A variational approach to stochastic nonlinear problems

A variational principle is formulated which enables the mean value and higher moments of the solution of a stochastic nonlinear differential equation to be expressed as stationary values of certain quantities. Approximations are generated by using suitable trial functions in this variational principle and some of these are investigated numerically for the case of a Bernoulli oscillator driven by white noise. Comparison with exact data available for this system shows that the variational approach to such problems can be quite effective.     

A variational approach to nonlinear dynamics of nanoscale surface modulations

In this paper, we propose a variational formulation to study the singular evolution equations that govern the dynamics of surface modulations on crystals below the roughening temperature. The basic idea of the formulation is to expand the surface shape in terms of a complete set of basis functions and to use a variational principle equivalent to the continuum evolution equations to obtain coupled nonlinear ordinary differential equations for the expansion coefficients. Unlike several earlier approaches that rely on ad hoc regularization procedures to handle the singularities in the evolution equations, the only inputs required in the present approach are the orientation dependent surface energies and the diffusion constants. The method is applied to study the morphological equilibration of patterned unidirectional and bidirectional sinusoidal modulations through surface diffusion. In the case of bidirectional modulations, particular attention is given to the analysis of the profile decay as a function of ratio of the modulating wavelengths in the coordinate directions. A key question that we resolve is whether the one-dimensional decay behavior is recovered as one of the modulating wavelengths of the two-dimensional profiles diverges, or whether one-dimensional decay has qualitatively distinct features that cannot be described as a limiting case of the two-dimensional behavior. In contrast to some earlier suggestions, our analytical and numerical studies clearly show that the former situation is true; we find that the one-dimensional profiles, like the highly elongated two-dimensional profiles, decay with formation of facets. While our results for the morphological equilibration of symmetric one-dimensional profiles are in agreement with the free-boundary formulation of Spohn, the present approach can also be used to study the evolution of asymmetric profile shapes where the free-boundary approach is difficult to apply. The variational method is also used to analyze the decay of unidirectional modulations in the presence of steps that arise in most experimental studies due to a small misorientation from the singular surface.     

Electrostatic potential: A new approach for some mixed boundary value problems

A new approach to the solution of problems of electrostatics, some of them with mixed boundary conditions, is presented. The proposed scheme can be used in cases were we have a formal solution in the form of a series in Legendre polynomials and the boundary or matching conditions are given not on the whole interval (0, π) of the polar variable, θ, but only over the interval (0, π/2) or (π/2, π). Truncation of the series after the Nth term and the projection on the subspace generated by the set of the first N even (or odd) Legendre polynomials allows us to determine the unknown coefficients of the approximate solution. The results show rapid convergence toward the exact values as we increase the number of terms, N, included in the approximate solutions. The procedure allows to solve approximately some problems whose exact solutions, we believe, are not yet known.     

A non-equilibrium molecular dynamics approach to fluid transfer through microporous membranes

The colour field non-equilibrium molecular dynamics method is applied to model fluid transfer between two fluid phases separated by microporous membranes (i.e. pore width ≤ 2 nm). The model is simple. All particles show short range spherical interactions, but reproduce the main features of the experimental phenomenology of molecular sieve membranes. An external force is applied to the fluid particles and, due to the presence of the membrane, a stationary flux across the system results. A range of mass transfer coefficients was investigated and the fluxes were found to be linear with force intensities. The density was found to have a small effect on molecular mobility, while temperature had a more significant effect, showing the features of an activated process. In addition, the changes in these fluxes with the membrane size, and the cross interaction parameters between the fluid and the membrane particles were examined.     

Finite-temperature nonlinear dynamics in cavity QED: A thermofield dynamics approach

Heath-bath effects in the dynamics of atom+cavity system are studied. The temperature effects are explored using thermofield dynamics formalism. It is found that the dynamics of the system is sensitive to small changes in the temperature and the thermal effects lead to increasing instabilities by causing transitions from regular to chaotic motion.     

A new application of Riccati equation to some nonlinear evolution equations

By means of symbolic computation, a new application of Riccati equation is presented to obtain novel exact solutions of some nonlinear evolution equations, such as nonlinear Klein-Gordon equation, generalized Pochhammer-Chree equation and nonlinear Schrödinger equation. Comparing with the existing tanh methods and the proposed modifications, we obtain the exact solutions in the form as a non-integer power polynomial of tanh (or tan) functions by using this method, and the availability of symbolic computation is demonstrated.     

New approach to nonlinear dynamics of fullerenes and fullerites

A new type of nonlinear (anharmonic) excitations—bushes of vibrational modes—in physical systems with point or space symmetry is discussed. All infrared-active and Raman-active bushes for C₆₀ fullerene are found by means of special group-theoretical methods.     

A simple approach to nonlinear oscillators

A very simple and effective approach to nonlinear oscillators is suggested. Anyone with basic knowledge of advanced calculus can apply the method to finding approximately the amplitude-frequency relationship of a nonlinear oscillator. Some examples are given to illustrate its extremely simple solution procedure and an acceptable accuracy of the obtained solutions.     

