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.     

Exact solutions of a Fokker-Planck equation

Exact explicit solutions are given for a one-dimensional Fokker-Planck equation with a particular potential form involving hyperbolic functions. This potential contains four arbitrary parameters that can be chosen so that the potential is bistable. The solutions also contain parameters that can be chosen so that the initial distribution is approximately Gaussian, centered either at the unstable potential maximum or in the neighborhood of the secondary minimum. The use of the solutions to approximate solutions for other potentials is considered.     

Exact solution to the cauchy problem for a generalized “linear” vectorial Fokker-Planck equation: Algebraic approach

The exact solution to the Cauchy problem for a generalized “linear” vectorial Fokker-Planck equation is found by using the disentangling techniques of Feynman and algebraic (operational) methods.     

Path integrals for Fokker-Planck equation described by generalized random walks

Path integral representations for Fokker-Planck (FP) equations, described by the random walks (RW) and the generalized random walks (GRW), are given. The GRW is a generalized one from the usual random walks to study non-linear, non-equilibrium processes. The GRW includes some memory effects and couplings through the jumping probabilities. To derive the path integrals of the processes, a transformation of probability, scalings of site (space) and step (time) are performed on the GRW. By a function in the exponent of the path integrals for the FP equation obtained by the RW or the GRW, a Lagrangian giving most probable path is introduced. From the Lagrangian, an effective Hamiltonian is deduced.     

(2+1)-维Wick类型随机广义KP方程的类孤子解

利用埃尔米特变换和特殊的截断展开法求出(2+1)-维Wick类型随机广义KP方程的类孤子解. 这种方法的基本思想是通过埃尔米特变换把(2+1)-维Wick类型随机广义KP方程变成的(2+1)-维广义变系数KP方程,利用特殊的截断展开方法求出方程的解,然后通过埃尔米特的逆变换求出方程的随机解.     

Exact equation for curved stationary flames with arbitrary gas expansion

An exact equation describing freely propagating stationary flames with arbitrary values of the gas expansion coefficient is obtained. This equation respects all conservation laws at the flame front, and provides a consistent nonperturbative account of the effect of vorticity produced by the curved flame on the front structure. It is verified that the new equation is in agreement with the approximate equations derived previously in the case of weak gas expansion.     

Exact and approximate generalized diffusion equation for the Lorentz gas

The Lorentz model of a rarefied gas is used for testing two different methods of solving the Boltzman kinetic equation. It is shown that the Zwanzig-Mori method gives the generalized diffusion equation which agrees with the exact Hauge solution. The Zubarev-Khonkin approach gives a series expansion of the exact generalized diffusion coefficient. Their method is compared with the Chapman-Enskog method.     

Exact solutions of the generalized Bretherton equation

The generalized Bretherton equation is studied. The Bäcklund transformations between traveling wave solutions of the generalized Bretherton equation and solutions of polynomial ordinary differential equation are constructed. The classification problem for meromorphic solutions of the latter equation is discussed. Several new families of exact solutions for the generalized Brethenton equation are given.     

Exact stationary solution of a Fokker-Planck equation for multimode laser action including phase locking

We first derive the exact stationary solution of a Fokker-Planck equation where the complex drift coefficients are nonlinear functions of the variables, provided the drift and diffusion coefficients fulfill certain conditions. Then we apply the solution to

1. normal multimode action where no phase locking occurs at all.
2. phase locking in a laser with many axial modes having a narrow frequency spacing.

The atomic line which supports laser action may be homogeneously or inhomogeneously broadened. In case 1 the modes may be completely arbitrary, i.e. for instance running or standing waves. In case 2 we assume axial modes described by running waves. The treatment is valid in a region not too far below and not too far above laser threshold where the atomic variables adiabatically follow the motion of the lightfield variables. The drift coefficients are taken from the multimode Langevin equations ofHaken andSauermann. The diffusion coefficients are taken from a paper ofArzt et al. The only essential assumption is that the diffusion coefficients may be considered constant over the frequency range where modes participate in the laser process. If our results are specialized to single mode action we obtain Risken's solution.     

