arXiv research

Using the symmetry group theory of second order PDEs, one finds the symmetry group associated to Tzitzeica surfaces partial differential equation. One studies the inverse problem and one shows that the Tzitzeica surfaces PDE is an Euler-Lagrange equation. One determines the variational symmetry group of the associated …

We classify the simplest rational elements in a twisted loop group, and prove that dressing actions of them on proper indefinite affine spheres give the classical Tzitzéica transformation and its dual. We also give the group point of view of the Permutability Theorem, construct complex Tzitzéica transformations, and di…

It is shown how solutions to the Tzitzéica equation can be used to construct a family of (pseudo) hyper-complex metrics in four dimensions.

One applies the symmetry group theory for study the partial differential equations of Tzitzeica surfaces theory. One finds infinitesimal symmetries, Lagrangians and a new solution of Titzeica equation.

A new hypersurface of Tzitzeica type is obtained in all three forms: parametric, implicit and explicit. Its two-dimensional version, although well-known from a theoretical point of view, is plotted with Matlab.

A discrete analog of the Tzitzeica equation is found in the form of quad-equation. Its continuous symmetry is an inhomogeneous Narita--Bogoyavlensky type lattice equation which defines a discretization of the Sawada--Kotera equation. The integrability of these discretizations is proven by construction of the Lax repres…

In this paper we show that all totally real superconformal minimal tori in $CP^2$ correspond with doubly-periodic finite gap solutions of the Tzitzeica equation $ω_{z{\overline z}}=e^{-2ω}-e^ω$ Using the results on the Tzitzeica equation in integrable system theory, we describe explicitly all these tori by Prym-theta…

It is shown that both the sinh--Gordon equation and the elliptic Tzitzeica equation can be interpreted as the Taubes equation for Abelian vortices on a CMC surface embedded in $\R^{2, 1}$, or on a surface conformally related to a hyperbolic affine sphere in $\R^3$. In both cases the Higgs field and the U(1) vortex conn…

Vortices on conical surfaces embedded in hyperbolic space.

We use an elliptic differential equation of Tzitzeica type to construct a minimal Lagrangian surface in CH2 from the data of a compact hyperbolic Riemann surface and a small holomorphic cubic differential. The minimal Lagrangian surface is invariant under an SU(2,1) action of the fundamental group. We further parameter…

The paper studies steady motions of fibre-reinforced fluids on curved surfaces.

We give a gauge invariant characterisation of the elliptic affine sphere equation and the closely related Tzitzéica equation as reductions of real forms of $SL(3, \C)$ anti--self--dual Yang--Mills equations by two translations, or equivalently as a special case of the Hitchin equation. We use the Loftin--Yau--Zaslow co…

This is an elementary and self--contained review of twistor theory as a geometric tool for solving non-linear differential equations. Solutions to soliton equations like KdV, Tzitzeica, integrable chiral model, BPS monopole or Sine-Gordon arise from holomorphic vector bundles over $T\CP^1$. A different framework is pro…

Paper solves a class of differential equations with specific solutions.

New method constructs translationally equivariant hyperbolic affine spheres.

The aim of this paper is to offer an overview of the most important applications of Jordan structures inside mathematics and also to physics, up-dated references being included. For a more detailed treatment of this topic see - especially - the recent book Iordanescu [364w], where sugestions for further developments ar…

We present a connection between the Killing fields that arise in the loop-group approach to integrable systems and conservation laws viewed as elements of the characteristic cohomology. We use the connection to generate the complete set of conservation laws (as elements of the characteristic cohomology) for the Tzitzei…

We present the first steps of a procedure which discretises surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adop…

We propose a natural discretisation scheme for classical projective minimal surfaces. We follow the classical geometric characterisation and classification of projective minimal surfaces and introduce at each step canonical discrete models of the associated geometric notions and objects. Thus, we introduce discrete ana…

Holomorphic map connects Hitchin components to character varieties.

This is the sixth in a series of papers constructing examples of special Lagrangian m-folds in C^m. We present a construction of special Lagrangian cones in C^3 involving two commuting o.d.e.s, motivated by the first two papers of the series. Then we generalize it to a construction of non-conical special Lagrangian 3-f…

The paper develops methods to derive mixed superposition rules for Lie systems and applies them to various physical systems.