Optimal Gaussian solutions of nonlinear stochastic partial differential equations

We present a linearization procedure of a stochastic partial differential equation for a vector field (X _i (t,x)) (t[0, ),xR ^d ,i=l,...,n): _t X _i (t,x)=b _i (X(t, x)) +D, X _i (t, x) + _i f _i (t, x). Here is the Laplace-Beltrami operator inR ^d , and (f _i (t,x)) is a Gaussian random field with f _i (t,x)f _j (t,x) = _ij (t – t)(x – x). The procedure is a natural extension of the equivalent linearization for stochastic ordinary differential equations. The linearized solution is optimal in the sense that the distance between true and approximate solutions is minimal when it is measured by the Kullback-Leibler entropy. The procedure is applied to the scalar-valued Ginzburg-Landau model in R¹ withb ₁(z) =z - vz ³. Stationary values of mean, variance, and correlation length are calculated. They almost agree with exact ones if 1.24 ( ² ₁ ⁴ /D ₁ ^1/3:= _c . When _c , there appear quasistationary states fluctuating around one of the bottoms of the potentialU(z) = b ₁(z)dz. The second moment at the quasistationary states almost agrees with the exact one. Transient phenomena are also discussed. Half-width at half-maximum of a structure function decays liket ^–1/2 for small t. The diffusion term _x ² X accelerates the relaxation from the neighborhood of an unstable initial stateX(0,x) 0.     

Studies in nonlinear stochastic processes. I. Approximate solutions of nonlinear stochastic differential equations by the method of statistical linearization

A method based on a variational procedure is presented which provides simple and useful approximate solutions to a wide variety of nonlinear stochastic differential equations. This method of statistical linearization is most successful when the stochasticity of the differential equation is due to excitations which are normally distributed or harmonic with random phase. Effects due to deviations from normality can be corrected for in a systematic fashion. Comments regarding existence and uniqueness are given and some error bounds arising from the use of statistical linearization are computed.Supported by the National Science Foundation under Grants NSF GP 32031X and MPS72-04353 A03.     

A stochastic return map for stochastic differential equations

A method is presented for constructing a stochastic return map from a stochastic differential equation containing a locally stable limit cycle and small-amplitude [O()] additive Gaussian colored noise. The construction is valid provided the correlation time isO() orO(1). The effective noise in the return map has nonzeroO( ²) mean and is state dependent. The method is applied to a model dynamical system, illustrating how the effective noise in the return map depends on both the original noise process and the local deterministic dynamics.     

简谐速度噪声与简谐噪声环境中谐振子的动力学共振

通过求解简谐势场中的广义量子朗之万方程，得到平均能量的精确表达式.由于简谐速度噪声与简谐噪声功率谱的不同特点，两种内部噪声驱动的谐振子在简谐外力的作用下具有不同的共振特征.这些特征可用来检验两种噪声.     

Numerical simulations of stochastic differential equations

A simple, very accurate algorithm for numerical simulation of stochastic differential equations is described. Its relationship to colored noise is elucidated and exhibited by explicit results. The especially delicate problem of mean first passage times is highlighted and highly accurate agreement between the numerical simulations and analytic results are shown.     

On the linearization of nonlinear langevin-type stochastic differential equations

We show that a logical extension of the piecewise optimal linearization procedure leads to the Gaussian decoupling scheme, where no iteration is required. The scheme is equivalent to solving a few coupled equations. The method is applied to models which represent (a) a single steady state, (b) passage from an initial unstable state to a final preferred stable state by virtue of a finite displacement from the unstable state, and (c) a bivariate case of passage from an unstable state to a final stable state. The results are shown to be in very good agreement with the Monte Carlo calculations carried out for these cases. The method should be of much value in multidimensional cases.     

A virial approach to soliton-like solutions of coupled non-linear differential equations including the ’t-Hooft-Polyakov monopole equations

A virial theorem for solitons derived by Friedberg, Lee and Sirlin is used to reduce a system of second order equations to an equivalent first order set. It is shown that this theorem, when used in conjunction with our earlier observation that soliton-like solutions lie on the separatrix, helps in obtaining soliton-like solutions of theories involving coupled fields. The method is applied to a system of equations studied extensively by Rajaraman. The ’t-Hooft-Polyakov monopole equations are then studied and we obtain the well-known monopole solutions in the Prasad-Sommerfeld limit (λ=0); for the case λ≠0, we succeed in obtaining a non-trivial algebraic constraint between the fields of the theory.     

Microdynamics and nonlinear stochastic processes of gross variables

Starting from classical Hamiltonian mechanics, we derive for the dynamics of gross variables in nonequilibrium systems exact nonlinear generalized Fokker-Planck and Langevin equations in which the effect of the initial preparation is taken into account explicitly. This latter concept allows for the construction of a uniquely determined projection operator. The memory functions occurring in the Langevin equations are related to the random forces by a fluctuation-dissipation theorem of the second kind. We discuss the connection with the generalized Fokker-Planck equation. The known results for equilibrium fluctuations are recovered as a special case.Supported in part by the National Science Foundation, Grant CHE78-21460.     

Equal-time second-order moments of a harmonic oscillator with stochastic frequency and driving force

Using a simple matrix method, we have obtained exact second-order equilibrium moments for a linearly damped harmonic oscillator with a fluctuating frequency (t) and driven by a fluctuating forcef(t). We have assumed each of the fluctuating quantities to be delta-correlated. We demonstrate that the final answers are identical whetherf(t) and (t) are statistically independent or delta-correlated. We have also established the region of parameter space in which the oscillator is energetically stable. The results are shown to be completely determined by the coefficients of the first and second cumulants of the fluctuations.Supported in part by the Office of Naval Research, NSF Grant # CHE 78-21460, and by a grant from Charles and Renée Taubman.     

