Negaton and Positon solutions of the soliton equation with self-consistent sources

The KdV equation with self-consistent sources (KdVES) is used as a model to illustrate the method. A generalized binary Darboux transformation (GBDT) with an arbitrary time-dependent function for the KdVES as well as the formula for $N$-times repeated GBDT are presented. This GBDT provides non-auto-B\"{a}cklund transformation between two KdV equations with different degrees of sources and enable us to construct more general solutions with $N$ arbitrary $t$-dependent functions. By taking the special $t$-function, we obtain multisoliton, multipositon, multinegaton, multisoliton-positon, multinegaton-positon and multisoliton-negaton solutions of KdVES. Some properties of these solutions are discussed.Comment: 13 pages, Latex, no figues, to be published in J. Phys. A: Math. Ge

Long-term variability of the optical spectra of NGC 4151: II. Evolution of the broad Ha and Hb emission-line profiles

Results of the long-term (11 years, from 1996 to 2006) H$\alpha$ and H$\beta$ line variations of the active galactic nucleus of NGC 4151 are presented. High quality spectra (S/N>50 and R~8A) of H$\alpha$ and H$\beta$ were investigated. We analyzed line profile variations during monitoring period. Comparing the line profiles of H$\alpha$ and H$\beta$, we studied different details (bumps, absorption features) in the line profiles. The variations of the different H$\alpha$ and H$\beta$ line profile segments have been investigated. Also, we analyzed the Balmer decrement for whole line and for line segments. We found that the line profiles were strongly changing during the monitoring period, showing blue and red asymmetries. This indicates a complex BLR geometry of NGC 4151 with, at least, three kinematically distinct regions: one that contributes to the blue line wing, one to the line core and one to the red line wing. Such variation can be caused by an accelerating outflow starting very close to the black hole, where the red part may come from the region {closer to the black hole than the blue part, which is coming} from the region having the highest outflow velocities. Taking into account the fact that the BLR of NGC 4151 has a complex geometry (probably affected by an outflow) and that a portion of the broad line emission seems to have not a pure photoionization origin, one can ask the question whether the study of the BLR by reverberation mapping may be valid in the case of this galaxy.Comment: 24 pages, 18 figures, accepted for publications in A&

Effective spacetime from multi-dimensional gravity

We study the effective spacetimes in lower dimensions that can be extracted from a multidimensional generalization of the Schwarzschild-Tangherlini spacetimes derived by Fadeev, Ivashchuk and Melnikov ({\it Phys. Lett,} {\bf A 161} (1991) 98). The higher-dimensional spacetime has $D = (4 + n + m)$ dimensions, where $n$ and $m$ are the number of "internal" and "external" extra dimensions, respectively. We analyze the effective $(4 + n)$ spacetime obtained after dimensional reduction of the $m$ external dimensions. We find that when the $m$ extra dimensions are compact (i) the physics in lower dimensions is independent of $m$ and the character of the singularities in higher dimensions, and (ii) the total gravitational mass $M$ of the effective matter distribution is less than the Schwarzshild mass. In contrast, when the $m$ extra dimensions are large this is not so; the physics in $(4 + n)$ does explicitly depend on $m$, as well as on the nature of the singularities in high dimensions, and the mass of the effective matter distribution (with the exception of wormhole-like distributions) is bigger than the Schwarzshild mass. These results may be relevant to observations for an experimental/observational test of the theory.Comment: A typo in Eq. (24) is fixe

The Solutions of the NLS Equations with Self-Consistent Sources

We construct the generalized Darboux transformation with arbitrary functions in time $t$ for the AKNS equation with self-consistent sources (AKNSESCS) which, in contrast with the Darboux transformation for the AKNS equation, provides a non-auto-B\"{a}cklund transformation between two AKNSESCSs with different degrees of sources. The formula for N-times repeated generalized Darboux transformation is proposed. By reduction the generalized Darboux transformation with arbitrary functions in time $t$ for the Nonlinear Schr\"{o}dinger equation with self-consistent sources (NLSESCS) is obtained and enables us to find the dark soliton, bright soliton and positon solutions for NLS$^{+}$ESCS and NLS$^{-}$ESCS. The properties of these solution are analyzed.Comment: 24 pages, 3 figures, to appear in Journal of Physics A: Mathematical and Genera

