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

Prevention of gastrointestinal cancer by surveillance endoscopy

The classification of the endoscopic appearance of superficial neoplastic lesions of the digestive mucosa aims to evaluate the risk of progression to advanced neoplasia in 3° (low, intermediate, high) and to predict appropriate treatment and corresponding surveillance. The privileged position of endoscopy results from its double impact on prevention of digestive cancer through reduction in incidence after early detection and eradication of precursors; and through reduction of mortality after detection and treatment of cancer at an early and curable stage. However the efficacy of diagnostic endoscopy still requires improvement and quality control on the following points: (1) technology, with a generalized use of the recently introduced high-resolution endoscopes. (2) diagnosis of poorly visible nonpolypoid precursors: this applies to small depressed lesions and large slightly elevated or sessile serrated and non-serrated precursors, particularly in the proximal colon. (3) treatment and training in therapeutic endoscopy, including the most recent techniques of mucosal resection of nonpolypoid lesions

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

Physical properties and fracture behavior of rocks in uniaxial and triaxial compression: A case-study of Zhdanovskoe deposit, Kola Mining and Metallurgical Company

Abstract The paper presents the test data on properties of rocks from Zhdanovskoe deposit located in the Pechenga area of the Murmansk Region. The nature of the elastic energy accumulation in rock specimens and their susceptibility to uncontrollable fracture in the dynamic mode are determined. The uniaxial and triaxial compression test data are used to find mechanical properties and energy characteristics of rock specimens. The tested rocks are assessed as hard and susceptible to dynamic fracture, which is one of the criteria of rockburst hazard.</jats:p