A sigma-model approach to glassy dynamics

In this contribution we review recent progress in understanding fluctuations in the aging process of macroscopic systems, and we propose further tests of these ideas. We discuss how the emergence of a symmetry in aging systems, global timereparametrization invariance, could be responsible for the observed ‘universal’ behavior of local and mesoscopic non-equilibrium fluctuations. We discuss (i) the two-time scaling and functional form of the distribution of local correlations and responses; (ii) the scaling of multi-time correlations and susceptibilities; (iii) how the above can be derived from a random surface effective action; (iv) the behavior of a diverging two-time dependent correlation length; (v) how these ideas apply to off-lattice particle systems.     

Post-buckling longterm dynamics of a forced nonlinear beam: A perturbation approach

The aim of this paper is to determine by a singular perturbation approach the dynamic response of a harmonically forced system experiencing a pitchfork bifurcation. The model of an extensible beam forced by a harmonic excitation and subject to an axial static buckling is space-discretized by a Galerkin approach and studied by the Normal Form Method for different values of equation parameters influencing the nonlinear dynamic behavior like damping coefficient, load amplitude and frequency. A relevant issue in the perturbation methods is the concept of small and zero divisors which are related to the possibility to build a transformation that simplifies the original studied problem, i.e. to obtain the Normal Form, by eliminating as much as possible nonlinearities in the equations. For nonconservative systems, like structural damped systems, there are no conditions in the prior literature that define what “small” means relatively to a divisor. In the present paper some conditions about the order of magnitude of the divisors with respect to the perturbation entity are given and related to some physical parameters in the governing equations in order to estimate the relevance of some nonlinear effects.     

非等温非牛顿黏性流体流动问题的修正光滑粒子动力学方法模拟

为准确、有效地模拟非等温非牛顿黏性流体的流动问题,本文基于一种不含核导数计算的核梯度修正格式和不可压缩条件给出了一种改进光滑粒子动力学(SPH)离散格式,它较传统SPH离散格式具有较高精度和较好稳定性.同时,为准确地描述温度场的演化过程,建立了非牛顿黏性的SPH温度离散模型.通过对等温Poiseuille流、喷射流和非等温Couette流、4:1收缩流进行模拟,并与其他数值结果作对比,分别验证了改进SPH方法模拟非牛顿黏性流动问题的可靠性和提出的SPH温度离散模型求解非等温流动问题的有效性和准确性.随后,运用改进SPH方法结合SPH温度离散模型对环形腔和C形腔内非等温非牛顿黏性流体的充模过程进行了试探性模拟研究,分析了数值模拟的收敛性,讨论了不同位置处热流参数对温度和流动的影响.     

A novel approach to Ising problems

In 2000 Istrail suggested that calculating the partition function of non‐planar Ising models is an NP‐complete problem, implying that these problems are intractable and thus essentially unsolvable. In this note we discuss the validity of this suggestion and introduce the idea of gauging on an exact equation. We illustrate how this method works by applying it to two non‐planar Ising models, namely the 2D model with nearest and weak next nearest neighbor interactions and the anisotropic 3D model.     

一种分解指数算符的简洁方法

提出一种新的分解指数算符的方法,这种方法不仅简单易懂,而且有助于找到更多的算符恒等式.     

Variational approaches to the problems of plasma stability and of nonlinear plasma dynamics

A variational method is developed in order to investigate the nonlinear dynamics and stability of plasma using hydrodynamic plasma models, namely, the one-fluid, Hall, and electron MHD models. The key idea of the method is to adequately take into account variational symmetries and the associated conservation laws inherent in these hydrodynamic models. This approach is applied to derive variational criteria for the stability of a steadily moving plasma and to propose a variational method of the adiabatic separation of fast and slow motions, which makes it possible to simplify (reduce) the basic hydrodynamic models.     

Precise fabrication of nanomaterials: a nonlinear dynamics approach

Modeling of the precise fabrication in the self-assembling of particles is studied using the nonlinear Langevin equation system. The numerical simulation showed a marked ordering of the particles as a function of time after some induction period. The abnormally enlarged fluctuation was found around the start of the evident ordering. After the fluctuation, a sudden increase of the cluster size was observed. The results corresponded well to the dynamics due to the formation of the critical cluster. The shape of the critical cluster around the enlarged fluctuation was not compact and showed fractal-like structures. The fluctuation of the cluster size around the formation of the critical cluster was explained by the anomalous fluctuation theorem for the generalized Langevin equation. The characterization of the stochastic dynamics of the critical clusters rationalized the concept of dynamic templating for the fabrication technique of the self-assembling of nanoparticles, that is, the structural constraint on the particle assembly by externally adding the resonance frequencies that match with the localized nonlinear vibrational modes of the target structures originating from thermal (Brownian) activation.     

A perspective on nonlinear dynamics

We present a brief report on the conference, a summary of the proceedings, and a discussion on the field of nonlinear science studies and its current frontiers.