Global weak solutions for the initial–boundary-value problems Vlasov–Poisson–Fokker–Planck System

This work is devoted to prove the existence of weak solutions of the kinetic Vlasov–Poisson–Fokker–Planck system in bounded domains for attractive or repulsive forces. Absorbing and reflection-type boundary conditions are considered for the kinetic equation and zero values for the potential on the boundary. The existence of weak solutions is proved for bounded and integrable initial and boundary data with finite energy. The main difficulty of this problem is to obtain an existence theory for the linear equation. This fact is analysed using a variational technique and the theory of elliptic–parabolic equations of second order. The proof of existence for the initial–boundary value problems is carried out following a procedure of regularization and linearization of the problem. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Dynamics and Stability of Bose-Einstein Condensates: The Nonlinear Schrödinger Equation with Periodic Potential

Summary. The cubic nonlinear Schr?dinger equation with a lattice potential is used to model a periodic dilute-gas Bose-Einstein condensate. Both two- and three-dimensional condensates are considered, for atomic species with either repulsive or attractive interactions. A family of exact solutions and corresponding potential is presented in terms of elliptic functions. The dynamical stability of these exact solutions is examined using both analytical and numerical methods. For condensates with repulsive atomic interactions, all stable, trivial-phase solutions are off-set from the zero level. For condensates with attractive atomic interactions, no stable solutions are found, in contrast to the one-dimensional case [8].     

Properties of Defect Modes in One-Dimensional Optically Induced Photonic Lattices

In this article, localized defect modes in one-dimensional optically induced photonic lattices are studied comprehensively. First, the origin of these defect modes is investigated analytically in the weak-defect limit by perturbation methods. It is shown that in an attractive defect where the lattice light intensity at the defect site is higher than that of nearby sites, a defect mode bifurcates from the left edge of every Bloch band; while in a repulsive defect, a defect mode bifurcates from the right edge of every Bloch band. When the defect is not weak, defect modes are examined by numerical methods. It is shown that in a repulsive defect, the strongest confinement of defect modes arises when the lattice light intensity at the defect site is nonzero rather than zero. In addition, as the potential strength increases, defect modes disappear from lower bandgaps and appear in higher bandgaps. In an attractive defect, however, defect modes persist in every bandgap as the potential strength increases. Using a piecewise-constant potential model, defect modes are calculated analytically for a general defect. The analytical results qualitatively explain the main features in numerical results.     

Theory of Dark Energy and Dark Matter

In (T. Ma and S. Wang. Gravitational field equations and theory of dark matter and dark energy, Discrete and Continuous Dynamical Systems, Ser. A, 34(2): 335-366, 2014; arXiv:1206.5078v2), a new set of gravitational field equations are derived based only on 1) the Einstein principle of general relativity, and 2) the principle of interaction dynamics, due to the the presence of dark energy and dark matter. With the field equations, we show that gravity can display both attractive and repulsive behavior, and the dark matter and dark energy are just a property of gravity caused by the nonlinear interactions of the gravitational potential $g_{μv}$and its dual field. The main objectives of this paper are two-fold. The first is to study the PID-induced cosmological model, and to show explicitly, as addressed in (T. Ma and S. Wang, Astrophysical dynamics and cosmology, Journal of Mathematical Study, 47(4): 305-378, 2014), that 1) dark matter is due to the curvature of space, and 2) dark energy corresponds to the negative pressure generated by the dual gravitational potential in the field equations, and maintains the stability of geometry and large scale structure of the Universe. Second, for the gravitational field outside of a ball of centrally symmetric matter field, there exist precisely two physical parameters dictating the two-dimensional stable manifold of asymptotically flat space-time geometry, such that, as the distance to the center of the ball of the matter field increases, gravity behaves as Newtonian gravity, then additional attraction due to the curvature of space (dark matter effect), and repulsive (dark energy effect). This also clearly demonstrates that both dark matter and dark energy are just a property of gravity.     

