Internal modes of discrete solitons near the anti-continuum limit of the dNLS equation

Discrete solitons of the discrete nonlinear Schrödinger (dNLS) equation are compactly supported in the anti-continuum limit of the zero coupling between lattice sites. Eigenvalues of the linearization of the dNLS equation at the discrete soliton determine its spectral stability. Small eigenvalues bifurcating from the zero eigenvalue near the anti-continuum limit were characterized earlier for this model. Here we analyze the resolvent operator and prove that it is bounded in the neighborhood of the continuous spectrum if the discrete soliton is simply connected in the anti-continuum limit. This result rules out the existence of internal modes (neutrally stable eigenvalues of the discrete spectrum) near the anti-continuum limit.     

Kinetic Thomas-Fermi solutions of the Gross-Pitaevskii equation

Approximate solutions of the Gross-Pitaevskii (GP) equation, obtained upon neglection of the kinetic energy, are well known as Thomas-Fermi solutions. They are characterized by the compensation of the local potential by the collisional energy. In this article we consider exact solutions of the GP-equation with this property and definite values of the kinetic energy, which suggests the term “kinetic Thomas-Fermi” (KTF) solutions. Despite their formal simplicity, KTF-solutions can possess complex current density fields with unconventional topology. We point out that a large class of light-shift potentials gives rise to KTF-solutions. As elementary examples, we consider one-dimensional and two-dimensional optical lattice scenarios, obtained by means of the superposition of two, three and four laser beams, and discuss the stability properties of the corresponding KTF-solutions. A general method is proposed to excite two-dimensional KTF-solutions in experiments by means of time-modulated light-shift potentials.     

Formal analytical solutions for the Gross-Pitaevskii equation

Considering the Gross-Pitaevskii integral equation we are able to formally obtain an analytical solution for the order parameter Φ(x) and for the chemical potential μ as a function of a unique dimensionless non-linear parameter Λ. We report solutions for different ranges of values for the repulsive and the attractive non-linear interactions in the condensate. Also, we study a bright soliton-like variational solution for the order parameter for positive and negative values of Λ. Introducing an accumulated error function we have performed a quantitative analysis with respect to other well-established methods as: the perturbation theory, the Thomas-Fermi approximation, and the numerical solution. This study gives a very useful result establishing the universal range of the Λ-values where each solution can be easily implemented. In particular, we showed that for Λ<−9, the bright soliton function reproduces the exact solution of GPE wave function.     

The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation

We consider the flow of a gas in a channel whose walls are kept at fixed (different) temperatures. There is a constant external force parallel to the boundaries which may themselves also be moving. The system is described by the stationary Boltzmann equation to which are added Maxwellian boundary conditions with unit accommodation coefficient. We prove that when the temperature gap, the relative velocity of the planes, and the force are all sufficiently small, there is a solution which converges, in the hydrodynamic limit, to a local Maxwellian with parameters given by the stationary solution of the corresponding compressible Navier-Stokes equations with no-slip voundary conditions. Corrections to this Maxwellian are obtained in powers of the Knudsen number with a controlled remainder.     

Riccati generalization of self-similar solutions of nonautonomous Gross-Pitaevskii equation

We present a systematic analytical approach to construct a family of self-similar waves, related through a free parameter, in quasi one-dimension Gross-Pitaevskii equation with time-varying parameters. This approach enables us to control the dynamics of dark and bright similaritons, and first- and second- order self-similar rogue waves in Bose-Einstein condensate through the modulation of time dependent trapping potential. The analysis is done for the sech²? type time-varying quadratic trapping potential for two different choices of linear potential.     

