Surface solitons in trilete lattices

Fundamental solitons pinned to the interface between three semi-infinite one-dimensional nonlinear dynamical chains, coupled at a single site, are investigated. The light propagation in the respective system with the self-attractive on-site cubic nonlinearity, which can be implemented as an array of nonlinear optical waveguides, is modeled by the system of three discrete nonlinear Schrödinger equations. The formation, stability and dynamics of symmetric and asymmetric fundamental solitons centered at the interface are investigated analytically by means of the variational approximation (VA) and in a numerical form. The VA predicts that two asymmetric and two antisymmetric branches exist in the entire parameter space, while four asymmetric modes and the symmetric one can be found below some critical value of the inter-lattice coupling parameter—actually, past the symmetry-breaking bifurcation. At this bifurcation point, the symmetric branch is destabilized and two new asymmetric soliton branches appear, one stable and the other unstable. In this area, the antisymmetric branch changes its character, getting stabilized against oscillatory perturbations. In direct simulations, unstable symmetric modes radiate a part of their power, staying trapped around the interface. Highly unstable asymmetric modes transform into localized breathers traveling from the interface region across the lattice without significant power loss.     

Theory of periodic and solitary space charge waves in extrinsic semiconductors

We present a theory of the existence and stability of traveling periodic and solitary space charge wave solutions to a standard rate equation model of electrical conduction in extrinsic semiconductors which includes effects of field-dependent impurity impact ionization. A nondimensional set of equations is presented in which the small parameter β = (dielectric relaxation time) / (characteristic impurity time) 1 plays a crucial role for our singular perturbation analysis. For a narrow range of wave velocities a phase plane analysis gives a set of limit cycle orbits corresponding to periodic traveling waves. while for a unique value of wave velocity we find a homoclinic orbit corresponding to a moving solitary space charge wave of the type experimentally observed in p-type germanium. A linear stability analysis reveals all waves to be unstable under current bias on the infinite one-dimensional line. Finally, we conjecture that solitary waves may be stable in samples of finite length under voltage bias.     

Kapitza pendulum effect in a continuously stirred tank reactor

The equations of a continuously stirred tank reactor in a wide range of the parameters have three stationary solutions describing the hot, cold, and intermediate unstable states. Similarity with the equations for one-dimensional motion indicates the possibility of the stabilization of instability by small high-frequency perturbation (Kapitza pendulum effect). This stabilization has been obtained in numerical simulation.     

Transverse wave propagation in relativistic two-fluid plasmas in de Sitter space

We investigate transverse electromagnetic waves propagating in a plasma near the horizon of the de Sitter space. Using the 3+1 formalism we derive the relativistic two-fluid equations to take account of the effects due to the horizon and describe the set of simultaneous linear equations for the perturbations. We use a local approximation to investigate the one-dimensional radial propagation of Alfvén and high frequency electromagnetic waves and solve the dispersion relation for these waves numerically.     

Equilibrium-torus bifurcation in nonsmooth systems

Considering a set of two coupled nonautonomous differential equations with discontinuous right-hand sides describing the behavior of a DC/DC power converter, we discuss a border-collision bifurcation that can lead to the birth of a two-dimensional invariant torus from a stable node equilibrium point. We obtain the chart of dynamic modes and show that there is a region of parameter space in which the system has a single stable node equilibrium point. Under variation of the parameters, this equilibrium may disappear as it collides with a discontinuity boundary between two smooth regions in the phase space. The disappearance of the equilibrium point is accompanied by the soft appearance of an unstable focus period-1 orbit surrounded by a resonant or ergodic torus.Detailed numerical calculations are supported by a theoretical investigation of the normal form map that represents the piecewise linear approximation to our system in the neighbourhood of the border. We determine the functional relationships between the parameters of the normal form map and the actual system and illustrate how the normal form theory can predict the bifurcation behaviour along the border-collision equilibrium-torus bifurcation curve.     

