Numerical methods for computing ground states and dynamics of nonlinear relativistic Hartree equation for boson stars

Efficient and accurate numerical methods are presented for computing ground states and dynamics of the three-dimensional (3D) nonlinear relativistic Hartree equation both without and with an external potential. This equation was derived recently for describing the mean field dynamics of boson stars. In its numerics, due to the appearance of pseudodifferential operator which is defined in phase space via symbol, spectral method is more suitable for the discretization in space than other numerical methods such as finite difference method, etc. For computing ground states, a backward Euler sine pseudospectral (BESP) method is proposed based on a gradient flow with discrete normalization; and respectively, for computing dynamics, a time-splitting sine pseudospectral (TSSP) method is presented based on a splitting technique to decouple the nonlinearity. Both BESP and TSSP are efficient in computation via discrete sine transform, and are of spectral accuracy in spatial discretization. TSSP is of second-order accuracy in temporal discretization and conserves the normalization in discretized level. In addition, when the external potential and initial data for dynamics are spherically symmetric, the original 3D problem collapses to a quasi-1D problem, for which both BESP and TSSP methods are extended successfully with a proper change of variables. Finally, extensive numerical results are reported to demonstrate the spectral accuracy of the methods and to show very interesting and complicated phenomena in the mean field dynamics of boson stars.     

Rogue waves, dissipation, and downshifting

We investigate the effects of dissipation on the development of rogue waves and downshifting by adding nonlinear and linear damping terms to the one-dimensional Dysthe equation. Significantly, rogue waves do not develop after the downshifting becomes permanent. Thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves. Using the inverse spectral theory of the NLS equation, simulations of the damped Dysthe equation for sea states characterized by JONSWAP spectrum consistently show that rogue wave events are well-predicted by proximity to homoclinic data, as measured by the spectral splitting distance δ. The cut off distance δ_cutoff decreases as the strength of the damping increases, indicating that for stronger damping the JONSWAP initial data must be closer to homoclinic data for rogue waves to occur.     

Higher-Dimensional KdV Equations and Their Soliton Solutions

A (2+1)-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and the Riccati equation, a type of bell-shape soliton solutions are produced via regarding the variable in the Riccati equation as the independent variable. Finally, we extend the above (2+1)-dimensional KdV equation into (3+1)-dimensional equation, the two-soliton solutions are given.     

New Exact Solutions of Multi-component mKdV Equation

In this paper, new Jacobi elliptic function solutions of multi-component mKdV equation are obtained directly in a unified way. When the modulus m→1, those periodic solutions degenerate as the corresponding hyperbolic function solutions. Then, to the three-component mKdV equation, five types of effective solution are presented in detail.     

Exact solutions for nonlinear Schr?dinger equation in phase space: applications to Bose－Einstein condensate

The stationary-state nonlinear Schr?dinger equation, which models the dilute-gas Bose－Einstein condensate, is introduced within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The exact solutions of equation are obtained in the phase space, by means of the wave-mechanics method. The eigenfunctions in position and momentum spaces are obtained through the ‘Fourier-like' projection transformation from the phase space eigenfunctions. The eigenfunction with a hypersecant part is discussed as an example.     

DNA纤维的X射线衍射分析与双螺旋结构的发现

概述了DNA化学结构式确立的过程,介绍了x射线衍射分析的概念、原理和方法,阐述了物理学家威尔金斯和物理化学家富兰克林在用实验事实证明DNA双螺旋结构方面所作出的巨大贡献。     

Birth–death process of local structures in defect turbulence described by the one-dimensional complex Ginzburg–Landau equation

Defect turbulence described by the one-dimensional complex Ginzburg–Landau equation is investigated and analyzed via a birth–death process of the local structures composed of defects, holes, and modulated amplitude waves (MAWs). All the number statistics of each local structure, in its stationary state, are subjected to Poisson statistics. In addition, the probability density functions of interarrival times of defects, lifetimes of holes, and MAWs show the existence of long-memory and some characteristic time scales caused by zigzag motions of oscillating traveling holes. The corresponding stochastic process for these observations is fully described by a non-Markovian master equation.     

Supersymmetry of FRW Barotropic Cosmologies

Barotropic FRW cosmologies are presented from the standpoint of nonrelativistic supersymmetry. First, we reduce the barotropic FRW system of differential equations to simple harmonic oscillator differential equations. Employing the factorization procedure, the solutions of the latter equations are divided into the two classes of bosonic (nonsingular) and fermionic (singular) cosmological solutions. We next introduce a coupling parameter denoted by K between the two classes of solutions and obtain barotropic cosmologies with dissipative features acting on the scale factors and spatial curvature of the universe. The K-extended FRW equations in comoving time are presented in explicit form in the low coupling regime. The standard barotropic FRW cosmologies correspond to the dissipationless limit K = 0.     

Estimating the Solutions of the Boltzmann Equation

We discuss some possible estimates of the solutions of the Boltzmann equation, which might permit a progress in the theory of existence of weak solutions.     

The Linear Fokker-Planck Equation for the Ornstein-Uhlenbeck Process as an (Almost) Nonlinear Kinetic Equation for an Isolated N-Particle System

It is long known that the Fokker-Planck equation with prescribed constant coefficients of diffusion and linear friction describes the ensemble average of the stochastic evolutions in velocity space of a Brownian test particle immersed in a heat bath of fixed temperature. Apparently, it is not so well known that the same partial differential equation, but now with constant coefficients which are functionals of the solution itself rather than being prescribed, describes the kinetic evolution (in the N→∞ limit) of an isolated N-particle system with certain stochastic interactions. Here we discuss in detail this recently discovered interpretation. An erratum to this article can be found at