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

Stability boundary approximation of periodic dynamics

We develop here the method for obtaining approximate stability boundaries in the space of parameters for systems with parametric excitation. The monodromy (Floquet) matrix of linearized system is found by averaging method. For system with 2 degrees of freedom (DOF) we derive general approximate stability conditions. We study domains of stability with the use of fourth order approximations of monodromy matrix on example of inverted position of a pendulum with vertically oscillating pivot. Addition of small damping shifts the stability boundaries upwards, thus resulting to both stabilization and destabilization effects.Comment: 9 pages, 2 figure

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the Engquist--Osher approximation for the flux and explicit time--stepping. An adaptivemultiresolution scheme with cell averages is then used to speed up CPU time and meet memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for the sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier--thickener model illustrate the efficiency of this method

Fluorpyromorphite, Pb5(PO4)3F, a new apatite-group mineral from Sukhovyaz Mountain, Southern Urals, and Tolbachik volcano, Kamchatka

Fluorpyromorphite, ideally Pb5(PO4)3F, a new apatite-group member, an F-dominant analog of pyromorphite and hydrox-ylpyromorphite. It is a supergene mineral found at two localities: Sukhovyaz Mountain, Ufaley District, Southern Urals (holotype) and Mountain 1004, Tolbachik volcano, Kamchatka (co-type), both in Russia. At Sukhovyaz, fluorpyromorphite forms anhedral grains up to 0.2 mm across (usually much smaller), filling cavities in quartz and sometimes partially replacing fluorapatite. Associated supergene minerals include pyromorphite, hydroxylpyromorphite, fluorphosphohedy-phane, mimetite, and nickeltsumcorite. At Tolbachik, fluorpyromorphite occurs in the oxidation zone of paleo-fumarolic deposits in close association with pyromorphite, fluorphosphohedyphane, wulfenite, cerussite, munakataite, vanadinite, chrysocolla, and opal. It forms crude long-prismatic to acicular crystals up to 0.1 mm long and up to 5 mu m thick com-bined in bunches and spherulites up to 0.2 mm. Fluorpyromorphite is colorless (Sukhovyaz) or yellow (Tolbachik), translucent to transparent and has a vitreous luster. It is brittle, with an uneven fracture and poor cleavage on (001). The calculated density values are 7.382 (holotype) and 6.831 (cotype) g/cm3. Fluorpyromorphite is optically uniaxial (-). In reflected light, it is light-grey, weakly anisotropic. The reflectance values (Rmin/Rmax, %) are: 15.8/16.6 (470 nm), 16.2/17.2 (546 nm), 15.9/16.9 (589 nm), 15.4/16.2 (650 nm). The chemical composition is (electron microprobe, wt. %; holotype/co-type): CaO 0.10/3.16, SrO 0.17/0.00, PbO 83.51/77.39, P2O516.13/16.35, CrO3 0.00/0.49, SeO3 0.00/0.98, F 1.00/1.35, Cl 0.29/0.40, H2Ocalc 0.13/0.00, -O=(F,Cl) -0.49/-0.66, total 100.84/99.46. The empirical formulae based on 13 anions (O +F + Cl+OH)pfu are Pb4.95Ca0.02Sr0.02P3.00O12F0.70(OH)0.19Cl0.11 (holotype) and Pb4.26Ca0.69P2.83Se6+0.09Cr6+0.06 O11.99F0.87Cl0.14 (co-type). Fluorpyromorphite is hexagonal, space group P63/m, unit-cell parameters (from powder X-ray diffraction data; holotype / co-type) are: a = 9.779(5) / 9.732(1), c = 7.241(9) / 7.242(1) angstrom, V = 599.6(7) / 594.0(2) angstrom 3, and Z = 2. The crystal structure was refined using the Rietveld method to Rp= 0.1764 (holotype). Fluorpyromorphite is isostructural with other members of the apatite group, a subdivision of the apatite supergroup