Applying asymptotic methods to a previously derived system of ordinary differential equations, we present an analytical description the slow (average) dynamics self-similar breathing pulses propagating in fiber links with dispersion management. We derive averaged quantities (adiabatic invariants) that characterize stable pulse propagation. In particular, but practically important, case when compensation period is much larger than amplification distance have found analytically fixed points...

We revisit the averaged equation, derived in Pelinovsky et al. [Phys. Rev. Lett. 91, 240201 (2003)] from nonlinear Schr\"odinger (NLS) equation with nonlinearity management. show that this is valid only at initial time interval, while a new Hamiltonian NLS can be used longer intervals. Using we construct numerically matter-wave solitons context of Bose-Einstein condensates under Feshbach resonance there no threshold on existence dark large amplitudes, whereas such exists for bright solitons.

In this paper the monotonic twist theorem is extended to quasiperiodic case and applied establish regularity of motion in a system particle bouncing elastically between two quasiperiodically moving walls. It shown that velocity uniformly bounded time if frequencies satisfy Diophantine inequality. This answers question recently asked Levi Zehnder (1995 SIAM J. Math. Anal. 26 1233-56).

Classical (Birkhoff) billiards with full 1-parameter families of periodic orbits are considered.It is shown that construction a convex billiard "rational" caustic (i.e.carrying only ) can be reformulated as the problem finding closed curve tangent to non-integrable distribution on manifold.The properties this described well consequences for rational caustics.A particular implication an ellipse infinitesimally perturbed so any chosen elliptic will persist.

Two dimensional resonators with a smooth strictly convex boundary are known to possess whispering gallery region supporting modes concentrated near the boundary. A new class of asymmetric resonant cavities is introduced, where gallery-like found deep inside resonator. The construction such novel application geometric control methods. results numerical simulations and experiments presented.

The motion of a classical particle bouncing elastically between two parallel walls, with one the walls undergoing periodic is considered. This problem, called Fermi-Ulam `ping-pong', known to possess only bounded solutions if wall sufficiently smooth , where p(t) position wall. It shown that stability result does not hold just continuous function by providing examples instability. second example also answers question posed in Levi M and Zehnder E (1995 Boundedness for quasiperiodic...

We consider the nonlinear Schrodinger equation with nonlinearity management which describes Bose-Einstein condensates under Feshbach resonance. By using an averaging theory, we derive Hamiltonian averaged and compare it other methods developed for this problem. The is used analytical approximations of nonlinearity-managed solitons.

We consider existence and stability of dispersion-managed solitons in the two approximations periodic nonlinear Schrödinger (NLS) equation: (i) a dynamical system for Gaussian pulse (ii) an average integral NLS equation. apply normal form transformations finite-dimensional infinite-dimensional Hamiltonian systems with coefficients. First-order corrections to leading-order averaged are derived explicitly both approximations. Bifurcations soliton solutions their stabilityare studied by...

An approach due to Wojtkovski [15], based on the Jacobi fields, is applied study sets of three-period orbits in billiards hyperbolic plane and two-dimensional sphere. It found that set plane, as planar case, has zero measure. For sphere, a new proof Baryshnikov's theorem obtained states can form positive measure if only certain natural condition orbit length satisfied.

A parametrically forced sine-Gordon equation with a fast periodic mean-zero forcing is considered. It shown that $\ensuremath{\pi}$ kinks represent class of solitary-wave solutions the equation. This result applied to quasi-one-dimensional ferromagnets an easy-plane anisotropy, in rapidly oscillating magnetic field. In this case $\ensuremath{\pi}$-kink solution we have introduced corresponds uniform ``true'' domain-wall motion, since magnetization directions on opposite sides wall are...

An exact pulse for the parametrically forced nonlinear Schrodinger equation (NLS) is isolated. The governs wave envelope propagation in dispersion-managed fiber lines with positive residual dispersion. obtained as a ground state of an averaged variational principle associated governing dynamics. solutions and original equations are shown to stay close sufficiently long time. A properly adjusted will therefore exhibit nearly periodic behavior time interval validity averaging procedure....

We study asymptotic stability properties of nonlinear systems in the presence "almost Lyapunov" functions which decrease along solutions a given region not everywhere but rather on complement set small volume. Nothing specific about structure this is assumed besides an upper bound its show that starting inside approach around origin whose volume depends where Lyapunov function does decrease, as well other system parameters. The result established by perturbation argument compares trajectory...

We study convergence properties of nonlinear systems in the presence "almost Lyapunov" functions which decrease along solutions a given region not everywhere but rather on complement set small volume. The structure is quite general except that system dynamics never vanishes away from equilibrium. It shown starting inside will approach around origin as long volume where Lyapunov function does fast enough sufficiently small. main theorem this paper established by tracking change value when...

We demonstrate that $\ensuremath{\pi}$ kinks exist in nonparametrically ac driven sine-Gordon systems if the drive is sufficiently fast. It found that, at a critical value of amplitude, there are two stable and unstable equilibria phase. The pairwise symmetry these implies existence one-parameter family $\ensuremath{\pi}$-kink solutions reduced system. In dissipative case systems, corresponding to Josephson junctions, velocity selected by balance between perturbations. results derived from...