Global weak solutions for the Vlasov–Poisson system with a point charge

We study the dynamics of three‐dimensional Vlasov‐Poisson system in the presence of a point charge with attractive interaction. Compared to the repulsive interaction,we cannot get a priori conversation law. Nevertheless,we are able to obtain bound of kinetic energy by introducing a Lyapunov functional. Combining this result with the concept of Diperna‐Lions flow, we establish global existence of weak solutions for this system. Copyright © 2014 John Wiley & Sons, Ltd.     

Derivation of macroscopic equations for individual cell‐based models: a formal approach

In this paper we review the theory of cells (particles) that evolve according to a dynamics determined by friction and that interact between themselves by means of suitable potentials. We derive by means of elementary arguments several macroscopic equations that describe the evolution of cell density. Some new results are also obtained—a formal derivation of a limit equation in the case of attractive potential as well as in the case of repulsive potential with a hard‐core part are presented. Finally we discuss the possible relevance of those results within the framework of individual cell‐based models. Several classes of potentials, including hard‐core, repulsive and potentials with attractive parts are discussed. The effect of noise terms in the equation is also considered. Copyright © 2005 John Wiley & Sons, Ltd.     

Identification of Piecewise-Constant Potentials by Fixed-Energy Phase Shifts

The identification of a spherically symmetric potential by its phase shifts is an important physical problem. Recent theoretical results assure that such a potential is uniquely defined by a sufficiently large subset of its phase shifts at any one fixed energy level. However, two different potentials can produce almost identical phase shifts. To resolve this difficulty we suggest the use of phase shifts corresponding to several energy levels. The identification is done by a nonlinear minimization of the appropriate objective function. It is based on a combination of probabilistic global and deterministic local minimization methods. The Multilevel Single-Linkage Method (MSLM) is used for the global minimization. A specially designed Local Minimization Method (LMM) with a Reduction Procedure is used for the local searches. Numerical results show the effectiveness of this procedure for potentials composed of a small number of spherical layers. Accepted 2 February 2001. Online publication 11 May 2001.     

Generalized model and optimum performance of an irreversible thermal Brownian microscopic heat pump

A generalized model of an irreversible thermal Brownian microscopic heat pump is established in this paper. It is composed of Brownian particles which are moving in a periodic sawtooth potential with external forces and contacting with alternating hot and cold reservoirs along the space coordinate. The generalized irreversible Brownian heat pump model incorporates heat flows driven by both the potential and kinetic energies of the particles as well as the heat leakage between the hot and cold reservoirs. This paper derives the expressions for heating load, power input and coefficient of performance (COP) of the Brownian heat pump. The optimum performance of the generalized heat pump model is analyzed by using the theory of finite time thermodynamics (FTT). Effects of the design parameters, i.e., the external force, the heat leakage coefficient, barrier height of the potential, asymmetry of the sawtooth potential and heat reservoir temperature ratio on the performance of the Brownian heat pump are discussed in detail. The performance of the Brownian heat pump depends strictly on the design parameters. Through the proper choice of these parameters, the Brownian heat pump can operate in the optimal regimes. The optimum COP performance and the fundamental optimal relations between COP and heating load are studied by detailed numerical examples. It is shown that due to the heat leakage between the heat reservoirs and heat flow via the change of kinetic energy of the particles, both the heating load and COP performances of the Brownian heat pump will decrease. The effective ranges of the external force and barrier height of the potential in which the Brownian motor system can operate as a heat pump are further determined.     

Estimation of the time a perturbed Lagrangian system stays in an assigned region

The problem of estimating the mean time a weakly perturbed dynamical system stays in a fixed region of the phase variables is investigated. The motion is described by Lagrange's equations with an attractive force potential and in the presence of additive dissipative forces. The corresponding Cauchy problem is obtained in Hamiltonian variables for a non-linear first-order partial differential equation. Its classical positive solution specifies the action functional and the estimate sought for the time interval. The structure of the equations that allows of an explicit solution in terms of expressions for the kinetic and potential energy, as well as dissipative and dispersion matrices for a random Wiener-type perturbation, is established. The phenomenon of the escape of a phase point from different parts of the boundary of the region is investigated. Interesting problems of estimating the time for the inversion of the inner gimbal of a gyroscope, the time taken to reach an assigned level or a potential barrier of a multidimensional oscillatory system that has central symmetry, and the time a non-linear system with two degrees of freedom takes to escape over a potential barrier for a Henon–Heiles potential are investigated as examples.     