Stability and collapse of localized solutions of the controlled three-dimensional Gross-Pitaevskii equation

On the basis of recent investigations, a newly developed analytical procedure is used for constructing a wide class of localized solutions of the controlled three-dimensional (3D) Gross-Pitaevskii equation (GPE) that governs the dynamics of Bose-Einstein condensates (BECs) in the presence of a spatio-temporally varying external potential. The controlled 3D GPE is decomposed into a two-dimensional (2D) linear Schr?dinger equation (called the `transverse equation’) and a one-dimensional (1D) nonlinear Schr?dinger equation (called the `longitudinal equation’), constrained by a variational condition for the controlling potential. The latter corresponds to the requirement for the minimization of the control operation in the transverse plane. Then, the above class of localized solutions are constructed as the product of the solutions of the transverse and longitudinal equations. A consistency condition between the transverse and longitudinal solutions yields a relationship between the transverse and longitudinal restoring forces produced by the external trapping potential well through a `controlling parameter’ (i.e. the average, with respect to the transverse profile, of the nonlinear inter-atomic interaction term of the GPE). It is found that the longitudinal profile supports localized solutions in the form of bright, dark or grey solitons with time-dependent amplitudes, widths and centroids. The related longitudinal phase is varying in space and time with time-dependent curvature radius and wavenumber. In turn, all the above parameters (i.e. amplitudes, widths, centroids, curvature radius and wavenumbers) can be easily expressed in terms of the controlling parameter. It is also found that the transverse profile has the form of Hermite-Gauss functions (depending on the transverse coordinates), and the explicit spatio-temporal dependence of the controlling potential is self-consistently determined. On the basis of these exact 3D analytical solutions, a stability analysis is carried out, focusing our attention on the physical conditions for having collapsing or non-collapsing solutions.     

Numerical and variational solutions of the dipolar Gross-Pitaevskii equation in reduced dimensions

Laser Physics - We suggest a simple Gaussian Lagrangian variational scheme for the reduced time-dependent quasi-one-and quasi-two-dimensional Gross-Pitaevskii (GP) equations of a dipolar...     

Rigorous derivation of the Gross-Pitaevskii equation

The time-dependent Gross-Pitaevskii equation describes the dynamics of initially trapped Bose-Einstein condensates. We present a rigorous proof of this fact starting from a many-body bosonic Schr?dinger equation with a short-scale repulsive interaction in the dilute limit. Our proof shows the persistence of an explicit short-scale correlation structure in the condensate.     

Strongly differentiable solutions of the discrete coagulation-fragmentation equation

We examine an infinite system of ordinary differential equations that models the binary coagulation and multiple fragmentation of clusters. In contrast to previous investigations, our analysis does not involve finite-dimensional truncations of the system. Instead, we treat the problem as an infinite-dimensional differential equation, posed in an appropriate Banach space, and apply perturbation results from the theory of strongly continuous semigroups of operators. The existence and uniqueness of physically meaningful solutions are established for uniformly bounded coagulation rates but with no growth restrictions imposed on the fragmentation rates.     

Mathematical and physical aspects of controlling the exact solutions of the 3D Gross-Pitaevskii equation

The possibility of the decomposition of the three-dimensional (3D) Gross-Pitaevskii equation (GPE) into a pair of coupled Schrödinger-type equations, is investigated. It is shown that, under suitable mathematical conditions, it is possible to construct the exact controlled solutions of the 3D GPE from the solutions of a linear 2D Schrödinger equation coupled with a 1D nonlinear Schrödinger equation (the transverse and longitudinal components of the GPE, respectively). The coupling between these two equations is the functional of the transverse and the longitudinal profiles. The applied method of nonlinear decomposition, called the controlling potential method (CPM), yields the full 3D solution in the form of the product of the solutions of the transverse and longitudinal components of the GPE. It is shown that the CPM constitutes a variational principle and sets up a condition on the controlling potential well. Its physical interpretation is given in terms of the minimization of the (energy) effects introduced by the control. The method is applied to the case of a parabolic external potential to construct analytically an exact BEC state in the form of a bright soliton, for which the quantitative comparison between the external and controlling potentials is presented.     

Some approximate solutions to the discrete master equation

Recent mathematical developments on approximate diffusionlike solutions to the master equation are summarized. The technique is applied to two master equations of physical interest-one that describes the phenomenon of superradiance and a second that characterizes generation-recombination noise in semiconductors. For this second case, some previously obtained equilibrium results are found and the extension of these results to finite times is given.     