On the derivation of the Fokker-Planck equation from generalized kinetic equations

The Fokker-Planck equation is obtained using the matrix representation of the Liouville equation introduced by Balescu in the general theory of irreversible processes developed by the Brussels group. It is shown that the phenomenological equation is valid when the mass and density of the Brownian particle are large compared to the mass and density of the bath. The relation with previous work is discussed.     

广义非对称Hartmann势Klein-Gordon方程的严格解(英文)

在标量势和矢量势相等的条件下,严格求解了在广义非对称Hartmann势场中粒子运动的Klein Gordon方程;并利用束缚态边界条件,获得了束缚态能谱表达式和由超几何函数表示出的波函数.     

Fractional Fokker-Planck equation for anomalous diffusion in a potential: Exact matrix continued fraction solutions

Methods for the exact solution of fractional Fokker-Planck equations for anomalous diffusion in an external potential are discussed using both ordinary and matrix continued fractions, whereby the scalar multi-term recurrence relations generated by such fractional diffusion equations are reduced to three-term matrix ones. The procedure is illustrated by solving various problems concerning the anomalous translational diffusion in both periodic and double-well potentials.     

Half-range completeness for the Fokker-Planck equation

Half-range completeness theorems are proved for eigenfunctions associated to the one-dimensional Fokker-Planck equation in a semi-infinite medium. Existence and uniqueness results for perfectly absorbing, partially absorbing, and purely specularly reflecting boundary conditions are deduced for the stationary and time-dependent problems. Similar results are obtained for a slab geometry.     

Fractional Fokker-Planck equation for fractal media

We consider the fractional generalizations of equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. Using fractional integrals, we derive the fractional generalization of the Chapman-Kolmogorov equation (Smolukhovski equation). In this paper fractional Fokker-Planck equation for fractal media is derived from the fractional Chapman-Kolmogorov equation. Using the Fourier transform, we get the Fokker-Planck-Zaslavsky equations that have fractional coordinate derivatives. The Fokker-Planck equation for the fractal media is an equation with fractional derivatives in the dual space.     

Fokker-Planck equation for a periodic potential

The Fokker-Planck equation in one dimension has been solved for a system in a periodic potential and linearly perturbed by a time- and space- varying external electric field. The solution is not exact, but we believe it is less approximate than any attempted previously on this problem. We have calculated the frequency and wave-vector dependent mobility and discussed its behaviour in some limiting cases.The application of our model to various physical situations is discussed.     

Kinetics of nonlinear systems with random and regular disturbances. Exact results versus Fokker-Planck equation

Using exact solutions of some nonlinear ordinary differential (kinetic) equations with time-dependent coefficients, it is found that different types of regular and random disturbances may have diverse effect on macroscopic kinetics. Among others, regular impulses nonlinearly coupled to linear (smooth) relaxation kinetics can lead to chaotic behaviour.     

Exact solutions to the generalized lienard equation and its applications

Some new exact solutions of the generalized Lienard equation are obtained, and the solutions of the equation are applied to solve nonlinear wave equations with nonlinear terms of any order directly. The generalized one-dimensional Klein-Gordon equation, the generalized Ablowitz (A) equation and the generalized Gerdjikov-Ivanov (GI) equation are investigated and abundant new exact travelling wave solutions are obtained that include solitary wave solutions and triangular periodic wave solutions.      

Exact periodic wave solutions to the generalized Nizhnik-Novikov-Veselov equation

The extended mapping method with symbolic computation is developed to obtain exact periodic wave solutions to the generalized Nizhnik-Novikov-Veselov equation. Limit cases are studied and new solitary wave solutions and triangular periodic wave solutions are obtained. The method is applicable to a large variety of non-linear partial differential equations, as long as odd-and even-order derivative terms do not coexist in the equation under consideration.