Numerical instability of the Akhmediev breather and a finite-gap model of it

In this paper we study the numerical instabilities of the NLS Akhmediev breather, the simplest space periodic, one-mode perturbation of the unstable background, limiting our considerations to the simplest case of one unstable mode. In agreement with recent theoretical findings of the authors, in the situation in which the round-off errors are negligible with respect to the perturbations due to the discrete scheme used in the numerical experiments, the split-step Fourier method (SSFM), the numerical output is well-described by a suitable genus 2 finite-gap solution of NLS. This solution can be written in terms of different elementary functions in different time regions and, ultimately, it shows an exact recurrence of rogue waves described, at each appearance, by the Akhmediev breather. We discover a remarkable empirical formula connecting the recurrence time with the number of time steps used in the SSFM and, via our recent theoretical findings, we establish that the SSFM opens up a vertical unstable gap whose length can be computed with high accuracy, and is proportional to the inverse of the square of the number of time steps used in the SSFM. This neat picture essentially changes when the round-off error is sufficiently large. Indeed experiments in standard double precision show serious instabilities in both the periods and phases of the recurrence. In contrast with it, as predicted by the theory, replacing the exact Akhmediev Cauchy datum by its first harmonic approximation, we only slightly modify the numerical output. Let us also remark, that the first rogue wave appearance is completely stable in all experiments and is in perfect agreement with the Akhmediev formula and with the theoretical prediction in terms of the Cauchy data.Comment: 27 pages, 8 figures, Formula (30) at page 11 was corrected, arXiv admin note: text overlap with arXiv:1707.0565

Moments of the Position of the Maximum for GUE Characteristic Polynomials and for Log-Correlated Gaussian Processes

We study three instances of log-correlated processes on the interval: the logarithm of the Gaussian unitary ensemble (GUE) characteristic polynomial, the Gaussian log-correlated potential in presence of edge charges, and the Fractional Brownian motion with Hurst index $H \to 0$ (fBM0). In previous collaborations we obtained the probability distribution function (PDF) of the value of the global minimum (equivalently maximum) for the first two processes, using the {\it freezing-duality conjecture} (FDC). Here we study the PDF of the position of the maximum $x_m$ through its moments. Using replica, this requires calculating moments of the density of eigenvalues in the $\beta$-Jacobi ensemble. Using Jack polynomials we obtain an exact and explicit expression for both positive and negative integer moments for arbitrary $\beta >0$ and positive integer $n$ in terms of sums over partitions. For positive moments, this expression agrees with a very recent independent derivation by Mezzadri and Reynolds. We check our results against a contour integral formula derived recently by Borodin and Gorin (presented in the Appendix A from these authors). The duality necessary for the FDC to work is proved, and on our expressions, found to correspond to exchange of partitions with their dual. Performing the limit $n \to 0$ and to negative Dyson index $\beta \to -2$, we obtain the moments of $x_m$ and give explicit expressions for the lowest ones. Numerical checks for the GUE polynomials, performed independently by N. Simm, indicate encouraging agreement. Some results are also obtained for moments in Laguerre, Hermite-Gaussian, as well as circular and related ensembles. The correlations of the position and the value of the field at the minimum are also analyzed.Comment: 64 page, 5 figures, with Appendix A written by Alexei Borodin and Vadim Gorin; The appendix H in the ArXiv version is absent in the published versio

Azimuthal anisotropy and correlations at large transverse momenta in p+pp+p and Au+Au collisions at sNN\sqrt{s_{_{NN}}}= 200 GeV

Results on high transverse momentum charged particle emission with respect to the reaction plane are presented for Au+Au collisions at $\sqrt{s_{_{NN}}}$= 200 GeV. Two- and four-particle correlations results are presented as well as a comparison of azimuthal correlations in Au+Au collisions to those in $p+p$ at the same energy. Elliptic anisotropy, $v_2$, is found to reach its maximum at $p_t \sim 3$ GeV/c, then decrease slowly and remain significant up to $p_t\approx 7$ -- 10 GeV/c. Stronger suppression is found in the back-to-back high-$p_t$ particle correlations for particles emitted out-of-plane compared to those emitted in-plane. The centrality dependence of $v_2$ at intermediate $p_t$ is compared to simple models based on jet quenching.Comment: 4 figures. Published version as PRL 93, 252301 (2004

