into a 3-dimensional Euclidean space, as well as then-dimensional generalization of this problem, is related to the problem ofExpand
Weingarten surfaces and nonlinear partial differential equations
The sine-Gordon equation has been known for a long time as the equation satisfied by the angle between the two asymptotic lines on a surface inR3 with constant Gauss curvature −1. In this paper, weExpand
On Closed Weingarten Surfaces
Abstract.We investigate closed surfaces in Euclidean 3-space satisfying certain functional relations κ = F(λ) between the principal curvatures κ, λ. In particular we find analytic closed surfaces ofExpand
Linear Weingarten Surfaces in ℝ3
Abstract. In this paper we study properties of linear Weingarten immersions and graphs related to non-existence problems and behaviour of its curvatures. The main results are obtained giving aExpand
Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory
Preface Acknowledgements General introduction and outline 1. Pseudospherical surfaces and the classical Backlund transformation: the Bianchi system 2. The motion of curves and surfaces. solitonExpand
Soliton surfaces and their applications (soliton geometry from spectral problems)
The paper contains a complete presentation of the ideas and results of the approach of soliton surfaces (manifolds). In this approach any n-dim. soliton system with a matrix real semi-simple LieExpand
...
1
2
3
4
5
...