A particle method for a self-gravitating fluid: A convergence result

We study an Hamiltonian system of N particles in ?³ interacting by a short-range repulsive and a long-range attractive potential. It is shown that the empirical measures associated to the positions and velocity of the system converge to the solutions of Euler equations for a self-gravitating fluid, in the limit as the particle number tends to infinity, for a suitable scaling of the interactions.     

A generalized model of an irreversible thermal Brownian refrigerator and its performance

A generalized model of an irreversible thermal Brownian refrigerator, which consists of Brownian particles moving in a periodic sawtooth potential with external forces and contacting with the alternating hot and cold reservoirs along the space coordinate, is established in this paper. The heat flows driven by both potential and kinetic energies of the particles as well as the heat leakage between the hot and cold reservoirs are taken into account. The optimum performance of the generalized model is analyzed using the theory and method of finite time thermodynamics. The analytical expressions for cooling load, coefficient of performance (COP) and power input of the Brownian refrigerator are derived. It is shown by numerical examples that due to the heat leakage between the heat reservoirs and heat flow via the change of kinetic energy of the particles, the Brownian refrigerator is always irreversible and the COP can never attain the Carnot COP. The influences of the heat leakage, the external force, barrier height of the potential, asymmetry of the sawtooth potential and temperature ratio of the heat reservoirs on the performance of the Brownian refrigerator are also investigated in detail. The effective regions of external force and barrier height of the potential in which the Brownian motor can operate as a refrigerator are determined. It is found that the performance of the Brownian refrigerator depends strictly on the design parameters. If these design parameters are properly chosen, the Brownian refrigerator can be controlled to operate in the optimal regimes. The results obtained herein about the general Brownian refrigerator model include those obtained in many previous literatures.     

Turbulent Energy Spectrum via an Interaction Potential

For a large system of identical particles interacting by means of a potential, we find that a strong large scale flow velocity can induce motions in the inertial range via the potential coupling. This forcing lies in special bundles in the Fourier space, which are formed by pairs of particles. These bundles are not present in the Boltzmann, Euler and Navier–Stokes equations, because they are destroyed by the Bogoliubov–Born–Green–Kirkwood–Yvon formalism. However, measurements of the flow can detect certain bulk effects shared across these bundles, such as the power scaling of the kinetic energy. We estimate the scaling effects produced by two types of potentials: the Thomas–Fermi interatomic potential (as well as its variations, such as the Ziegler–Biersack–Littmark potential), and the electrostatic potential. In the near-viscous inertial range, our estimates yield the inverse five-thirds power decay of the kinetic energy for both the Thomas–Fermi and electrostatic potentials. The electrostatic potential is also predicted to produce the inverse cubic power scaling of the kinetic energy at large inertial scales. Standard laboratory experiments confirm the scaling estimates for both the Thomas–Fermi and electrostatic potentials at near-viscous scales. Surprisingly, the observed kinetic energy spectrum in the Earth atmosphere at large scales behaves as if induced by the electrostatic potential. Given that the Earth atmosphere is not electrostatically neutral, we cautiously suggest a hypothesis that the atmospheric kinetic energy spectra in the inertial range are indeed driven by the large scale flow via the electrostatic potential coupling.

     

On the inverse problem at fixed energy in the "Iterative" range

The scattering amplitude by a spherically symmetric potential at fixed energy is given in the Born approximation by a filtered Fourier transform, whose inverse is not unique. It is well known that matrix methods enable one to study exactly the problem at fixed energy in classes of potentials parametrised by sequences of numbers. In the range of potentials (or of phase shifts) where these methods can be managed by iteration, Born case is a limit. This article is a brief survey of the inverse problem (scattering amplitude?→?potential?) recalling how the nonuniqueness predicted in the Born approximation appears in these exact methods, showing henceforth that the inverse problem ill-posedness corresponds to physical features of the potential on which experiments at finite energy are unable to give information.     