Azimuthal anisotropy in Au+Au collisions at sqrtsNN = 200 GeV

The results from the STAR Collaboration on directed flow (v_1), elliptic flow (v_2), and the fourth harmonic (v_4) in the anisotropic azimuthal distribution of particles from Au+Au collisions at sqrtsNN = 200 GeV are summarized and compared with results from other experiments and theoretical models. Results for identified particles are presented and fit with a Blast Wave model. Different anisotropic flow analysis methods are compared and nonflow effects are extracted from the data. For v_2, scaling with the number of constituent quarks and parton coalescence is discussed. For v_4, scaling with v_2^2 and quark coalescence is discussed.Comment: 26 pages. As accepted by Phys. Rev. C. Text rearranged, figures modified, but data the same. However, in Fig. 35 the hydro calculations are corrected in this version. The data tables are available at http://www.star.bnl.gov/central/publications/ by searching for "flow" and then this pape

Rapidity and Centrality Dependence of Proton and Anti-proton Production from Au+Au Collisions at sqrt(sNN) = 130GeV

We report on the rapidity and centrality dependence of proton and anti-proton transverse mass distributions from Au+Au collisions at sqrt(sNN) = 130GeV as measured by the STAR experiment at RHIC. Our results are from the rapidity and transverse momentum range of |y|<0.5 and 0.35 <p_t<1.00GeV/c. For both protons and anti-protons, transverse mass distributions become more convex from peripheral to central collisions demonstrating characteristics of collective expansion. The measured rapidity distributions and the mean transverse momenta versus rapidity are flat within |y|<0.5. Comparisons of our data with results from model calculations indicate that in order to obtain a consistent picture of the proton(anti-proton) yields and transverse mass distributions the possibility of pre-hadronic collective expansion may have to be taken into account.Comment: 4 pages, 3 figures, 1 table, submitted to PR

Disappearance of back-to-back high-p(T) hadron correlations in central Au+Au collisions at root s(NN)=200 GeV

Azimuthal correlations for large transverse momentum charged hadrons have been measured over a wide pseudorapidity range and full azimuth in Au+Au and p+p collisions at roots(NN)=200 GeV. The small-angle correlations observed in p+p collisions and at all centralities of Au+Au collisions are characteristic of hard-scattering processes previously observed in high-energy collisions. A strong back-to-back correlation exists for p+p and peripheral Au+Au. In contrast, the back-to-back correlations are reduced considerably in the most central Au+Au collisions, indicating substantial interaction as the hard-scattered partons or their fragmentation products traverse the medium

Azimuthal anisotropy and correlations in the hard scattering regime at RHIC

Azimuthal anisotropy (v(2)) and two-particle angular correlations of high p(T) charged hadrons have been measured in Au+Au collisions at roots(NN) = 130 GeV for transverse momenta up to 6 GeV/c, where hard processes are expected to contribute significantly. The two-particle angular correlations exhibit elliptic flow and a structure suggestive of fragmentation of high p(T) partons. The monotonic rise of v(2)(p(T)) for p(T) 3 GeV/c, a saturation of v(2) is observed which persists up to p(T) = 6 GeV/c

Transverse-momentum and collision-energy dependence of high-p(T) hadron suppression in Au+Au collisions at ultrarelativistic energies

We report high statistics measurements of inclusive charged hadron production in Au+Au and p+p collisions at rootS(NN)=200 GeV. A large, approximately constant hadron suppression is observed in central Au+Au collisions for 5<p(T)<12 GeV/c. The collision energy dependence of the yields and the centrality and p(T) dependence of the suppression provide stringent constraints on theoretical models of suppression. Models incorporating initial-state gluon saturation or partonic energy loss in dense matter are largely consistent with observations. We observe no evidence of p(T)-dependent suppression, which may be expected from models incorporating jet attenuation in cold nuclear matter or scattering of fragmentation hadrons

Cross sections and transverse single-spin asymmetries in forward neutral-pion production from proton collisions at root s=200 GeV

Measurements of the production of forward high-energy pi(0) mesons from transversely polarized proton collisions at root200 GeV are reported. The cross section is generally consistent with next-to-leading order perturbative QCD calculations. The analyzing power is small at x(F) below about 0.3, and becomes positive and large at higher x(F), similar to the trend in data at roots less than or equal to20 GeV. The analyzing power is in qualitative agreement with perturbative QCD model expectations. This is the first significant spin result seen for particles produced with p(T)>1 GeV/c at a polarized proton collider

Azimuthal anisotropy at the relativistic heavy ion collider: The first and fourth harmonics

We report the first observations of the first harmonic (directed flow, v(1)) and the fourth harmonic (v(4)), in the azimuthal distribution of particles with respect to the reaction plane in Au+Au collisions at the BNL Relativistic Heavy Ion Collider (RHIC). Both measurements were done taking advantage of the large elliptic flow (v(2)) generated at RHIC. From the correlation of v(2) with v(1) it is determined that v(2) is positive, or in-plane. The integrated v(4) is about a factor of 10 smaller than v(2). For the sixth (v(6)) and eighth (v(8)) harmonics upper limits on the magnitudes are reported