Searcharxiv⌕ Search

arXiv subjects

Andrey E. Mironov

Publications and source records attributed to Andrey E. Mironov.

34 records · Page 2Linked to original sources
arXiv2016

Algebraic non-integrability of magnetic billiards

We consider billiard ball motion in a convex domain of the Euclidean plane bounded by a piece-wise smooth curve influenced by the constant magnetic field. We show that if there exists a polynomial in velocities integral of the magnetic billiard flow then every smooth piece $γ$ of the boundary must be algebraic and either is a circle or satisfies very strong restrictions. In particular in the case of ellipse it follows that magnetic billiard is algebraically not integrable for all magnitudes of the magnetic field. We conjecture that circle is the only integrable magnetic billiard not only in the algebraic sense, but for a broader meaning of integrability. We also introduce the model of Outer magnetic billiards. As an application of our method we prove analogous results on algebraically integrable Outer magnetic billiards.

math.DG↗
arXiv2016

Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane

We consider a convex curve $γ$ lying on the Sphere or Hyperbolic plane. We study the problem of existence of polynomial in velocities integrals for Birkhoff billiard inside the domain bounded by $γ$. We extend the result by S. Bolotin (1992) and get new obstructions on polynomial integrability in terms of the dual curve $Γ$. We follow a method which was introduced by S. Tabachnikov for Outer billiards in the plane and was applied later on in our recent paper to Birkhoff billiards with the help of a new the so called Angular billiard.

math.DG↗
arXiv2016

Angular Billiard and Algebraic Birkhoff conjecture

In this paper we introduce a new dynamical system which we call Angular billiard. It acts on the exterior points of a convex curve in Euclidean plane. In a neighborhood of the boundary curve this system turns out to be dual to the Birkhoff billiard. Using this system we get new results on algebraic Birkhoff conjecture on integrable billiards.

math.DG↗
arXiv2015

Spectral Curve of the Halphen Operator

The Halphen operator is a third-order operator of the form $$ L_3=\partial_x^3-g(g+2)\wp(x)\partial_x-\frac{1}{2}g(g+2)\wp'(x), $$ where $g\ne 2\,\mbox{mod(3)}$, the Weierstrass $\wp$-function satisfies the equation $$ (\wp'(x))^2=4\wp^3(x)-g_2\wp(x)-g_3. $$ In the equianharmonic case, i.e., $g_2=0$ the Halphen operator commutes with some ordinary differential operator $L_n$ of order $n\ne 0\,\mbox{mod(3)}.$ In this paper we find the spectral curve of the pair $L_3,L_n$.

math-ph↗
arXiv2015

One-point commuting difference operators of rank one

We consider one-point commuting difference operators of rank one. The coefficients of these operators depend on a functional parameter, shift operators being included only with positive degrees. We study these operators in the case of hyperelliptic spectral curve when the marked point coincides with the branch point. We construct examples of operators with polynomial and trigonometric coefficients. Moreover, difference operators with polynomial coefficients can be embedded in the differential ones with polynomial coefficients. This construction provides a new way of constructing commutative subalgebras in the first Weyl algebra.

math.AG↗
arXiv2015

Commuting ordinary differential operators with polynomial coefficients and automorphisms of the first Weyl algebra

In this paper we study rank two commuting ordinary differential operators with polynomial coefficients and the orbit space of the automorphisms group of the first Weyl algebra on such operators. We prove that for arbitrary fixed spectral curve of genus one the space of orbits is infinite. Moreover, we prove in this case that for for any $n\ge 1$ there is a pair of self-adjoint commuting ordinary differential operators of rank two $L_4=(\partial_x^2+V(x))^2+W(x)$, $L_{6}$, where $W(x),V(x)$ are polynomials of degree $n$ and $n+2$. We also prove that there are hyperelliptic spectral curves with the infinite spaces of orbits.

math-ph↗
arXiv2014

Difference Krichever-Novikov operators

In this paper we study commuting difference operators of rank two. We introduce an equation on potentials $V(n),W(n)$ of the difference operator $L_4=(T+V(n)T^{-1})^2+W(n)$ and some additional data. With the help of this equation we find the first examples of commuting difference operators of rank two corresponding to spectral curves of higher genus.

nlin.SI↗
arXiv2014

Integrable geodesic flows on 2-torus: formal solutions and variational principle

In this paper we study quasi-linear system of partial differential equations which describes the existence of the polynomial in momenta first integral of the integrable geodesic flow on 2-torus. We proved in [3] that this is a semi-Hamiltonian system and we show here that the metric associated with the system is a metric of Egorov type. We use this fact in order to prove that in the case of integrals of degree three and four the system is in fact equivalent to a single remarkable equation of order 3 and 4 respectively. Remarkably the equation for the case of degree four has variational meaning: it is Euler-Lagrange equation of a variational principle. Next we prove that this equation for $n=4$ has formal double periodic solutions as a series in a small parameter.

math.DG↗
arXiv2013

Periodic and rapid decay rank two self-adjoint commuting differential operators

Self-adjoint rank two commuting ordinary differential operators are studied in this paper. Such operators with trigonometric, elliptic and rapid decay coefficients corresponding to hyperelliptic spectral curves are constructed. Some problems related to the Lame operator and rank two solutions of soliton equations are discussed.

math-ph↗
arXiv2012

Self-adjoint commuting differential operators and commutative subalgebras of the Weyl algebra

In this paper we study self-adjoint commuting ordinary differential operators. We find sufficient conditions when an operator of fourth order commuting with an operator of order $4g+2$ is self-adjoint. We introduce an equation on coefficients of the self-adjoint operator of order four and some additional data. With the help of this equation we find the first example of commuting differential operators of rank two corresponding to a spectral curve of arbitrary genus. These operators have polynomial coefficients and define commutative subalgebras of the first Weyl algebra.

math-ph↗
arXiv2010

Baker-Akhiezer Modules on Rational Varieties

The free Baker-Akhiezer modules on rational varieties obtained from ${\mathbb C}P^{1}\times{\mathbb C}P^{n-1}$ by identification of two hypersurfaces are constructed. The corollary of this construction is the existence of embedding of meromorphic function ring with some fixed pole into the ring of matrix differential operators in $n$ variables.

math.AG↗
arXiv2009

Rich quasi-linear system for integrable geodesic flows on 2-torus

Consider a Riemannian metric on two-torus. We prove that the question of existence of polynomial first integrals leads naturally to a remarkable system of quasi-linear equations which turns out to be a Rich system of conservation laws. This reduces the question of integrability to the question of existence of smooth (quasi-) periodic solutions for this Rich quasi-linear system.

math.DG↗