Nonlinear time-dependent Schrödinger equations: the Gross-Pitaevskii equation with double-well potential

We consider a class of Schrödinger equations with a symmetric double-well potential and an external, both repulsive and attractive, nonlinear perturbation. We show that, under certain conditions and in the limit of large barrier between the two wells, the reduction of the time-dependent equation to a two-mode equation gives the dominant term of the solution with a precise estimate of the error.     

Piecewise constant positive potentials with practically the same fixed energy phase shifts

It has recently been shown that spherically symmetric potentials of finite range are uniquely determined by the part of their phase shifts at a fixed energy level k² > 0. However, numerical experiments show that two quite different potentials can produce almost identical phase shifts. It has been guessed by physicists that such examples are possible only for “less physical” oscillating and changing sign potentials. In this note it is shown that the above guess is incorrect: we give examples of four positive spherically symmetric compactly supported quite different potentials having practically identical phase shifts. The note also describes a hybrid stochastic deterministic method for global minimization for the construction of these potentials.     

Generalized Choquet-like aggregation functions for handling bipolar scales

We are interested in modeling interaction between criteria in multi-criteria decision making when underlying scales are bipolar. Interacting phenomena involving behavioral bias between attractive and repulsive values are in particular considered here. We show in an example that both the Choquet integral and the cumulative prospect theory (CPT) model fail to represent these interacting phenomena. Axioms that enable the construction of the preferences of the decision maker over each attribute, and the representation of his preferences about aggregation of criteria are introduced and justified. We show there is a unique aggregation operator that fits with these axioms. It is based on the notion of bi-capacity and generalizes both the Choquet integral and the CPT model.     

Decay to bound states of a soliton in a well

The decay of a soliton in a trapped state inside a well is shown numerically. Bound states of a kink in an attractive well, both centered and off centered are found. Their stability is studied. Unstable soliton solutions inside a repulsive barrier are also found.     

HIGH-ORDER WENO SIMULATIONS OF THREE-DIMENSIONAL RESHOCKED RICHTMYER-MESHKOV INSTABILITY TO LATE TIMES:DYNAMICS,DEPENDENCE ON INITIAL CONDITIONS,AND COMPARISONS TO EXPERIMENTAL DATA

The dynamics of the reshocked multi-mode Richtmyer-Meshkov instability is investigated using 513×257 2three-dimensional ninth-order weighted essentially nonoscillatory shock-capturing simulations.A two-mode initial perturbation with superposed random noise is used to model the Mach 1.5 air/SF6 Vetter-Sturtevant shock tube experiment. The mass fraction and enstrophy isosurfaces,and density cross-sections are utilized to show the detailed flow structure before,during,and after reshock.It is shown that the mixing layer growth agrees well with the experimentally measured growth rate before and after reshock.The post-reshock growth rate is also in good agreement with the prediction of the Mikaelian model.A parametric study of the sensitivity of the layer growth to the choice of amplitudes of the short and long wavelength initial interfacial perturbation is also presented.Finally,the amplification effects of reshock are quantified using the evolution of the turbulent kinetic energy and turbulent enstrophy spectra,as well as the evolution of the baroclinic enstrophy production,buoyancy production,and shear production terms in the enstrophy and turbulent kinetic transport equations.     

Do peaked solitary water waves indeed exist?

Many models of shallow water waves, such as the famous Camassa–Holm equation, admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa–Holm equation. Besides, it is proved that Kelvin’s theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, unlike the traditional smooth waves that are dispersive with wave height, the phase speed of the peaked solitary waves has nothing to do with wave height, but depends (for a fixed wave height) on its decay length, i.e., the actual wavelength: in fact, the peaked solitary waves are dispersive with the actual wavelength when wave height is fixed. In addition, unlike traditional smooth waves whose kinetic energy decays exponentially from free surface to bottom, the kinetic energy of the peaked solitary waves either increases or almost keeps the same. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.