On the three-body Coulomb scattering problem

We describe the smoothness properties and the asymptotic form of the Green's function (in configuration space) for three charged particles. We also discuss the integral equations and the boundary value problems for the Coulomb wavefunctions and we show that they form a complete set. Finally, we study the singularities of the Coulomb scattering operator, and we investigate the connection between the Dollard wave operators and the Coulomb wavefunctions.     

Galerkin methods for natural frequencies of high-speed axially moving beams

In this paper, natural frequencies of planar vibration of axially moving beams are numerically investigated in the supercritical ranges. In the supercritical transport speed regime, the straight equilibrium configuration becomes unstable and bifurcate in multiple equilibrium positions. The governing equations of coupled planar is reduced to two nonlinear models of transverse vibration. For motion about each bifurcated solution, those nonlinear equations are cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform. The natural frequencies are investigated for the beams via the Galerkin method to truncate the corresponding governing equations without nonlinear parts into an infinite set of ordinary-differential equations under the simple support boundary. Numerical results indicate that the nonlinear coefficient has little effects on the natural frequency, and the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters and the integro-partial-differential equation yields results quantitatively closer to those of the coupled equations.     

Nonlinearly saturated dynamical state of a three-wave mode-coupled dissipative system with linear instability

Linearly unstable dissipative systems with quadratic nonlinearity occurring in plasma physics, optics, fluid mechanics, etc. are often modeled by a general set of three-wave mode-coupled ordinary differential equations for complex variables. Bounded attractors of the set approximate nonlinearly saturated turbulent states of real physical systems. Exact criteria for boundedness of the attractors are found. Fundamentally different kinds of asymptotic behavior of the wave triad are classified in the parameter space and quantitatively assessed.     

On diffraction and dispersion effect on three wave interaction

The effect of diffraction and dispersion on three wave coupling is investigated. The instability leading to space modulation of packets is obtained. This instability and its nonlinear stage is similar to modulational instability of quasimonochromatic waves. The localized structures (solitons and waveguide) can be a result of the instability. We demonstrate that one-dimensional and two-dimensional such structures are unstable. The existence of stable three-dimensional solitons is shown.     

Semiclassical quantization rules near separatrices

We derive semiclassical quantization equations with uniform estimate of the error term near unstable equilibria of the classical system for the one-dimensional Schrödinger operator.     

A model of an open universe with a variable equation of state

An approach is proposed allowing one to find exact solutions of the equations of GTR, written for a conformally-planar metric with a specific dependence of the coordinates. The approach is based on an analogy between the equation determining the pressure, and the one-dimensional Newton's equation with a potential right side. The equation of state is not fixed, but is found by giving the form of the potential. The Friedmann solution corresponds to the free motion of a Newtonian particle. Exact solutions are derived describing the mixture of dust and ultrarelativistic matter. The analysis shows that a model containing only radiation is unstable with respect to small dustlike additions. In all the models the equation of state is a function of a definite combination of space — time coordinates.The work was carried out with the financial support of the Krasnoyarsk Territorial Science Fund, grant D(1F0019).Krasnoyarsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 89–94, January, 1994.     

A study of the asymmetric Malkus waterwheel: the biased Lorenz equations

In this work, the asymmetric case of the Malkus waterwheel is studied, where the water inflow to the system is biasing the system toward stable motion in one direction, like a Pelton wheel. The governing equations of this system, when expressed in Fourier space and decoupled to form a closed set, can be mapped into a four-dimensional space where they form a quasi-Lorenz system. This set of equations is analyzed in light of analogues of the Rayleigh Bernard convection and conclusions are drawn. The properties and behavior of the equations are studied and correlated to the physical model. Phase space behavior and linear stability analysis are used for this. Spectral analysis is used as a qualitative measure of chaos. Chaotic behavior is quantified through the calculation of the Lyapunov exponents and these are further correlated to the bifurcation diagrams for a conclusive analysis of the dynamical behavior of the system.     

