Integrable discretizations of the sine-Gordon equation

The inverse scattering theory for the sine-Gordon equation discretized in space and both in space and time is considered.Comment: 18 pages, LaTeX2

Raman solitons in transient SRS

We report the observation of Raman solitons on numerical simulations of transient stimulated Raman scattering (TSRS) with small group velocity dispersion. The theory proceeds with the inverse scattering transform (IST) for initial-boundary value problems and it is shown that the explicit theoretical solution obtained by IST for a semi-infinite medium fits strikingly well the numerical solution for a finite medium. We understand this from the rapid decrease of the medium dynamical variable (the potential of the scattering theory). The spectral transform reflection coefficient can be computed directly from the values of the input and output fields and this allows to see the generation of the Raman solitons from the numerical solution. We confirm the presence of these nonlinear modes in the medium dynamical variable by the use of a discrete spectral analysis.Comment: LaTex file, to appear in Inverse Problem

On the equivalence of different approaches for generating multisoliton solutions of the KPII equation

The unexpectedly rich structure of the multisoliton solutions of the KPII equation has been explored by using different approaches, running from dressing method to twisting transformations and to the tau-function formulation. All these approaches proved to be useful in order to display different properties of these solutions and their related Jost solutions. The aim of this paper is to establish the explicit formulae relating all these approaches. In addition some hidden invariance properties of these multisoliton solutions are discussed

Ablowitz-Ladik system with discrete potential. I. Extended resolvent

Ablowitz-Ladik linear system with range of potential equal to {0,1} is considered. The extended resolvent operator of this system is constructed and the singularities of this operator are analyzed in detail.Comment: To be published in Theor. Math. Phy

Prostaglandin receptors and role of G protein-activated pathways on corpora lutea of pseudopregnant rabbit in vitro

Studies were conducted to characterize receptors for prostaglandin (PG) F(2alpha) (PGF(2alpha)) and PGE(2), and the signalling pathways regulating total nitric oxide synthase activity and progesterone production in rabbit corpora lutea (CL) of different luteal stages. CL were obtained at days 4, 9 and 13 of pseudopregnancy and cultured in vitro for 2 h with PGF(2alpha) or PGE(2) and with activators and inhibitors of G protein (Gp), phospholipase C (PLC), protein kinase C (PKC), adenylate cyclase (AC) and protein kinase A (PKA). High affinity PGF(2alpha) receptor (K(d)=1.9+/-0.6 nM mean+/-s.e.m. ) concentrations increased (P< or =0.01) four- to five-fold from early to mid- and late-luteal phases (50.6+/-8.5, 188.3+/-36.1 and 231.4+/-38.8 fmol/mg protein respectively). By contrast, PGE(2) receptor (K(d)=1.6+/-0.5 nM) concentrations decreased (P< or =0.01) from day 4 to day 9 and 13 (27.5+/-7.7, 12.4+/-2.4 and 16.5+/-3.0 fmol/mg protein respectively). The Gp-dependent AC/PKA pathway was triggered only on day 4 CL, mimicking the PGE(2) treatment and increasing progesterone production. In both day 9 and day 13 CL, the Gp-activated PLC/PKC pathway evoked a luteolytic effect similar to that induced by PGF(2alpha). The time-dependent selective resistance to PGF(2alpha) and PGE(2) by rabbit CL is mediated by factors other than a lack of luteal receptor-ligand interactions

Expression patterns of cytokines, p53 and nitric oxide synthase enzymes in corpora lutea of pseudopregnant rabbits during spontaneous luteolysis

