The time dependent amplitude equation for the Swift-Hohenberg problem

Precise estimates for the validity of the amplitude approximation for the swift-Hohenberg equation are given, in a fully time dependent framework. It is shown that small solutions of orderO() which are modulated like stationary solutions have an evolution which is well described in the amplitude approximation for a time of orderO(^-2). For the proofs, we use techniques for nonlinear semigroups and oscillatory integrals.Dedicated to Res Jost and Arthur Wightman     

Control-parameter-dependent Swift-Hohenberg equation as a model for bioconvection patterns

We consider a complex Swift-Hohenberg equation with control-parameter-dependent coefficients and use it as a model to describe dynamical features seen in an experimental bacterial bioconvection pattern. In particular, we give numerical results showing the development of a phase-unstable pattern behind a moving front.     

Additive noise-induced Turing transitions in spatial systems with application to neural fields and the Swift-Hohenberg equation

This work studies the spatio-temporal dynamics of a generic integral-differential equation subject to additive random fluctuations. It introduces a combination of the stochastic center manifold approach for stochastic differential equations and the adiabatic elimination for Fokker-Planck equations, and studies analytically the systems’ stability near Turing bifurcations. In addition two types of fluctuation are studied, namely fluctuations uncorrelated in space and time, and global fluctuations, which are constant in space but uncorrelated in time. We show that the global fluctuations shift the Turing bifurcation threshold. This shift is proportional to the fluctuation variance. Applications to a neural field equation and the Swift-Hohenberg equation reveal the shift of the bifurcation to larger control parameters, which represents a stabilization of the system. All analytical results are confirmed by numerical simulations of the occurring mode equations and the full stochastic integral-differential equation. To gain some insight into experimental manifestations, the sum of uncorrelated and global additive fluctuations is studied numerically and the analytical results on global fluctuations are confirmed qualitatively.     

First-order optical systems with real eigenvalues

It is shown that a lossless first-order optical system whose real symplectic ray transformation matrix can be diagonalized and has only real eigenvalues, is similar to a separable hyperbolic expander in the sense that the respective ray transformation matrices are related by means of a similarity transformation. Moreover, it is shown how eigenfunctions of such a system can be determined, based on the fact that simple powers are eigenfunctions of a separable magnifier. As an example, a set of eigenfunctions of a hyperbolic expander is determined and the resemblance between these functions and the well-known Hermite-Gauss modes is exploited.     

On small stationary localized solutions for the generalized 1-D Swift-Hohenberg equation

We prove the existence of small localized stationary solutions for the generalized Swift-Hohenberg equation and find under some assumption a part of a boundary of their existence in the parameter plane. The related stationary equation creates a reversible Hamiltonian system with two degrees of freedom that undergoes the Hamiltonian-Hopf bifurcation with an additional degeneracy. We investigate this bifurcation in a two-parameter unfolding by means of the sixth-order normal form for the related Hamiltonian. The region where no localized solutions exist has been pointed out as well. (c) 1995 American Institute of Physics.     

Strings, anisometric gels, and solutions in chemical and biological systems

This article presents a review of the works that are devoted to the formation of supramolecular anisotropic structures in a wide class of model solutions of homochiral compounds and biological fluids. Spontaneous formation of crystalline rigid strings takes place in the study solutions at low concentrations (10^?3-10^?2 M) and continues to the stage of solution solidification. This is preceded, as revealed by dynamic light scattering, by the formation of linear associates, isometric granules, and strings themselves, which strongly and nonmonotonously affect chiroptical properties and the IR and UV absorption in solutions. Strings and similar objects can be considered as model elements of intercellular switches. The described structure apparently characterizes many biological fluids.     

Coarsening and frozen faceted structures in the supercritical complex Swift-Hohenberg equation

The supercritical complex Swift-Hohenberg equation models pattern formation in lasers, optical parametric oscillators and photorefractive oscillators. Simulations of this equation in one spatial dimension reveal that much of the observed dynamics can be understood in terms of the properties of exact solutions of phase-winding type. With real coefficients these states take the form of time-independent spatial oscillations with a constant phase difference between the real and imaginary parts of the order parameter and may be unstable to a longwave instability. Depending on parameters the evolution of this instability may or may not conserve phase. In the former case the system undergoes slow coarsening described by a Cahn-Hilliard equation; in the latter it undergoes repeated phase-slips leading either to a stable phase-winding state or to a faceted state consisting of an array of frozen defects connecting phase-winding states with equal and opposite phase. The transitions between these regimes are studied and their location in parameter space is determined.     