Large-time behavior of solutions of the discrete Boltzmann equation

Large-time behavior of solutions of the one-dimensional discrete Boltzmann equation is studied. Under suitable assumptions it is proved that as time tends to infinity, the solution approaches a function which is constructed explicitly in terms of the self-similar solutions of the Burgers equation and the linear heat equation.     

Stability of stationary solutions of the discrete self-trapping equation

The discrete self-trapping equation is a model coupled oscillator system with applications in many areas including the dynamics of small molecules and the study of solitions on alpha-helix proteins. Some simple stability criteria for stationary solutions of this equation are presented, together with some example calculations.     

一维Tonks-Girardeau原子气区域中 Gross-Pitaevskii方程简化模型的精确行波解

利用动力系统方法研究一维Tonks-Girardeau原子气区域中Gross-Pitaevskii (GP)方程简化模型的一些精确行波解以及这些精确行波解的动力学行为, 研究系统的参数对行波解的动力学行为的影响. 在不同的参数条件下, 获得了一维Tonks-Girardeau原子气区域中GP方程简化模型的六个行波解的精确参数表达式. 关键词： 动力系统方法 孤立波解 周期波解 扭波解     

A discrete KdV equation hierarchy: continuous limit,diverse exact solutions and their asymptotic state analysis

In this paper, a discrete KdV equation that is related to the famous continuous KdV equation is studied. First, an integrable discrete KdV hierarchy is constructed, from which several new discrete KdV equations are obtained. Second, we correspond the first several discrete equations of this hierarchy to the continuous KdV equation through the continuous limit. Third, the generalized (m, 2N − m)-fold Darboux transformation of the discrete KdV equation is established based on its known Lax pair. Finally, the diverse exact solutions including soliton solutions, rational solutions and mixed solutions on non-zero seed background are obtained by applying the resulting Darboux transformation, and their asymptotic states and physical properties such as amplitude, velocity, phase and energy are analyzed. At the same time, some soliton solutions are numerically simulated to show their dynamic behaviors. The properties and results obtained in this paper may be helpful to understand some physical phenomena described by KdV equations.     

Vortex Solutions of the Liouville Equation

The most general vortex solution of the Liouville equation (which arises in nonrelativistic Chern–Simons theory) is associated with rational functions, f(z)=P(z) Qz), where P(z) and Q(z) are both polynomials, deg P相似文献   

Nonisotropic solutions of the Boltzmann equation

We consider the relaxation to equilibrium of a spatially uniform Maxwellian gas. We expand the solution of the nonlinear Boltzmann equation in a truncated series of orthogonal functions. We integrate numerically the equation for non-isotropic initial conditions. For certain simple conditions we find interesting proximity effects and other transient relaxation phenomena at thermal energies. Furthermore, we define a resummation of the orthogonal expansion which is more convenient than the original one for the numerical analysis of the relaxation process.     

The Navier-Stokes limit of the stationary boltzmann equation for hard potentials

In this paper we extend recent results on the hydrodynamic Navier-Stokes limit of the stationary Boltzmann equation for the flow of a gas of hard spheres in a channel in the presence of an external force to the case of a hard intermolecular potential with Grad angular cutoff. We prove the convergence of the solution, for small Knudsen numbers, to the Maxwellian with parameters solving the corresponding Navier-Stokes equation. In the present case we only get polynomial decay of the solution for large velocities, instead of the exponential decay which holds for hard spheres.     

Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrödinger equation

We investigate the properties of modulational instability and discrete breathers in the cubic-quintic discrete nonlinear Schrödinger equation. We analyze the regions of modulational instabilities of nonlinear plane waves. Using the Page approach [J.B. Page, Phys. Rev. B 41 (1990) 7835], we derive the conditions for the existence and stability for bright discrete breather solutions. It is shown that the quintic nonlinearity brings qualitatively new conditions for stability of strongly localized modes. The application to the existence of localized modes in the Bose-Einstein condensate (BEC) with three-body interactions in an optical lattice is discussed. The numerical simulations agree with the analytical predictions.