随机脉冲微分方程的p阶矩稳定性和参激白噪声作用下Lorenz系统的脉冲同步

考察了随机脉冲微分系统的p阶矩稳定性问题,在更符合脉冲系统一般假设的情况下,建立了条件更弱的随机脉冲微分系统p阶矩稳定性判定定理.并应用该判定定理,考察了参激白噪声作用下Lorenz系统的脉冲同步问题,证明了同步误差系统的p阶矩稳定性,从而说明在p阶矩的意义下,两个系统是可以用脉冲方法实现同步的.数值模拟验证了随机Lorenz系统脉冲同步的可行性. 关键词： 随机脉冲微分方程 p阶矩稳定性')" href="#">p阶矩稳定性 脉冲 同步     

Analysis of nonstationary,Gaussian and non-Gaussian,generalized Langevin equations using methods of multiplicative stochastic processes

Using the methods of multiplicative stochastic processes, a thorough analysis of non-Markovian, generalized Langevin equations is presented. For the Gaussian case, these methods are used to show that the nonstationary Fokker-Planck equation already found by Adelman and others is also obtainable from van Kampen's lemma for stochastic probability flows. Here, results applicable to an arbitraryn-component process are obtained and the specific two-component case of the Brownian harmonic oscillator is presented in detail in order to explicitly exhibit the matrix algebraic methods. The non-Gaussian case is presented at the end of the paper and shows that the methods already used in the Gaussian case lead directly to results for the non-Gaussian case. In order to use the methods of multiplicative stochastic processes analysis, it is necessary to transform the non-Markovian, generalized Langevin equation using a stochastic extension of a transformation discussed by Adelman. This transformation removes the memory kernel term in the usual generalized Langevin equation and in the Gaussian case leads to the result that the original process was in fact not non-Markovian but actually nonstationary,Markovian.Supported through a fellowship from the Alfred P. Sioan Foundation.     

Closure approximation error in the mean solution of stochastic differential equations by the hierarchy method

The iterative or stochastic Green's function method of solution of stochastic differential equations is used to find the error terms in the solution and mean solution due to truncation in the hierarchy method. A comparison is made of solutions by the iterative and the hierarchy method.Supported by a grant from the Sloan Foundation.     

Studies in nonlinear stochastic processes. II. The duffing oscillator revisited

We have applied the approximation method of statistical linearization and various higher order corrections thereto to the study of a nonlinear oscillator perturbed by Gaussian, delta-correlated noise. We compute the second-order statistics of the response, i.e., the variances, autocorrelation functions, and spectral densities for various forms of the nonlinearity and compare our results with the few more exact calculations which are available in the literature. We show that a very simple modification of statistical linearization, based upon the use of the variance as obtained from the appropriate Fokker-Planck equation, yields results which are in better agreement with the exact literature results than either statistical linearization or first-order corrections thereto. This modified method of statistical linearization has the significant advantage of great computational simplicity as compared to other attempts of accurate calculations of second-order statistics of nonlinear stochastic equations now in the literature.This work was supported in part by the National Science Foundation under Grants MPS 72-04363 and CHE 75-20624.     

Generalized Fokker-Planck equation for a class of stochastic dynamical systems driven by additive Gaussian and Poissonian fractional white noises of order <Emphasis Type="Italic">α</Emphasis>

In a first stage, the paper deals with the derivation and the solution of the equation of the probability density function of a stochastic system driven simultaneously by a fractional Gaussian white noise and a fractional Poissonian white noise both of the same order. The key is the Taylor’s series of fractional order f(x + h) = E _α(h^αD _x ^α)f(x) where E _α() denotes the Mittag-Leffler function, and D _x ^α is the so-called modified Riemann-Liouville fractional derivative which removes the effects of the non-zero initial value of the function under consideration. The corresponding fractional linear partial differential equation is solved by using a suitable extension of the Lagrange’s technique involving an auxiliary set of fractional differential equations. As an example, one considers a half-oscillator of fractional order driven by a fractional Poissonian noise.      

Asymptotic properties of coupled nonlinear langevin equations in the limit of weak noise. II: Transition to a limit cycle

We apply the singular perturbation technique, developed in the companion paper, to the study of the fluctuations at the onset of a limit cycle, both for the cases of a soft and a hard transition. The technique and results are illustrated on the Poincaré model (soft transition) and on the Van der Pol oscillator (hard transition).     

Stochastic differential equations in quantum statistical mechanics. Observables and multiple wiener integrals

The physical and mathematical framework for quantum mechanical stochastic differential equations is discussed as the quantization ofc-number equations that typically describe Brownian motion in polynomial potentials.     

A new method for constructing soliton solutions and periodic solutions of nonlinear evolution equations

With the aid of the symbolic computation, we improve Xie's algorithm [F. Xie, Z.Y. Yan, H. Zhang, Phys. Lett. A 285 (2001) 76], and present a new extended method. Based on the new general ansatz (3), we successfully solve a compound KdV-MKdV equation, and obtain some special solutions which contain soliton solutions, and periodic solutions. The method can also be applied to other nonlinear partial differential equations.     

对“求一类非线性偏微分方程解析解的一种简洁方法”一文的一点注记

利用文献中所引入的变换，将一个非线性偏微分方程化为一个非线性常微分方程，再直接求解该常微分方程，从而简洁地求得了Burgers方程的几个精确解析解.所得结果与已有结果完全符合. 关键词： 非线性偏微分方程 非线性常微分方程 解析解