Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials

In this paper we consider a one-dimensional non-linear Schrödinger equation with a periodic potential. In the semiclassical limit we prove the existence of stationary solutions by means of the reduction of the non-linear Schrödinger equation to a discrete non-linear Schrödinger equation. In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential. A connection of these results with the Mott insulator phase for Bose–Einstein condensates in a one-dimensional periodic lattice is also discussed.     

Effective potential for complex Langevin equations

We construct an effective potential for the complex Langevin equation on a lattice. We show that the minimum of this effective potential gives the space–time and Langevin time average of the complex Langevin field. The loop expansion of the effective potential is matched with the derivative expansion of the associated Schwinger–Dyson equation to predict the stationary distribution to which the complex Langevin equation converges.     

Hydrodynamics of stationary non-equilibrium states for some stochastic lattice gas models

We consider discrete lattice gas models in a finite interval with stochastic jump dynamics in the interior, which conserve the particle number, and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials. The unique stationary measures of these processes support a steady particle current from the reservoir of higher chemical potential into the lower and are non-reversible. We study the structure of the stationary measure in the hydrodynamic limit, as the microscopic lattice size goes to infinity. In particular, we prove as a law of large numbers that the empirical density field converges to a deterministic limit which is the solution of the stationary transport equation and the empirical current converges to the deterministic limit given by Fick's law.Dedicated to Res Jost and Arthur WightmanSupported in part by NSF Grants DMR 89-18903 and INT 8521407. H.S. also supported by the Deutsche Forschungsgemeinschaft     

Amplitude Equation for Lattice Maps, A Renormalization Group Approach

We consider the development of instabilities of homogeneous stationary solutions of discrete-time lattice maps. Under some generic hypotheses we derive an amplitude equation which is the space-time-continuous Ginzburg–Landau equation. Using dynamical renormalization group methods, we control the accuracy of this approximation in a large ball of its basin of attraction.     

Dynamic chaos in the solution of the Gross-Pitaevskii equation for a periodic potential

We analytically and numerically investigate the solution to the stationary Gross-Pitaevskii equation for a one-dimensional potential of the optical lattice in the case of repulsive nonlinearity. From the mathematical viewpoint, this problem is similar to the well-known problem of the classical mathematical Kapitza pendulum perturbed by a weak high-frequency force. At certain values of the parameters, dynamic chaos is produced in the considered problem. It is modeled analytically by a nonlinear diffusion equation.     

Next-nearest neighbor interaction and localized solutions of polymer chains

We study localization in polymer chains modeled by the nonlinear discrete Schr?dinger equation (DNLS) with next-nearest-neighbor (n-n-n) interaction extending beyond the usual nearest-neighbor exchange approximation. Modulational instability of plane carrier waves is discussed and it is shown that localization gets amplified under the influence of an enhanced interaction radius. Furthermore, we construct exact localized solitonlike solutions of the n-n-n interaction DNLS. To this end the stationary lattice system is cast into a nonlinear map. The homoclinic orbits of unstable equilibria of this map are attributed to standing solitonlike solutions of the lattice system. We note that in comparison with the standard next-neighbor interaction DNLS which bears only one type of static soliton-like states (either staggering or unstaggering) the one with n-n-n interaction radius can support unstaggering as well as staggering stationary localized states with frequencies lying above respectively below the linear band. Generally, the stronger the n-n-n interaction on the DNLS lattice the smaller are the maximal amplitudes of the standing solitonlike solutions and the less rapid are their exponential decays. Received 4 October 2000     

