Analysis of Traveling and Standing Waves in the DNA Model by Peyrard-Bishop-Dauxois

The model by Peyrard - Bishop - Dauxois (the PBD model), which describes the DNA molecule nonlinear dynamics, is considered. This model represents two chains of rigid disks connected by nonlinear springs. An interaction between opposite disks of different chains is modeled by the Morse potential. Solutions of equations of motion are obtained analytically in two approximations of the small parameter method for two limit cases. The first one is the long-wavelength limit of traveling waves, when frequencies of vibrations are small. Dispersion relations are obtained also for the long-wavelength limit by the small parameter method. The second case is a limit of high frequency standing waves in the form of out-of-phase vibration modes. Two such out-of-phase modes are obtained; it is selected one of them, which has the larger frequency. In both cases systems of nonlinear ODEs are obtained. Nonlinear terms are presented by the Tailor series expansion, where terms up to third degree by displacement are saved. The analytical solutions are compared with checking numerical simulation obtained by the Runge - Kutta method of the 4-th order. The comparison shows a good exactness of these approximate analytical solutions. Stability of the standing localized modes is analyzed by the numerical-analytical approach, which is connected with the Lyapunov definition of stability

Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with variable coefficient

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2012 Taylor & Francis.In this paper, a numerical implementation of a direct united boundary-domain integral equation (BDIE) related to the Neumann boundary value problem for a scalar elliptic partial differential equation with a variable coefficient is discussed. The BDIE is reduced to a uniquely solvable one by adding an appropriate perturbation operator. The mesh-based discretization of the BDIEs with quadrilateral domain elements leads to a system of linear algebraic equations (discretized BDIE). Then, the system is solved by LU decomposition and Neumann iterations. Convergence of the iterative method is discussed in relation to the distribution of eigenvalues of the corresponding discrete operators calculated numerically.The work was supported by the grant EP/H020497/1 "Mathematical analysis of localised boundary-domain integral equations for BVPs with variable coefficients" of the EPSRC, UK

Banach Analytic Sets and a Non-Linear Version of the Levi Extension Theorem

We prove a certain non-linear version of the Levi extension theorem for meromorphic functions. This means that the meromorphic function in question is supposed to be extendable along a sequence of complex curves, which are arbitrary, not necessarily straight lines. Moreover, these curves are not supposed to belong to any finite dimensional analytic family. The conclusion of our theorem is that nevertheless the function in question meromorphically extends along an (infinite dimensional) analytic family of complex curves and its domain of existence is a pinched domain filled in by this analytic family.Comment: 19 pages, This is the final version with significant corrections and improvements. To appear in Arkiv f\"or matemati

Self-Induced Quasistationary Magnetic Fields

The interaction of electromagnetic radiation with temporally dispersive magnetic solids of small dimensions may show very special resonant behaviors. The internal fields of such samples are characterized by magnetostatic-potential scalar wave functions. The oscillating modes have the energy orthogonality properties and unusual pseudo-electric (gauge) fields. Because of a phase factor, that makes the states single valued, a persistent magnetic current exists. This leads to appearance of an eigen-electric moment of a small disk sample. One of the intriguing features of the mode fields is dynamical symmetry breaking

Bose-Einstein condensation in an optical lattice: A perturbation approach

We derive closed analytical expressions for the order parameter $\Phi (x)$ and for the chemical potential $\mu$ of a Bose-Einstein Condensate loaded into a harmonically confined, one dimensional optical lattice, for sufficiently weak, repulsive or attractive interaction, and not too strong laser intensities. Our results are compared with exact numerical calculations in order to map out the range of validity of the perturbative analytical approach. We identify parameter values where the optical lattice compensates the interaction-induced nonlinearity, such that the condensate ground state coincides with a simple, single particle harmonic oscillator wave function

Breakdown of Universality in Random Matrix Models

We calculate smoothed correlators for a large random matrix model with a potential containing products of two traces \tr W_1(M) \cdot \tr W_2(M) in addition to a single trace \tr V(M). Connected correlation function of density eigenvalues receives corrections besides the universal part derived by Brezin and Zee and it is no longer universal in a strong sense.Comment: 16 pages, LaTex, references and footnote adde

Dynamical interaction of an elastic system and a vibro-impact absorber

The nonlinear two-degree-of-freedom system under consideration consists of the linear oscillator with a relatively big mass, which is an approximation of some continuous elastic system, and of the vibro-impact oscillator with a relatively small mass, which is an absorber of the linear system vibrations. Analysis of nonlinear normal vibration modes shows that a stable localized vibration mode, which provides the vibration regime appropriate for the elastic vibration absorption, exists in a large region of the system parameters. In this regime, amplitudes of vibrations of the linear system are small, simultaneously vibrations of the absorber are significant

Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators

We prove Sobolev-type p((.)) -> q ((.))-theorems for the Riesz potential operator I-alpha in the weighted Lebesgue generalized spaces L-p(.)(R-n, p) with the variable exponent p (x) and a two-parametrical power weight fixed to an arbitrary finite point and to infinity, as well as similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces L-p(.)(S-n, p) on the unit sphere S-n in Rn+1. (c) 2005 Elsevier Inc. All rights reserved.info:eu-repo/semantics/publishedVersio

The anapole moments in disk-form MS-wave ferrite particle

The anapole moments describe the parity-violating parity-odd, time-reversal-even couplings between elementary particles and the electromagnetic (EM) field. Surprisingly, the anapole-like moment properties can be found in certain artificially engineered physical systems. In microwaves, ferrite resonators with multi-resonance magnetostatic-wave (MS-wave) oscillations may have sizes two-four orders less than the free-space EM wavelength at the same frequency. MS-wave oscillations in a ferrite sample occupy a special place between the pure electromagnetic and spin-wave (exchange) processes. The energy density of MS-wave oscillations is not the electromagnetic-wave density of energy and not the exchange energy density as well. These microscopic oscillating objects -- the particles -- may interact with the external EM fields by a very specific way, forbidden for the classical description. To describe such interactions, the quantum mechanical analysis should be used. The presence of surface magnetic currents is one of the features of MS oscillations in a normally magnetized ferrite disk resonator. Because of such magnetic currents, MS oscillations in ferrite disk resonators become parity violating. The parity-violating couplings between disk-form ferrite particles and the external EM field should be analyzed based on the notion of an anapole moment.Comment: 20 pages, 2 figures, PDF (created from MS-Word

On the Causality and Stability of the Relativistic Diffusion Equation

This paper examines the mathematical properties of the relativistic diffusion equation. The peculiar solution which Hiscock and Lindblom identified as an instability is shown to emerge from an ill-posed initial value problem. These do not meet the mathematical conditions required for realistic physical problems and can not serve as an argument against the relativistic hydrodynamics of Landau and Lifshitz.Comment: 6 page