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)
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)
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)
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)