On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources

Non-holonomic deformations of integrable equations of the KdV hierarchy are studied by using the expansions over the so-called "squared solutions" (squared eigenfunctions). Such deformations are equivalent to perturbed models with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV equation. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV hierarchy with generic self-consistent sources are analyzed. This allows one to formulate a set of conditions on the perturbation terms that preserve the integrability. The perturbation corrections to the scattering data and to the corresponding action-angle variables are studied. The analysis shows that although many nontrivial solutions of KdV equations with generic self-consistent sources can be obtained by the Inverse Scattering Transform (IST), there are solutions that, in principle, can not be obtained via IST. Examples are considered showing the complete integrability of KdV6 with perturbations that preserve the eigenvalues time-independent. In another type of examples the soliton solutions of the perturbed equations are presented where the perturbed eigenvalue depends explicitly on time. Such equations, however in general, are not completely integrable.Comment: 16 pages, no figures, LaTe

Energy transmission in the forbidden bandgap of a nonlinear chain

A nonlinear chain driven by one end may propagate energy in the forbidden band gap by means of nonlinear modes. For harmonic driving at a given frequency, the process ocurs at a threshold amplitude by sudden large energy flow, that we call nonlinear supratransmission. The bifurcation of energy transmission is demonstrated numerically and experimentally on the chain of coupled pendula (sine-Gordon and nonlinear Klein-Gordon equations) and sustained by an extremely simple theory.Comment: LaTex file, 6 figures, published in Phys Rev Lett 89 (2002) 13410

Spectral optical monitoring of the double peaked emission line AGN Arp 102B: II. Variability of the broad line properties

We investigate a long-term (26 years, from 1987 to 2013) variability in the broad spectral line properties of the radio galaxy Arp 102B, an active galaxy with broad double-peaked emission lines. We use observations presented in Paper I (Shapovalova et al. 2013) in the period from 1987 to 2011, and a new set of observations performed in 2012--2013. To explore the BLR geometry, and clarify some contradictions about the nature of the BLR in Arp 102B we explore variations in the H$\alpha$ and H$\beta$ line parameters during the monitored period. We fit the broad lines with three broad Gaussian functions finding the positions and intensities of the blue and red peaks in H$\alpha$ and H$\beta$. Additionally we fit averaged line profiles with the disc model. We find that the broad line profiles are double-peaked and have not been changed significantly in shapes, beside an additional small peak that, from time to time can be seen in the blue part of the H$\alpha$ line. The positions of the blue and red peaks { have not changed significantly during the monitored period. The H$\beta$ line is broader than H$\alpha$ line in the monitored period. The disc model is able to reproduce the H$\beta$ and H$\alpha$ broad line profiles, however, observed variability in the line parameters are not in a good agreement with the emission disc hypothesis. It seems that the BLR of Arp 102B has a disc-like geometry, but the role of an outflow can also play an important role in observed variation of the broad line properties.Comment: 17 pages, Accepted for publication in A&

Implementation of the Backlund transformations for the Ablowitz-Ladik hierarchy

The derivation of the Backlund transformations (BTs) is a standard problem of the theory of the integrable systems. Here, I discuss the equations describing the BTs for the Ablowitz-Ladik hierarchy (ALH), which have been already obtained by several authors. The main aim of this work is to solve these equations. This can be done in the framework of the so-called functional representation of the ALH, when an infinite number of the evolutionary equations are replaced, using the Miwa's shifts, with a few equations linking tau-functions with different arguments. It is shown that starting from these equations it is possible to obtain explicit solutions of the BT equations. In other words, the main result of this work is a presentation of the discrete BTs as a superposition of an infinite number of evolutionary flows of the hierarchy. These results are used to derive the superposition formulae for the BTs as well as pure soliton solutions.Comment: 20 page