We discuss various dualities, relating integrable systems and show that these dualities are explained in the framework of Hamiltonian and Poisson reductions. The dualities we study shed some light on the known integrable systems as well as allow to construct new ones, double elliptic among them. We also discuss applications to the (supersymmetric) gauge theories in various dimensions.

We present approximate analytical solutions to the Hamiltonian and momentum constraint equations, corresponding to systems composed of two black holes with arbitrary linear and angular momentum. The analytical nature of these initial data solutions makes them easier to implement in numerical evolutions than the traditional numerical approach of solving the elliptic equations derived from the Ei...

In this paper we study superlinear elliptic systems in Hamiltonian form. Using an Orlicz-space setting, we extend the notion of critical growth to superlinear nonlinearities which do not have a polynomial growth. Existence of nontrivial solutions is proved for superlinear nonlinearities which are subcritical in this generalized sense.

It is proved that periodic and homoclinic trajectories which are tangent to the boundary of any scattering (ergodic) billiard produce elliptic islands in thènearby' Hamiltonian ows; i.e., in a family of two-degrees-of-freedom smooth Hamiltonian ows which converge to the singular billiard ow smoothly where the billiard ow is smooth and continuously where it is continuous. Such Hamiltonians exist...

It has been long observed that area-preserving maps and reversible maps share similar results. This was certainly known to G.D. Birkhoff [5] who showed that these two types of maps have periodic orbits near a general elliptic fixed point. The KAM theory, developed by Kolmogorov-ArnoldMoser for Hamiltonian systems [9], [1] and area preserving maps [15], has also been extended a great deal to rev...

Abs t rac t , Sufficient conditions are found so that a family of smooth Hamiltonian flows limits to a billiard flow as a parameter e --~ 0. This limit is proved to be C 1 near non-singular orbits and C o near orbits tangent to the billiard boundary. These results are used to prove that scattering (thus ergodic) billiards with tangent periodic orbits or tangent homoclinic orbits produce nearby ...

We discuss various dualities, relating integrable systems and show that these dualities are explained in the framework of Hamiltonian and Poisson reductions. The dualities we study shed some light on the known integrable systems as well as allow to construct new ones, double elliptic among them. We also discuss applications to the (supersymmetric) gauge theories in various dimensions.

We discuss various dualities, relating integrable systems and show that these dualities are explained in the framework of Hamiltonian and Poisson reductions. The dualities we study shed some light on the known integrable systems as well as allow to construct new ones, double elliptic among them. We also discuss applications to the (supersymmetric) gauge theories in various dimensions.