The Fokker-Planck equation for arbitrary nonlinear noise

We present the Fokker-Planck equation for arbitrary nonlinear noise terms. The white noise limit is taken as the zero correlation time limit of the Ornstein-Uhlenbeck process. The drift and diffusion coefficients of the Fokker-Planck equation are given by triple integrals of the fluctuations. We apply the Fokker-Planck equation to the active rotator model with a fluctuating potential barrier which depends nonlinearly on an additive noise. We show that the nonlinearity may be transformed into the correlation of linear noise terms.     

Effect of the dissipation, dispersion, and diffraction on the amplitude and the transverse stability of solitons

The transverse stability and the amplitude variations of soliton-like wave motions in the presence of nonlinearity, dispersion, diffraction, and dissipation in the medium are studied. The wave process is described by a quintic nonlinear evolution equation. It is demonstrated that the stability of the solution does not depend on the dissipation when the dissipation, diffraction, and dispersion are of the same order of magnitude. It depends on the sign of the ratio of the diffraction and dispersion coefficients. When the sign is positive, the soliton is stable. This result coincides with the stability condition for a nonlinear modulation wave. For the case of strong dissipation, an expression describing the soliton amplitude is obtained and the dissipation is shown to have no effect on the soliton stability.     

Theory of nonlinear external noise

We investigate a stochastic differential equation (SDE) with general nonlinearity in the noise. We derive an integro-differential equation for the probability density (PD) and an approximate equation for the mean first-passage time (MFPT). The approximate Fokker-Planck equation (AFPE) and MFPT are obtained for some examples.     

Heavy-and light-nuclei acoustic dressed shock waves in white dwarfs

In this investigation, the evolution of heavy-and light-nuclei acoustic (HLNA) dressed shock waves (DSWs) due to the contribution of higher order of nonlinearity and dissipation effects has been examined in a degenerate quantum plasma composed of inertial heavy as well as light nuclei and inertia-less ultra-relativistic degenerate electrons. By employing the reductive perturbation method, the nonlinear Burgers equation is derived. Further, an inhomogeneous Burgers-type equation accounting for the higher order contributions of nonlinearity and dissipation is also derived. With the insertion of higher order effects, a new humped type or dressed shock structures are evolved. The influence of different plasma parameters on the dynamical evolution of the HLNA-DSWs is examined. It is observed that these plasma parameters play significant role on the characteristics of HLNA-DSWs and their corresponding electric fields. The findings of present investigation may be applicable to provide a new insight to understand the evolution of HLNA-DSWs in different dense astrophysical regions such as white dwarfs.     

On the chaotic nature of turbulence observed in benchmark analysis of nonlinear plasma simulation

Simulational results of two dissipative interchange turbulence (Rayleigh-Taylor-type instability with dissipation) models with the same physics are compared. The convective nonlinearity is the nonlinear mechanism in the models. They are shown to have different time evolutions in the nonlinear phase due to the different initial value which is attributed to the initial noise. In the first model (A), a single pressure representing the sum of the ion and electron components is used (one-fluid model). In the second model (B) the ion and electron components of the pressure fields are independently solved (two-fluid model). Both models become physically identical if we set ion and electron pressure fields to be equal in the model (B). The initial conditions only differ by the infinitesimally small initial noise due to the roundoff errors which comes from the finite difference but not the differentiation. This noise grows in accordance with the nonlinear development of the turbulence mode. Interaction with an intrinsic nonlinearity of the system makes the noise grow, whose contribution becomes almost the same magnitude of the fluctuation itself in the results. The instantaneous deviation shows the chaotic characteristics of the turbulence. (c) 1997 American Institute of Physics.     

Effect of the interaction of spatial modes on the nonlinear dynamics of sound beams

Propagation of an acoustic beam in a medium with a combined second-and third-order nonlinearity is studied. The derivation of the dynamics equations and the determination of modes is performed using the orthogonal-projection operator technique. The problem on the beam evolution considered with allowance for weak nonlinearity, diffraction, and dissipation leads to a set of equations describing the interaction of directed waves and a quasi-stationary (thermal) mode. In the conditions of a directed beam, the inclusion of the interaction leads to a modified Khokhlov-Zabolotskaya-Kuznetsov equation with quadratic and cubic nonlinearities. The solutions to the problem are obtained in the region near the beam axis, in the form of series expansions in the transverse coordinate up to the focal point. The results of calculations are represented in graphical form for different nonlinearity combinations.     

Effects of strong noise on attractors of dynamical systems

Stationary solutions of the Fokker-Planck equation are found by expansions of the probability distribution with respect to the reciprocal noise strength. It is shown that this expansion is convergent. Explicit representations for the probability distribution are obtained by numerical simulations for the Lorenz model and for a model of generators with inertial nonlinearity (GIN). The obtained distributions show an increasing amount of fine structure with decreasing noise which more and more reflects the fractal attractor structure. Results of measurements of the power spectrum of the GIN and of the distribution in the phase space are presented in dependence on the noise strengths.     

Video solitons in a dispersive transmission line with the nonlinear capacitance of a p-n junction

Possible types of low-frequency electromagnetic solitary waves in a dispersive LC transmission line with a quadratic or cubic capacitive nonlinearity are investigated. The fourth-order nonlinear wave equation with ohmic losses is derived from the differential-difference equations of the discrete line in the continuum approximation. For a zero-loss line, this equation can be reduced to the nonlinear equation for a transmission line, the double dispersion equation, the Boussinesq equations, the Korteweg-de Vries (KdV) equation, and the modified KdV equation. Solitary waves in a transmission line with dispersion and dissipation are considered.     

