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

  title={Classification of integrable Weingarten surfaces possessing an \mathfrak\{sl\} (2)-valued zero curvature representation},
  author={H. Baran and M. Marvan},
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

Figures and Tables from this paper

Soliton surfaces and generalized symmetries of integrable systems
In this paper, we discuss some specific features of symmetries of integrable systems which can be used to contruct the Fokas-Gel'fand formula for the immersion of 2D-soliton surfaces, associated withExpand
Soliton surfaces associated with generalized symmetries of integrable equations
In this paper, based on the Fokas et al approach (Fokas and Gel'fand 1996 Commun. Math. Phys. 177 203–20; Fokas et al 2000 Sel. Math. 6 347–75), we provide a symmetry characterization of continuousExpand
Soliton surfaces via a zero-curvature representation of differential equations
The main aim of this paper is to introduce a new version of the Fokas–Gel’fand formula for immersion of soliton surfaces in Lie algebras. The paper contains a detailed exposition of the technique forExpand
On the invariant theory of Weingarten surfaces in Euclidean space
On any Weingarten surface in Euclidean space (strongly regular or rotational), we introduce locally geometric principal parameters and prove that such a surface is determined uniquely up to a motionExpand
Soliton surfaces associated with CP^{N-1} sigma models
Soliton surfaces associated with CP^{N-1} sigma models are constructed using the Generalized Weierstrass and the Fokas-Gel'fand formulas for immersion of 2D surfaces in Lie algebras. The consideredExpand
Space-like Weingarten surfaces in the three-dimensional Minkowski space and their natural partial differential equations
On any space-like Weingarten surface in the three-dimensional Minkowski space we introduce locally natural principal parameters and prove that such a surface is determined uniquely up to motion by aExpand
Lax Triples for Integrable Surfaces in Three-Dimensional Space
We study Lax triples (i.e., Lax representations consisting of three linear equations) associated with families of surfaces immersed in three-dimensional Euclidean space . We begin with a naturalExpand
Evolving to Non-round Weingarten Spheres: Integer Linear Hopf Flows
In the 1950's Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper weExpand
Integrable dispersive chains and energy dependent Schrödinger operator
In this paper we consider integrable dispersive chains associated with the so-called ‘energy dependent’ Schrodinger operator. In a general case multi-component reductions of these dispersive chainsExpand
Lie symmetry analysis, conservation laws and analytical solutions for the constant astigmatism equation
Abstract In this paper, the constant astigmatism equation is investigated, which finds numerous applications in geometry and physics describing surfaces of constant astigmatism. Utilizing a set ofExpand


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