1,928 research outputs found
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 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
Finite-gap minimal Lagrangian surfaces in
In this paper we suggest a method for constructing minimal Lagrangian
immersions of in with induced diagonal metric in terms of
Baker-Akhiezer functions of algebraic curves.Comment: 12 pages, 1 figur
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.Comment: 19 pages, 12 figure
New Semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces
We consider magnetic geodesic flows on the 2-torus. We prove that the
question of existence of polynomial in momenta first integrals on one energy
level leads to a Semi-Hamiltonian system of quasi-linear equations, i.e. in the
hyperbolic regions the system has Riemann invariants and can be written in
conservation laws form
Hamiltonian minimal Lagrangian submanifolds in toric varieties
Hamiltonian minimality (H-minimality) for Lagrangian submanifolds is a
symplectic analogue of Riemannian minimality. A Lagrangian submanifold is
called H-minimal if the variations of its volume along all Hamiltonian vector
fields are zero. This notion was introduced in the work of Y.-G. Oh in
connection with the celebrated Arnold conjecture on the number of fixed points
of a Hamiltonian symplectomorphism. In the previous works the authors defined
and studied a family of H-minimal Lagrangian submanifolds in complex space
arising from intersections of Hermitian quadrics. Here we extend this
construction to define H-minimal submanifolds in toric varieties.Comment: 2 pages, minor changes in version
In search of periodic solutions for a reduction of the Benney chain
We search for smooth periodic solutions for the system of quasi-linear PDEs
known as the Lax dispersionless reduction of the Benney moments chain. It is
naturally related to the existence of a polynomial in momenta integral for a
Classical Hamiltonian system with 1,5 degrees of freedom. For the solution in
question it is not known a-priori if the system is elliptic or hyperbolic or of
mixed type. We consider two possible regimes for the solution. The first is the
case of only one real eigenvalue, where we can completely classify the
solutions. The second case of strict Hyperbolicity is really a challenge. We
find a remarkable 2 by 2 reduction which is strictly Hyperbolic but violates
the condition of genuine non-linearity.Comment: 10
Discretization of Baker-Akhiezer Modules and Commuting Difference Operators in Several Discrete Variables
We introduce the notion of discrete Baker-Akhiezer (DBA) modules, which are
modules over the ring of difference operators, as a certain discretization of
Baker-Akhiezer modules which are modules over the ring of differential
operators. We use it to construct commuting difference operators with matrix
coefficients in several discrete variables.Comment: 23 page
CFT exercises for the needs of AGT
An explicit check of the AGT relation between the W_N-symmetry controlled
conformal blocks and U(N) Nekrasov functions requires knowledge of the
Shapovalov matrix and various triple correlators for W-algebra descendants. We
collect simplest expressions of this type for N=3 and for the two lowest
descendant levels, together with the detailed derivations, which can be now
computerized and used in more general studies of conformal blocks and AGT
relations at higher levels.Comment: 29 page
Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane
We consider a convex curve 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 . We extend the result
by S. Bolotin (1992) and get new obstructions on polynomial integrability in
terms of the dual curve . 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.Comment: 10
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 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
- …