arXiv subjects

A. I. Bobenko

Publications and source records attributed to A. I. Bobenko.

11 recordsLinked to original sources

arXiv2014

Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems

Pluri-Lagrangian systems are variational systems with the multi-dimensional consistency property. This notion has its roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics, in the theory of variational symmetries going back to Noether and in the theory of discrete integrable systems. A $d$-dimensional pluri-Lagrangian problem can be described as follows: given a $d$-form $L$ on an $m$-dimensional space, $m > d$, whose coefficients depend on a function $u$ of $m$ independent variables (called field), find those fields $u$ which deliver critical points to the action functionals $S_Σ=\int_ΣL$ for any $d$-dimensional manifold $Σ$ in the $m$-dimensional space. We investigate discrete 2-dimensional linear pluri-Lagrangian systems, i.e. those with quadratic Lagrangians $L$. The action is a discrete analogue of the Dirichlet energy, and solutions are called discrete pluriharmonic functions. We classify linear pluri-Lagrangian systems with Lagrangians depending on diagonals. They are described by generalizations of the star-triangle map. Examples of more general quadratic Lagrangians are also considered.

math-ph↗

arXiv2007

Curvature line parametrization from circle patterns

We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with infinitesimal quads. It is shown that choosing the points of discrete nets on the smooth surface one can obtain second-order approximation globally. Also a simple geometric construction for approximate determination of principal directions of smooth surfaces is given.

math.DG↗

arXiv2003

Geometry of Yang--Baxter maps: pencils of conics and quadrirational mappings

Birational Yang-Baxter maps (`set-theoretical solutions of the Yang-Baxter equation') are considered. A birational map $(x,y)\mapsto(u,v)$ is called quadrirational, if its graph is also a graph of a birational map $(x,v)\mapsto(u,y)$. We obtain a classification of quadrirational maps on $\CP^1\times\CP^1$, and show that all of them satisfy the Yang-Baxter equation. These maps possess a nice geometric interpretation in terms of linear pencil of conics, the Yang-Baxter property being interpreted as a new incidence theorem of the projective geometry of conics.

math.QA↗

arXiv2003

Discrete and smooth orthogonal systems: $C^\infty$-approximation

Discrete conjugate systems are quadrilateral nets with all planar faces. Discrete orthogonal systems are defined by the additional property of all faces being concircular. Their geometric properties allow one to consider them as proper discretization of conjugate, resp. orthogonal coordinate systems of classical differential geometry. We develop techniques that allow us to extend this known qualitative analogy to rigorous convergence results. In particular, we prove the $C^\infty$-convergence of discrete conjugate/orthogonal coordinate systems to smooth ones. We also show how to construct the approximating discrete nets. Coordinate systems and their transformations are treated on an equal footing, and the approximation results hold for transformations as well.

math.DG↗

arXiv2003

Hexagonal circle patterns with constant intersection angles and discrete Painleve and Riccati equations

Hexagonal circle patterns with constant intersection angles mimicking holomorphic maps z^c and log(z) are studied. It is shown that the corresponding circle patterns are immersed and described by special separatrix solutions of discrete Painleve and Riccati equations. The general solution of the Riccati equation is expressed in terms of the hypergeometric function. Global properties of these solutions, as well as of the discrete z^c and log(z), are established.

math.CV↗

arXiv2002

Nonlinear hyperbolic equations in surface theory: integrable discretizations and approximation results

A numerical scheme is developed for solution of the Goursat problem for a class of nonlinear hyperbolic systems with an arbitrary number of independent variables. Convergence results are proved for this difference scheme. These results are applied to hyperbolic systems of differential-geometric origin, like the sine-Gordon equation describing the surfaces of the constant negative Gaussian curvature (K-surfaces). In particular, we prove the convergence of discrete K--surfaces and their Backlund transformations to their continuous counterparts. This puts on a firm basis the generally accepted belief (which however remained unproved untill this work) that the classical differential geometry of integrable classes of surfaces and the classical theory of transformations of such surfaces may be obtained from a unifying multi-dimensional discrete theory by a refinement of the coordinate mesh-size in some of the directions

math.NA↗

arXiv2002

Classification of integrable equations on quad-graphs. The consistency approach

A classification of discrete integrable systems on quad-graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three-dimensional consistency. This property yields, among other features, the existence of the discrete zero curvature with a spectral parameter. For all integrable systems of the obtained exhaustive list, the so called three-leg forms are found. This establishes Lagrangian and symplectic structures for these systems, and the connection to discrete systems of the Toda type on arbitrary graphs. Generalizations of these ideas to the three-dimensional integrable systems and to the quantum context are also discussed.

nlin.SI↗

arXiv2002

Integrable non-commutative equations on quad-graphs. The consistency approach

We extend integrable systems on quad-graphs, such as the Hirota equation and the cross-ratio equation, to the non-commutative context, when the fields take values in an arbitrary associative algebra. We demonstrate that the three-dimensional consistency property remains valid in this case. We derive the non-commutative zero curvature representations for these systems, based on the latter property. Quantum systems with their quantum zero curvature representations are particular cases of the general non-commutative ones.

nlin.SI↗

arXiv2001

Hexagonal circle patterns and integrable systems: Patterns with the multi-ratio property and Lax equations on the regular triangular lattice

Hexagonal circle patterns are introduced, and a subclass thereof is studied in detail. It is characterized by the following property: For every circle the multi-ratio of its six intersection points with neighboring circles is equal to -1. The relation of such patterns with an integrable system on the regular triangular lattice is established. A kind of a B"acklund transformation for circle patterns is studied. Further, a class of isomonodromic solutions of the aforementioned integrable system is introduced, including circle patterns analogons to the analytic functions $z^α$ and $\log z$.

math.CV↗

arXiv1998

Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top

We develop the theory of discrete time Lagrangian mechanics on Lie groups, originated in the work of Veselov and Moser, and the theory of Lagrangian reduction in the discrete time setting. The results thus obtained are applied to the investigation of an integrable time discretization of a famous integrable system of classical mechanics, -- the Lagrange top. We recall the derivation of the Euler--Poinsot equations of motion both in the frame moving with the body and in the rest frame (the latter ones being less widely known). We find a discrete time Lagrange function turning into the known continuous time Lagrangian in the continuous limit, and elaborate both descriptions of the resulting discrete time system, namely in the body frame and in the rest frame. This system naturally inherits Poisson properties of the continuous time system, the integrals of motion being deformed. The discrete time Lax representations are also found. Kirchhoff's kinetic analogy between elastic curves and motions of the Lagrange top is also generalised to the discrete context.

solv-int↗

arXiv1998

Integrable discretizations of the Euler top

Discretizations of the Euler top sharing the integrals of motion with the continuous time system are studied. Those of them which are also Poisson with respect to the invariant Poisson bracket of the Euler top are characterized. For all these Poisson discretizations a solution in terms of elliptic functions is found, allowing a direct comparison with the continuous time case. We demonstrate that the Veselov--Moser discretization also belongs to our family, and apply our methods to this particular example.

solv-int↗