Vadim Zharnitsky

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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)
In this note, we present a simple proof that three bugs involved in cyclic evasion converge to an equilateral triangle configuration. The approach relies on an energy-type estimate that 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(More)
An exact pulse for the parametrically forced nonlinear Schrödinger equation ~NLS! is isolated. The equation governs wave envelope propagation in dispersion-managed fiber lines with positive residual dispersion. The pulse is obtained as a ground state of an averaged variational principle associated with the equation governing pulse dynamics. The solutions of(More)
In this paper the monotonic twist theorem is extended to the quasiperiodic case and applied to establish regularity of motion in a system of a particle bouncing elastically between two quasiperiodically moving walls. It is shown that the velocity of the particle is uniformly bounded in time if the frequencies satisfy a Diophantine inequality. This answers a(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)
The motion of a classical particle bouncing elastically between two parallel walls, with one of the walls undergoing a periodic motion is considered. This problem, called Fermi–Ulam ‘ping-pong’, is known to possess only bounded solutions if the motion of the wall is sufficiently smooth p(t) ∈ C4+ , where p(t) is the position of the wall. It is shown that(More)
Applying asymptotic methods to a previously derived system of ordinary differential equations, we present an analytical description of the slow ~average! dynamics of self-similar breathing pulses propagating in fiber links with dispersion management. We derive asymptotic averaged quantities ~adiabatic invariants! that characterize the stable pulse(More)