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S. I. Agafonov

Publications and source records attributed to S. I. Agafonov.

7 recordsLinked to original sources
arXiv2016

Quasilinear systems with linearizable characteristic webs

We classify quasilinear systems in Riemann invariants whose characteristic webs are linearizable on every solution. Although the linearizability of an individual web is a rather nontrivial differential constraint, the requirement of linearizability of characteristic webs on all solutions imposes simple second-order constraints for the characteristic speeds of the system. It is demonstrated that every such system with n>3 components can be transformed by a reciprocal transformation to n uncoupled Hopf equations. All our considerations are local.

math.DG↗
arXiv2008

On implicit ODEs with hexagonal web of solutions

Solutions of an implicit ODE form a web. Already for cubic ODEs the 3-web of solutions has a nontrivial local invariant, namely the curvature form. Thus any local classification of implicit ODEs necessarily has functional moduli if no restriction on the class of ODEs is imposed. Here the most symmetric case of hexagonal 3-web of solutions is discussed, i.e. the curvature is supposed to vanish identically. A finite list of normal forms is established under some natural regularity assumptions. Geometrical meaning of these assumptions is that the surface, defined by ODE in the space of 1-jets, is smooth as well as the criminant, which is the critical set of this surface projection to the plane.

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↗
arXiv2003

Discrete $Z^γ$: embedded circle patterns with the combinatorics of the square grid and discrete Painlevé equations

A discrete analog of the holomorphic map $z^γ$ is studied. It is given by Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding circle patterns are embedded and described by special separatrix solutions of discrete Painlevé equations. Global properties of these solutions, as well as of the discrete $z^γ$, are established.

nlin.SI↗
arXiv2003

Discrete Riccati equation, hypergeometric functions and circle patterns of Schramm type

Square grid circle patterns with prescribed intersection angles, mimicking holomorphic maps z^a and log(z) are studied. It is shown that the corresponding circle patterns are imbedded and described by special separatrix solutions of discrete Painleve and Riccati equations. General solution of this Riccati equation is expressed in terms of the hypergeometric function. Global properties of these solutions, as well as of the discrete z^a and log(z), are established.

math.CV↗
arXiv2001

Systems of conservation laws of Temple class, equations of associativity and linear congruences in projective space

We propose a geometric correspondence between (a) linearly degenerate systems of conservation laws with rectilinear rarefaction curves and (b) congruences of lines in projective space whose developable surfaces are planar pencils of lines. We prove that in projective 4-space such congruences are necessarily linear. Based on the results of Castelnuovo, the classification of three-component systems is obtained, revealing a close relationship of the problem with projective geometry of the Veronese variety and the theory of associativity equations of two-dimensional topological field theory.

math.DG↗