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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)

- Evan VanderZee, Anil N. Hirani, Vadim Zharnitsky, Damrong Guoy
- Comput. Geom.
- 2010

It is shown that there exists a dihedral acute triangulation of the three-dimensional cube. The method of constructing the acute triangulation is described, and symmetries of the triangulation are discussed.

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)

- Maxim Arnold, Vadim Zharnitsky
- The American Mathematical Monthly
- 2015

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)

It is shown that a large class of solutions in two-degree-of-freedom Hamiltonian systems of billiard type can be described by slowly varying one-degree-of-freedom Hamiltonian systems. Under some non-degeneracy conditions such systems are found to possess a large set of quasiperiodic solutions filling out two dimensional tori, which correspond to caustics in… (More)

- Yuliy Baryshnikov, Pascal Heider, Wolfgang Parz, Vadim Zharnitsky
- Physical review letters
- 2004

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)

- Vadim Zharnitsky
- 2000

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)

- Vadim Zharnitsky
- 1997

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)

It is shown that the set of 4-period orbits in outer billiard with piecewise smooth convex boundary has an empty interior, provided the boundary is not a parallelogram.