Evolution equation of the optical tomogram for arbitrary quantum Hamiltonian and optical tomography of relativistic classical and quantum systems

The dynamic equation for the optical tomogram of nonrelativistic quantum system with an arbitrary Hamiltonian is obtained. The kinetic equation in the classical relativistic kinetics is discussed, and its optical tomography representation is obtained. Dynamic equations for the Wigner functions of relativistic spinless quantum particles in electromagnetic and scalar fields are obtained. Optical tomographic-distribution functions of weakly relativistic spinless quantum particles are introduced, and dynamic equations for these functions in weak electric and scalar fields are obtained.     

Intercalation effect on the optical properties,structure, and chemical composition of thin film systems

The effect of gaseous ammonia intercalation (IC) on the optical and structural properties of two-layer Ag-PbI₂ systems and PbI₂ films, both freshly prepared and irradiated by laser beams with wavelengths λ _a = 442 nm and λ₀ = 633 nm, is studied. The changes in the spectra of optical transmittance are shown to be caused by changes in the structure and chemical composition of the samples. The results of the performed electron diffraction studies and their comparison with the known data of X-ray analysis give grounds for believing that intercalation compounds of several types appear in thin film samples     

Nonparaxial equation for linear and nonlinear optical propagation

The formalism of coupled-mode theory, specialized to the continuum of radiation modes, allows us to extend the standard parabolic wave equation to include nonparaxial terms and vectorial effects, and, in particular, to generalize the nonlinear Schr?dinger equation that describes propagation in the presence of an intensity-dependent refractive index.     

Silver nanocrystals modified microstructured polymer optical fibres for chemical and optical sensing

In-fibre chemical and optical sensors based on silver nanocrystals modified microstructured polymer optical fibres (MPOFs) were demonstrated. The silver nanocrystals modified MPOFs were formed by direct chemical reduction of silver ammonia complex ions on the templates of array holes in the microstructure polymer optical fibres. The nanotube-like and nanoisland-like Ag-modified MPOFs could be obtained by adjusting the conditions of Ag-formation in the air holes of MPOFs. SEM images showed that the higher concentration of the reaction solution (silver ammonia 0.5 mol/L, glucose 0.25 mol/L), gave rise to a tubular silver layer in MPOF, while the lower concentration (silver ammonia 0.1 M, glucose 0.05 M) produced an island-like Ag nanocrystal modified MPOF. The tubular Ag-MPOF composite fibre was conductive and could be directly used as array electrodes in electrochemical analyses. It displayed high electrochemical activity on sensing nitrate or nitrite ions. The enhanced fluorescence of dye molecules was observed when the island-like Ag-modified MPOF was inserted into a fluorescent dye solution.     

Global and local optimization for optical systems

Issues concerning the efficiency of the Global Explorer (GE) algorithm are studied. Global optimization with GE consists of many local optimization runs with or without the escape function using the damped-least-squares method. In order to improve the efficiency of the local optimization and the GE run, two search schemes are incorporated. The first one searches for the best damping factor which effectively determines the optimum direction of the solution vector in the multi-dimensional variable space, and the second search is conducted along that direction to find the optimum length of the solution vector. Experiments are also made to determine the optimum default values for the parameters of the escape function. The algorithm with the improvements is successfully implemented in the GOLD program developed by Beijing Institute of Technology. Design examples using the software are given, which show that the global optimization feature is practical and powerful.     

Singular Lagrangian, Hamiltonization and Jacobi last multiplier for certain biological systems

We study the construction of singular Lagrangians using Jacobi’s last multiplier (JLM). We also demonstrate the significance of the last multiplier in Hamiltonian theory by explicitly constructing the Hamiltonian of the Host-Parasite model and a Lotka-Volterra mutualistic system, both of which are well known first-order systems of differential equations arising in biology.     

Nonlocal complex Ginzburg-Landau equation for electrochemical systems

By means of an extended center-manifold reduction, we derive the nonlocal complex Ginzburg-Landau equation (NCGLE) valid for electrochemical systems with migration coupling. We carry out the stability analysis of the uniform oscillation, elucidating the role of the nonlocal coupling in electrochemical systems at the vicinity of a supercritical Hopf bifurcation. We apply the NCGLE to an experimental system, an N-type negative differential resistance electrochemical oscillator, which is shown to exhibit electrochemical turbulence for wide parameter ranges.     

Boussinesq方程组的广义变分原理

半反推法是何吉欢为了寻求物理问题的变分原理而提出的,可避免由拉氏乘子法引起的临界变分现象. 应用半反推法分别获得了描述水波运动的两类Boussinesq方程组的一族广义变分原理,并验证了它们的正确性. 关键词： 半反推法 广义变分原理 Boussinesq方程组