Low-energy elastic scattemng of pions by the deuteron

The problem of the elastic scattering of pions by a deuteron is considered using the separable representation of the two-body t-matrix. The Faddeev equations are reduced to a set of one-dimensional integral equations by separating the angular variables. The dependence of the π-d scattering length on the form of two-body interaction and on the values of the π-N scattering lengths is studied in the case of a one-term nonlocal potential with separable variables. The π-d scattering length proves to be practically independent of the two-body interaction form, and is essentially dependent on the values of the π-N scattering lengths.     

Structure quantum numbers and unstable leptons

Wave equations make it possible to ascribe the quantum numbers to leptons theoretically. Minimal extension of the groups describing stable leptons led to three new types of groups. Each of them has an appropriate structure invariant ±1.0 and its own set of substructures having a physical interpretation in terms of stable leptons. The properties of the above-mentioned groups allow one to generate new types of equations and to relate them with two doublets (particle-antiparticle) of massive, charged unstable leptons and a doublet of massive unstable neutrinos.     

Faddeev equations and the method of hyperspherical harmonics in the problem of three-nucleon continuum

A method for solving the Faddeev equations in configuration space is developed for a three-nucleon system in the continuum by using the decomposition over a hyperspherical basis. The wave functions of Nd-system, phase shifts, and cross sections of Nd-scattering at subthreshold energies are calculated. Also, within the framework of this method, one-dimensional integral equations are formulated for the problem of infinite motion of all three strongly interacting particles, and the Faddeev equations for a system of three hadrons with Coulomb interaction in the continuum are modified. Similar methods of investigation of three-particle systems are reviewed.     

A note on chaotic unimodal maps and applications

Based on the word-lift technique of symbolic dynamics of one-dimensional unimodal maps, we investigate the relation between chaotic kneading sequences and linear maximum-length shift-register sequences. Theoretical and numerical evidence that the set of the maximum-length shift-register sequences is a subset of the set of the universal sequence of one-dimensional chaotic unimodal maps is given. By stabilizing unstable periodic orbits on superstable periodic orbits, we also develop techniques to control the generation of long binary sequences.     

Discrete dynamical equations in Minkowski space

Discrete difference equations in Minkowski space are obtained and the discrete Minkowski force is shown to be a four-vector. A transformation from a discrete dynamical equation in Minkowski space to a Lorentz-invariant difference equation in one-dimensional space is given.     

变分法研究一维Bose-Fermi系统的稳定性

利用能量泛函变分法研究了一维Bose-Fermi系统稳定基态的存在条件.根据Bose-Fermi系统的Lagrange量可以得到三维Bose-Fermi体系所满足的非线性动力学方程组.当外势阱的横向囚禁频率远大于轴向囚禁频率时,体系可以当作一维模型来处理.从描述三维体系的动力学方程可以得到描述一维体系的动力学方程,选取适当的无量纲参数,可以对一维动力学方程组进行无量纲处理,得到数值计算和理论分析中常用到的无量纲方程.选择高斯型试探解(简单孤立子解),利用能量泛函变分法得到一维Bose-Fermi体系稳定的高斯型孤立子存在条件.分析了两种特殊情况下孤立子能够稳定存在的区域以及原子数的临界条件,最后得出了一般情况下稳定基态存在时临界散射长度与原子数以及波包宽度之间的关系.     

R-matrix approach to the 3-body problem

The asymptotic form of the Faddeev amplitude in coordinate space is derived in various orders. This form and the structure of the Faddeev equations allow by aR-matrix method to establish a set of equations directly for the 3-body on-shellT-matrix elements. The procedure is equally well suited for local and nonlocal interactions.     

The backreaction for conformally invariant fields on a conformally flat background

We study conformally invariant fields within the context of semi-classical gravity. We claim that, generically, conformal flatness implies Friedmann-Robertson-Walker behaviour. A proof is presented here for the case in which the Ricci tensor is of the perfect fluid type. We also rewrite the field equations as a quadratic three dimensional autonomous system of ordinary differential equations, the critical points of which are Minkowski space and de Sitter space. Both these critical points are unstable in the linear as well as in the non-linear theory.This essay received an honorable mention from the Gravity Research Foundation, 1990 —Ed.