Vadim Zharnitsky

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A cubic nonlinear Schrödinger equation (NLS) with periodically varying dispersion coefficient, as it arises in the context of fiber-optics communication, is considered. For sufficiently strong variation, corresponding to the so-called strong dispersion management regime, the equation possesses pulse-like solutions which evolve nearly periodically. This(More)
We consider existence and stability of dispersion-managed solitons in the two approximations of the periodic nonlinear Schrödinger (NLS) equation: (i) a dynamical system for a Gaussian pulse and (ii) an average integral NLS equation. We apply normal form transformations for finite-dimensional and infinite-dimensional Hamiltonian systems with periodic(More)
An n-simplex is said to be n-well-centered if its circumcenter lies in its interior. We introduce several other geometric conditions and an algebraic condition that can be used to determine whether a simplex is n-well-centered. These conditions, together with some other observations, are used to describe restrictions on the local combinatorial structure of(More)
The evolution of optical pulses in fiber optic communication systems with strong, higher order dispersion management is modeled by a cubic nonlinear Schrödinger equation with periodically varying linear dispersion at second and third order. Through an averaging procedure, we derive an approximate model for the slow evolution of such pulses, and show that(More)
In this note we present a simple proof that 3 bugs involved in cyclic evasion, converge to an equilateral triangle configuration. The approach relies on energy type estimate which makes use of a new inequality for the triangle. The problem of the cyclic pursuit or n−bug problem is a classical one, see e.g. an article by Klamkin and Newman, " cyclic pursuit(More)
A family of discontinuous symplectic maps arising naturally in the study of non-smooth switched Hamiltonian systems is considered. This family depends on two parameters and is a canonical model for the study of bounded and unbounded behavior in discontinuous area-preserving transformations due to nonlinear resonances. This paper provides a general(More)
Two dimensional resonators with a smooth strictly convex boundary are known to possess a whispering gallery region supporting modes concentrated near the boundary. A new class of asymmetric resonant cavities is introduced, where a whispering gallery-like region is found deep inside the resonator. The construction of such resonators is a novel application of(More)
Classical (Birkhoff) billiards with full 1-parameter families of periodic orbits are considered. It is shown that construction of a convex billiard with a " rational " caustic (i.e. carrying only periodic orbits) can be reformulated as the problem of finding a closed curve tangent to a non-integrable distribution on a manifold. The properties of this(More)