Bessel型光晶格中自旋-轨道耦合极化激元凝聚的稳态结构

Bessel型光晶格是一种非空间周期性的柱对称的光晶格势场,其兼具无限深势阱和环状势阱的特征,在0阶Bessel光晶格势场中央形成深势阱,而在非0阶Beseel光晶格势场中能形成具有中央势垒的环状浅势阱.极化激元是一种半光半物质的准粒子,该准粒子甚至可以在室温条件下发生玻色-爱因斯坦凝聚相变,形成极化激元凝聚.另外,通过极化激元能级的腔诱导TE-TM分裂能在极化激元凝聚中实现足够强的自旋-轨道耦合作用.极化激元凝聚能在室温条件下实现,在其中又存在自旋-轨道耦合作用,其为量子物理的研究提供了全新的平台.本文把Bessel光晶格势场引入到极化激元凝聚系统,研究了存在自旋-轨道耦合作用下的旋量双组分极化激元凝聚系统的稳态结构.通过求解Gross-Pitaevskii方程给出了极化激元凝聚系统在实验室坐标系和旋转坐标系中极化激元凝聚系统的稳态结构,由于Bessel势场的引入,使得稳态结构更具有多样性.给出了实验室坐标系中在中央深势阱中存在的基础型高斯孤立子、多极孤立子和在环状浅势阱中存在环状孤立子和多极孤立子的稳态结构;给出了旋转坐标系中存在的涡旋环状孤立子,及其由于自旋-轨道相互作用引起的组...     

Lattice Gas Model in Random Medium and Open Boundaries: Hydrodynamic and Relaxation to the Steady State

We consider a lattice gas interacting by the exclusion rule in the presence of a random field given by i.i.d. bounded random variables in a bounded domain in contact with particles reservoir at different densities. We show, in dimensions d≥3, that the rescaled empirical density field almost surely, with respect to the random field, converges to the unique weak solution of a quasilinear parabolic equation having the diffusion matrix determined by the statistical properties of the external random field and boundary conditions determined by the density of the reservoir. Further we show that the rescaled empirical density field, in the stationary regime, almost surely with respect to the random field, converges to the solution of the associated stationary transport equation.     

一类两维光学格子中稳定的复合孤子

对两维光学格子中非线性薛定谔方程一类新的稳态解做了数值分析，发现其在传播过程中逐渐衰变为一种稳定的复合孤子，孤子的两分量在传播过程中不断交换能量，总能量守恒. 关键词： 稳态解 两维格子 孤子     

High-frequency thermal processes in harmonic crystals

We consider two high frequency thermal processes in uniformly heated harmonic crystals relaxing towards equilibrium: (i) equilibration of kinetic and potential energies and (ii) redistribution of energy among spatial directions. Equation describing these processes with deterministic initial conditions is derived. Solution of the equation shows that characteristic time of these processes is of the order of ten periods of atomic vibrations. After that time the system practically reaches the stationary state. It is shown analytically that in harmonic crystals temperature tensor is not isotropic even in the stationary state. As an example, harmonic triangular lattice is considered. Simple formula relating the stationary value of the temperature tensor and initial conditions is derived. The function describing equilibration of kinetic and potential energies is obtained. It is shown that the difference between the energies (Lagrangian) oscillates around zero. Amplitude of these oscillations decays inversely proportional to time. Analytical results are in a good agreement with numerical simulations.     

Non-Equilibrium Behaviour in Unidirectionally Coupled Map Lattices

A spatially one dimensional coupled map lattice with a local and unidirectional coupling is introduced. This model is studied analytically by a perturbation theory that is valid for small coupling strength. In parameter space three phases with different ergodic behaviour are observed. Via coarse graining the deterministic model is mapped to a stochastic spin model that can be described by a master equation. Because of the anisotropic coupling non-equilibrium behaviour is found on the coarse grained level. However, the stationary statistical properties of the spin dynamics can still be described with a nearest neighbour Ising model whereby the ordering is predominantly antiferromagnetic.     

The Phonon Boltzmann Equation, Properties and Link to Weakly Anharmonic Lattice Dynamics

For low density gases the validity of the Boltzmann transport equation is well established. The central object is the one-particle distribution function, f, which in the Boltzmann-Grad limit satisfies the Boltzmann equation. Grad and, much refined, Cercignani argue for the existence of this limit on the basis of the BBGKY hierarchy for hard spheres. At least for a short kinetic time span, the argument can be made mathematically precise following the seminal work of Lanford. In this article a corresponding program is undertaken for weakly nonlinear, both discrete and continuum, wave equations. Our working example is the harmonic lattice with a weakly nonquadratic on-site potential. We argue that the role of the Boltzmann f-function is taken over by the Wigner function, which is a very convenient device to filter the slow degrees of freedom. The Wigner function, so to speak, labels locally the covariances of dynamically almost stationary measures. One route to the phonon Boltzmann equation is a Gaussian decoupling, which is based on the fact that the purely harmonic dynamics has very good mixing properties. As a further approach the expansion in terms of Feynman diagrams is outlined. Both methods are extended to the quantized version of the weakly nonlinear wave equation.The resulting phonon Boltzmann equation has been hardly studied on a rigorous level. As one novel contribution we establish that the spatially homogeneous stationary solutions are precisely the thermal Wigner functions. For three phonon processes such a result requires extra conditions on the dispersion law. We also outline the reasoning leading to Fourier’s law for heat conduction.     

Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian

We study the stationary points of what is known as the lattice Landau gauge fixing functional in one-dimensional compact U(1) lattice gauge theory, or as the Hamiltonian of the one-dimensional random phase XY model in statistical physics. An analytic solution of all stationary points is derived for lattices with an odd number of lattice sites and periodic boundary conditions. In the context of lattice gauge theory, these stationary points and their indices are used to compute the gauge fixing partition function, making reference in particular to the Neuberger problem. Interpreted as stationary points of the one-dimensional XY Hamiltonian, the solutions and their Hessian determinants allow us to evaluate a criterion which makes predictions on the existence of phase transitions and the corresponding critical energies in the thermodynamic limit.     

Bifurcations in globally coupled map lattices

The dynamics of globally coupled map lattices can be described in terms of a nonlinear Frobenius-Perron equation in the limit of large system size. This approach allows for an analytical computation of stationary states and their stability. The bifurcation behavior of coupled tent maps near the chaotic band merging point is presented. Furthermore, the time-independent states of coupled logistic equations are analyzed. The bifurcation diagram of the uncoupled map carries over to the map lattice. The analytical results are supplemented with numerical simulations     

Stability Diagrams of a Bose-Einstein Condensate in a Periodic Array of Quantum Wells

With the help of a set of exact closed-form solutions to the stationary Gross Pitaevskii equation, we compre-hensively investigate Landau and dynamical instabilities of a Bose-Einstein condensate in a periodic array of quantum wells. In the tight-binding limit, the anaiyticai expressions for both Landau and dynamical instabilities are obtained in terms of the compressibility and effective mass of the BEC system. Then the stability phase diagrams are shown to be similar to the one in the case of the sinusoidal optical lattice.     

Onset of motion and dynamic reordering of a vortex lattice

Time resolved transport measurements on a driven vortex lattice in an undoped 2H-NbSe2 crystal show that the response to a current pulse is governed by healing of defects as the lattice evolves from a stationary to a moving steady state and that the response time reflects the degree of order in the initial vortex state. We find that stationary field cooled vortex lattices become more ordered with decreasing temperature and identify a temperature below which a qualitative change in the response signals the disappearance of topological defects.     

Steady state convergence acceleration of the generalized lattice Boltzmann equation with forcing term through preconditioning

Several applications exist in which lattice Boltzmann methods (LBM) are used to compute stationary states of fluid motions, particularly those driven or modulated by external forces. Standard LBM, being explicit time-marching in nature, requires a long time to attain steady state convergence, particularly at low Mach numbers due to the disparity in characteristic speeds of propagation of different quantities. In this paper, we present a preconditioned generalized lattice Boltzmann equation (GLBE) with forcing term to accelerate steady state convergence to flows driven by external forces. The use of multiple relaxation times in the GLBE allows enhancement of the numerical stability. Particular focus is given in preconditioning external forces, which can be spatially and temporally dependent. In particular, correct forms of moment projections of source/forcing terms are derived such that they recover preconditioned Navier–Stokes equations with non-uniform external forces. As an illustration, we solve an extended system with a preconditioned lattice kinetic equation for magnetic induction field at low magnetic Prandtl numbers, which imposes Lorentz forces on the flow of conducting fluids. Computational studies, particularly in three-dimensions, for canonical problems show that the number of time steps needed to reach steady state is reduced by orders of magnitude with preconditioning. In addition, the preconditioning approach resulted in significantly improved stability characteristics when compared with the corresponding single relaxation time formulation.     

一维定态薛定谔方程的宏观模拟解法

将实验模拟法引入量子力学.设计了一个弦振动系统,这个系统的定态方程与定态薛定谔方程数学形式一样,为定态薛定谔方程的模拟解法提供了理论和实验途径.宏观模拟的结果为理解薛定谔方程提供了宏观类比. 关键词： 薛定谔方程 宏观模拟解法