Misha Bialy

Cubic and quartic integrals for geodesic flow on 2-torus via a system of the hydrodynamic type

In this paper, we deal with the classical question of the existence of polynomials in momenta integrals for geodesic flows on the 2-torus. For the quasilinear system on the coefficients of the polynomial integral, we investigate the region (so-called elliptic region) where two of the eigenvalues are complex conjugate. We show that for quartic integrals the other two eigenvalues are real and necessarily genuinely nonlinear. This observation, together with the property of the system to be rich (semi-Hamiltonian), enables us to classify elliptic regions completely. We prove that on these regions the integral is always reducible. The case of complex-conjugate eigenvalues for the system corresponding to the integral of degree 3 is done similarly. These results show that if new integrable examples exist, they can be found only within the region of hyperbolicity of the quasilinear system.

Symplectic twist maps without conjugate points

For sequences of symplectic twist maps without conjugate points, an invariant Lagrangian subbundle is constructed. This allows one to deduce that absence of conjugate points is a rare property in some classes of maps.

Polynomial integrals for a Hamiltonian system and breakdown of smooth solutions for quasi-linear equations

The purpose of this paper is to relate the non-existence of polynomial integrals for a Hamiltonian system to the breakdown phenomenon of smooth solutions in quasi-linear equations. Using this relation it is shown that for the classical Hamiltonian system with 1.5 degrees of freedom there are no non-trivial third power integrals of motion. The main tool used in the proof is the Lax analysis on formation of singularities in quasi-linear equations. Some results and perspectives for the case of higher degrees are discussed.

Shock formation for the forced Burgers equation and an application

Abstract. We study the inviscid Burgers equation in the presence of spatially periodic non-zero potential force. We prove that for the foliated initial value problem there are always solutions developing shocks in a finite time. We give an application of this result to a quasi-linear system of conservation laws which appeared in the study of integrable Hamiltonian systems with 1.5 degrees of freedom.

Hamiltonian form and infinitely many conservation laws for a quasilinear system

In the present paper we show the closed relation between the existence of smooth periodic solutions for a certain quasilinear system and the existence question for the integral of motion of a classical Hamiltonian system with 1.5 degrees of freedom. It turns out that this quasilinear system has infinitely many additional conservation laws and moreover can be written in the Hamiltonian form of the hydrodynamic type. There is the hope that these results will help to classify those classical Hamiltonian systems with 1.5 degrees of freedom which are completely integrable.

Nonsmooth convex caustics for Birkhoff billiards

This paper is devoted to the examination of the properties of the string construction for the Birkhoff billiard. Based on purely geometric considerations, string construction is suited to provide a table for the Birkhoff billiard, having the prescribed caustic. Exploiting this framework together with the properties of convex caustics, we give a geometric proof of a result by Innami first proved in 2002 by means of Aubry-Mather theory. In the second part of the paper we show that applying the string construction one can find a new collection of examples of

Variational properties of a nonlinear elliptic equation and rigidity

C^2

In search of periodic solutions for a reduction of the Benney chain

-smooth convex billiard tables with a non-smooth convex caustic.

Geodesics of Hofer’s metric on the group of Hamiltonian diffeomorphisms

We consider in this paper elliptic equations which are perturbations of Laplaces equation by a compactly supported potential. We show that in dimension greater than three for a wide class of potentials all the solutions are globally minimising. However, in dimension two the situation is different. We show that for radially symmetric potentials there always exist solutions which are not locally minimal unless the potential vanishes identically. We discuss the relations of this with the so-called Hopf rigidity phenomenon.

Convex billiards and a theorem by E. Hopf

We search for smooth periodic solutions for the system of quasi-linear equations 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 × 2 reduction which is strictly hyperbolic with one umbilic point but violates the condition of genuine non-linearity.