The gene expressions for macrophage chemoattractant protein-1 (MCP-1), interleukin (IL)-1 beta, IL-2 and p53 were examined by semi-quantitative RT-PCR in corpora lutea (CL) of rabbits during spontaneous luteolysis at days 13, 15, 18 and 22 of pseudopregnancy. In the same luteal tissue, total activity of nitric oxide (NO) synthase (NOS) and genes for both endothelial (eNOS) and inducible (iNOS) isoforms were also analysed. From day 13 to 15, MCP-1 and IL-1 beta mRNA levels rose (P &lt; or = 0.01) almost 2-fold, and the transcript for p53 almost 8-fold, but then all dropped (P &lt; or = 0.05) from day 18 onward. IL-2 mRNA abundance was higher (P &lt; or = 0.01) on day 13 and then gradually declined. During luteolysis, eNOS mRNA decreased 40% (P &lt; or = 0.05) by day 15, but thereafter remained unchanged, while iNOS mRNA was barely detectable and did not show any clear age-related pattern throughout the late luteal stages. Total NOS activity progressively increased (P &lt; or = 0.01) from day 13 to 18 of pseudopregnancy and then dropped to the lowest (P &lt; or = 0.01) levels on day 22. Luteal progesterone content also declined during CL regression from 411 to 17 pg/mg found on days 13 and 22 respectively, in parallel with the decrease in blood progesterone concentrations. These data further support a physiological role of NO as modulator of luteal demise in rabbits. Locally, luteal cytokines may be involved in the up-regulation of NOS activity, while downstream NO may inhibit steroroidogenesis and induce expression of p53 gene after removal of the protective action of progesterone

Multiscale expansion and integrability properties of the lattice potential KdV equation

We apply the discrete multiscale expansion to the Lax pair and to the first few symmetries of the lattice potential Korteweg-de Vries equation. From these calculations we show that, like the lowest order secularity conditions give a nonlinear Schroedinger equation, the Lax pair gives at the same order the Zakharov and Shabat spectral problem and the symmetries the hierarchy of point and generalized symmetries of the nonlinear Schroedinger equation.Comment: 10 pages, contribution to the proceedings of the NEEDS 2007 Conferenc

An Inverse Scattering Transform for the Lattice Potential KdV Equation

The lattice potential Korteweg-de Vries equation (LKdV) is a partial difference equation in two independent variables, which possesses many properties that are analogous to those of the celebrated Korteweg-de Vries equation. These include discrete soliton solutions, Backlund transformations and an associated linear problem, called a Lax pair, for which it provides the compatibility condition. In this paper, we solve the initial value problem for the LKdV equation through a discrete implementation of the inverse scattering transform method applied to the Lax pair. The initial value used for the LKdV equation is assumed to be real and decaying to zero as the absolute value of the discrete spatial variable approaches large values. An interesting feature of our approach is the solution of a discrete Gel'fand-Levitan equation. Moreover, we provide a complete characterization of reflectionless potentials and show that this leads to the Cauchy matrix form of N-soliton solutions

The lattice Schwarzian KdV equation and its symmetries

In this paper we present a set of results on the symmetries of the lattice Schwarzian Korteweg-de Vries (lSKdV) equation. We construct the Lie point symmetries and, using its associated spectral problem, an infinite sequence of generalized symmetries and master symmetries. We finally show that we can use master symmetries of the lSKdV equation to construct non-autonomous non-integrable generalized symmetries.Comment: 11 pages, no figures. Submitted to Jour. Phys. A, Special Issue SIDE VI

Multidimensional Inverse Scattering of Integrable Lattice Equations

We present a discrete inverse scattering transform for all ABS equations excluding Q4. The nonlinear partial difference equations presented in the ABS hierarchy represent a comprehensive class of scalar affine-linear lattice equations which possess the multidimensional consistency property. Due to this property it is natural to consider these equations living in an N-dimensional lattice, where the solutions depend on N distinct independent variables and associated parameters. The direct scattering procedure, which is one-dimensional, is carried out along a staircase within this multidimensional lattice. The solutions obtained are dependent on all N lattice variables and parameters. We further show that the soliton solutions derived from the Cauchy matrix approach are exactly the solutions obtained from reflectionless potentials, and we give a short discussion on inverse scattering solutions of some previously known lattice equations, such as the lattice KdV equation.Comment: 18 page