Nonmarkovian dynamics of stochastic differential equations with quadratic noise

We consider a stochastic differential equation with a quadratic nonlinearity in the noise. We derive equations for the steady state probability density and joint probability distribution valid beyond a markovian approximation. We do not assume that the strength of the random term is small. The equations are derived for the case of an Ornstein-Uhlenbeck noise and also for a dichotomic noise. A comparison is made. We discuss some examples for which correlation functions and the associated relaxation times are calculated.     

Widely tunable, nondegenerate three-wave mixing microwave device operating near the quantum limit

We present the first experimental realization of a widely frequency tunable, nondegenerate three-wave mixing device for quantum signals at gigahertz frequency. It is based on a new superconducting building block consisting of a ring of four Josephson junctions shunted by a cross of four linear inductances. The phase configuration of the ring remains unique over a wide range of magnetic fluxes threading the loop. It is thus possible to vary the inductance of the ring with flux while retaining a strong, dissipation-free, and noiseless nonlinearity. The device has been operated in amplifier mode, and its noise performance has been evaluated by using the noise spectrum emitted by a voltage-biased tunnel junction at finite frequency as a test signal. The unprecedented accuracy with which the crossover between zero-point fluctuations and shot noise has been measured provides an upper bound for the noise and dissipation intrinsic to the device.     

From Lagrangian to Brownian motion

We present a Lagrangian describing an idealized liquid interacting with a particle immersed in it. We show that the equation describing the motion of the particle as a functional of the initial conditions of the liquid incorporates noise and friction, which are attributed to specific dynamical processes. The equation is approximated to yield a Langevin equation with parameters depending on the Lagrangian and the temperature of the liquid. The origin of irreversibility and dissipation is discussed.     

Turbulent bursts,spots and slugs in a generalized Ginzburg-Landau equation

The time-dependent generalized Ginzburg-Landau equation is generalized by adding a higher-order nonlinearity and by allowing the coefficients to be a function of space. With the addition of external noise, this equation exhibits turbulent bursts, spots and slugs similar to those observed in many open-flow fluid systems. The external noise plays an important role in the spatio-temporal intermittency that is observed. Ideas presented in this paper provide a conceptual framework in which to better understand behaviour occuring in pipe flow and other open-flow systems. A few experiments to support some of these ideas and a mechanism for slug production in pipe flow are suggested.     

Group Analysis,Fractional Explicit Solutions and Conservation Laws of Time Fractional Generalized Burgers Equation

The generalized fractional Burgers equation is studied in this paper. Using the classical Lie symmetry method, all of the vector fields and symmetry reduction of the equation with nonlinearity are constructed. In particular,an exact solution is provided by using the ansatz method. In addition, other types of exact solution are obtained via the invariant subspace method. Finally, conservation laws for this equation are derived.     

Augmented Langevin description of multiplicative noise and nonlinear dissipation in Hamiltonian systems

The augmented Langevin approach described in a previous article is applied to the problem of introducing multiplicative noise and nonlinear dissipation into an arbitrary Hamiltonian system in a thermodynamically consistent way, so that a canonical equilibrium distribution is approached asymptotically at long times. This approach leads to a general nonlinear fluctuation-dissipation relation which, for a given form of the multiplicative noise (chosen on physical grounds), uniquely determines the form of the nonlinear dissipative terms needed to balance the fluctuations. In addition to the noise and dissipation terms, the augmented Langevin equation contains an additional term whose form depends on the stochastic interpretation rule used. This term vanishes when the Stratonovich rule is chosen and the noise itself is of a Hamiltonian origin. This development provides a simple phenomenological route to results previously obtained by detailed analysis of microscopic system-bath models. The procedure is illustrated by applications to a mechanical oscillator with fluctuating frequency, a classical spin in a fluctuating magnetic field, and the Brownian motion of a rigid rotor.     

A Semiclassical Theory of a Dissipative Henon—Heiles System

A semiclassical theory of a dissipative Henon—Heiles system is proposed. Based on -scaling of an equation for the evolution of the Wigner quasiprobability distribution function in the presence of dissipation and thermal diffusion, we derive a semiclassical equation for quantum fluctuations, governed by the dissipation and the curvature of the classical potential. We show how the initial quantum noise gets amplified by classical chaotic diffusion, which is expressible in terms of a correlation of stochastic fluctuations of the curvature of the potential due to classical chaos, and ultimately settles down to equilibrium under the influence of dissipation. We also establish that there exists a critical limit to the expansion of phase space. The limit is set by chaotic diffusion and dissipation. Our semiclassical analysis is corroborated by numerical simulation of a quantum operator master equation.     

Multiple singular manifold method and extended direct method: Application to the burgers equation

This paper considers the relationship between the multiple singular manifold method (MSMM) and the extended direct method (EDM) for studying partial differential equations. It is shown that the similarity reductions using EDM can be obtained by MSMM. The prototype example for illustrating the approach is the Burgers equation, which is the simplest evolution equation to embody nonlinearity and dissipation. As a conclusion of the MSMM, we obtain a set of Bäcklund transformations of the Burgers equation.     

金属1/f噪声理论

本文对已有四十多年实验观察历史的金属中的1/f噪声现象提出了新的物理机制:金属中局域电子的带有红外发散的扩散迁移引起1/f噪声。本文基于低频涨落、耗散、弛豫的普适理论建立了非马尔科夫扩散方程,由此求得的噪声谱能解释1/f噪声的四个基本特征,并在参数合理取值的情况下和实验结果符合得较好。本文还对已有理论等做了若干讨论。 关键词：     

Parametric generation of low-frequency sound in the propagation of high-intensity modulated noise

With the use of the one-dimensional Burgers equation, the evolution of a high-intensity noise with periodically modulated intensity is analyzed. The nonlinearity is shown to lead to partial suppression of the amplitude modulation and to the generation of a regular low-frequency component. The probability distributions and the power spectra of the field are studied.