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

Randomised clinical trial: the efficacy and safety of propionyl-L-carnitine therapy in patients with ulcerative colitis receiving stable oral treatment.

Ulcerative colitis (UC) is characterised by impaired fatty-acid oxidation; l-carnitine has a key role in fatty-acid metabolism and short-chain fatty acids such as butyrate and propionate are important energy source for intestinal epithelial cells. To evaluate efficacy and safety of colon-release propionyl-L-carnitine (PLC) in patients with mild-to-moderate UC receiving stable oral aminosalicylate or thiopurine therapy. In a multicentre, phase II, double-blind, parallel-group trial, patients were randomised to receive PLC 1 g/day, PLC 2 g/day or placebo. Main inclusion criteria were as follows: age 18-75; disease activity index (DAI) score 3-10 inclusive, be under oral stable treatment with aminosalicylate or thiopurine. The primary endpoint was clinical/endoscopic response, defined as a decrease in DAI score ≥ 3 points or remission, defined as a DAI score ≤ 2 with no individual sub-score > 1

Advent and Recent Advances in Research on the Role of Pituitary Adenylate Cyclase-activating Polypeptide (PACAP) in the Regulation of Gonadotropic Hormone Secretion of Female Rats.

PACAP (ADCYAP1) was isolated from ovine hypothalami. PACAP activates three distinct receptor types: G-protein coupled PAC1, VPAC1, and VPAC2 with seven transmembrane domains. Eight splice variants of PAC1 receptor are described. A part of the hypothalamic PACAP is released into the hypophyseal portal circulation. Both hypothalamic and pituitary PACAP are involved in the dynamic control of gonadotropic hormone secretion. In female rats, PACAP in the paraventricular nucleus is upregulated in the morning and pituitary PACAP is upregulated in the late evening of the proestrus stage of the reproductive cycle. PACAP mRNA peak in the hypothalamic PVN precedes the LHRH release into the portal circulation. It is supposed that PACAP peak is evoked by the elevated estrogen on proestrous morning. At the beginning of the so-called critical period of the same day, PACAP level starts to decline allowing LHRH release into the portal circulation, resulting in the LH surge that evokes ovulation. Just before the critical period, icv-administered exogenous PACAP blocks the LH surge and ovulation. The blocking effect of PACAP is mediated through CRF and endogenous opioids. The effect of the pituitary-born PACAP depends on the intracellular cross-talk between PACAP and LHRH