Factorization and Lie point symmetries of general Lienard-type equation in the complex plane

We present a variational approach to a general Lienard-type equation in order to linearize it and, as an example, the Van der Pol oscillator is discussed. The new equation which is almost linear is factorized. The point symmetries of the deformed equation are also discussed and the two-dimensional Lie algebraic generators are obtained

On astrophysical solution to ultra high energy cosmic rays

We argue that an astrophysical solution to UHECR problem is viable. The pectral features of extragalactic protons interacting with CMB are calculated in model-independent way. Using the power-law generation spectrum $\propto E^{-\gamma_g}$ as the only assumption, we analyze four features of the proton spectrum: the GZK cutoff, dip, bump and the second dip. We found the dip, induced by electron-positron production on CMB, as the most robust feature, existing in energy range $1\times 10^{18} - 4\times 10^{19}$ eV. Its shape is stable relative to various phenomena included in calculations. The dip is well confirmed by observations of AGASA, HiRes, Fly's Eye and Yakutsk detectors. The best fit is reached at $\gamma_g =2.7$, with the allowed range 2.55 - 2.75. The dip is used for energy calibration of the detectors. After the energy calibration the fluxes and spectra of all three detectors agree perfectly, with discrepancy between AGASA and HiRes at $E> 1\times 10^{20}$ eV being not statistically significant. The agreement of the dip with observations should be considered as confirmation of UHE proton interaction with CMB. The dip has two flattenings. The high energy flattening at $E \approx 1\times 10^{19}$ eV automatically explains ankle. The low-energy flattening at $E \approx 1\times 10^{18}$ eV provides the transition to galactic cosmic rays. This transition is studied quantitatively. The UHECR sources, AGN and GRBs, are studied in a model-dependent way, and acceleration is discussed. Based on the agreement of the dip with existing data, we make the robust prediction for the spectrum at $1\times 10^{18} - 1\times 10^{20}$ eV to be measured in the nearest future by Auger detector.Comment: Revised version as published in Phys.Rev. D47 (2006) 043005 with a small additio

Small Scale Anisotropy Predictions for the Auger Observatory

We study the small scale anisotropy signal expected at the Pierre Auger Observatory in the next 1, 5, 10, and 15 years of operation, from sources of ultra-high energy (UHE) protons. We numerically propagate UHE protons over cosmological distances using an injection spectrum and normalization that fits current data up to \sim 10^{20}\eV. We characterize possible sources of ultra-high energy cosmic rays (UHECRs) by their mean density in the local Universe, $\bar{\rho} = 10^{-r}$ Mpc$^{-3}$, with $r$ between 3 and 6. These densities span a wide range of extragalactic sites for UHECR sources, from common to rare galaxies or even clusters of galaxies. We simulate 100 realizations for each model and calculate the two point correlation function for events with energies above 4 \times 10^{19}\eV and above 10^{20}\eV, as specialized to the case of the Auger telescope. We find that for r\ga 4, Auger should be able to detect small scale anisotropies in the near future. Distinguishing between different source densities based on cosmic ray data alone will be more challenging than detecting a departure from isotropy and is likely to require larger statistics of events. Combining the angular distribution studies with the spectral shape around the GZK feature will also help distinguish between different source scenarios.Comment: 15 pages, 6 figures, 6 tables, submitted to JCA

Diffusion of Cosmic Rays in the Expanding Universe. II. Energy Spectra of Ultra-High Energy Cosmic Rays

We consider the astrophysical implications of the diffusion equation solution for Ultra-High Energy Cosmic Rays (UHECR) in the expanding universe, obtained in paper I (V.Berezinsky & A.Gazizov, ApJ 643 (2006) 8). The UHECR spectra are calculated in a model with sources located in vertices of the cubic grid with a linear constant (source separation) d. The calculations are performed for various magnetic field configurations (B_c,l_c), where l_c is the basic scale of the turbulence and B_c is the coherent magnetic field on this scale. The main purpose of these calculations is to demonstrate the validity of the solution obtained in paper I and to compare this solution with the Syrovatsky solution used in previous works. The Syrovatsky solution must be necessarily embedded in the static cosmological model. The formal comparison of the two solutions with all parameters being fixed identically reveals the appreciable discrepancies between two spectra. These discrepancies are less if in both models the different sets of the best-fit parameters are used.Comment: 17 pages, 10 figure

GZK Photons Above 10 EeV

We calculate the flux of "GZK-photons", namely the flux of photons produced by extragalactic nucleons through the resonant photoproduction of pions, the so called GZK effect. This flux depends on the UHECR spectrum on Earth, of the spectrum of nucleons emitted at the sources, which we characterize by its slope and maximum energy, on the distribution of sources and on the intervening cosmological backgrounds, in particular the magnetic field and radio backgrounds. For the first time we calculate the GZK photons produced by nuclei. We calculate the possible range of the GZK photon fraction of the total UHECR flux for the AGASA and the HiRes spectra. We find that for nucleons produced at the sources it could be as large as a few % and as low as 10^{-4} above 10 EeV. For nuclei produced at the sources the maximum photon fraction is a factor of 2 to 3 times smaller above 10 EeV but the minimum could be much smaller than for nucleons. We also comment on cosmogenic neutrino fluxes.Comment: 20 pages, 9 figures (21 panels), iopart.cls and iopart12.clo needed to typese