Analytical Solutions for the Two-Dimensional Gross-Pitaevskii Equation with a Harmonic Trap

Both the homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions of the two-dimensional and time-independent Gross-Pitaevskii equation,a nonlinear Schrdinger equation used in describing the system of Bose-Einstein condensates trapped in a harmonic potential.The approximate analytical solutions are obtained successfully.Comparisons between the analytical solutions and the numerical solutions have been made.The results indicate that they are agreement very well with each other when the atomic interaction is not too strong.     

Application of the homotopy analysis method for the Gross-Pitaevskii equation with a harmonic trap

<正>The Homotopy analysis method(HAM) is adopted to find the approximate analytical solutions of the GrossPitaevskii equation,a nonlinear Schrodinger equation is used in simulation of Bose-Einstein condensates trapped in a harmonic potential.Comparisons between the analytical solutions and the numerical solutions have been made.The results indicate that they fit very well with each other when the atomic interaction is weak.     

Weak Nonlinear Matter Waves in a Trapped Spin-1 Condensates

The dynamics of the weak nonlinear matter solitary waves in a spin-1condensates with harmonic external potential are investigated analytically by a perturbation method. It is shown that, in the small amplitude limit, the dynamics of the solitary waves are governed by a variable-coefficient Korteweg-de Vries (KdV) equation. The reduction to the (KdV) equation may be useful tounderstand the dynamics of nonlinear matter waves in spinor BECs. The analytical expressions for the evolution of soliton show that the small-amplitude vector solitons of the mixed types perform harmonic oscillations in the presence of the trap. Furthermore, the emitted radiation profiles and the soliton oscillation frequency are also obtained.     

Solid-Effect Rate for Dynamic Nuclear Polarization of Spin-l/2 and Spin-1 Nuclei

The solid-effect rate equations for the dynamic nuclear polarization of spin-1/2 (tritium or hydrogen) or spin-1 (deuterium) nuclei are derived in the limiting case of the electronspin-resonance line width being much smaller than the nuclear-magnetic-resonance frequency. The exact analytical solutions of the rate equations for spin-1/2 nuclei are given, while the rate equations for spin-1 nuclei, which are nonlinear, are solved in decoupling approximation. The nuclear polarization and the electronic polarization as well as the times to polarization are calculated in low temperature limit. It is found that there is an optimal radio frequency pumping rate for the nuclear polarizations, which is not considered until now.     

Spinless Duffin-Kemmer-Petiau Oscillator in a Galilean Non-commutative Phase Space

We examine Galilei-invariant linear wave equations in a non-commutative phase space. Specifically, we establish and solve the Galilean covariant Duffin-Kemmer-Petiau equation for spin-0 fields in a harmonic oscillator potential. We obtain these wave equations with a Galilean covariant approach, based on a (4+1)-dimensional manifold with light-cone coordinates followed by a reduction to a (3+1)-dimensional spacetime. We find the exact wave functions and their energy levels, and we examine the effects of non-commutativity.     

受限一维无自旋费米子系统的性质研究

本文借助于一维自旋1/2-XXZ模型的Bethe-ansatz精确解, 利用局域密度近似(LDA), 讨论了谐振势中一维无自旋费米子的密度分布, 得出了ρ-u相图(这里的ρ为无量纲的粒子数密度 变量u为相互作用强度)对相图的分析表明, 随着原子密度和近邻相互作用的变化, 系统出现五个不同的混合量子相通过对热力学硬度S_ρ的计算, 发现其可作为体系的序参量, 其奇异点可用以度量受限体系中量子相变的发生     

Magnetic ordering of the bond—diluted two—dimensional mixed spin Ising system

Numerical and analytical results are presented for the magnetic ordering in a bond-diluted spin-1/2 and spin-1 mixed transverse Ising system with a single-ion anisotropy on a honeycomb lattice.Special emphsis is placed on the magnetic ordering under the bond dilution and percolation threshold.We discuss in detail the influence of transverse fields of different sublattics on the normal magnetic ordering and on the magnetic ordering induced by single-lon anisotropy.We find that the magnetic ordering of a system exhibits an explicit difference when receiving the transverse field.This phenomenon has not been revealed in previous reports.     

Exact surface energy and elementary excitations of the XXX spin-1/2 chain with arbitrary non-diagonal boundary fields

We study the physical properties of the XXX spin-1/2 chain with arbitrary non-diagonal boundary fields. By using a combination of numerical analysis and analytical method, we obtain the surface energy and elementary excitations of the model. It shows that the contributions of the two boundary fields to the surface energy are additive. We also find that there exists a kind of excitations related to the boundary string.     

Higher harmonic generation in nonlinear waveguides of arbitrary cross-section

This article concerns the generation and properties of double harmonics in nonlinear isotropic waveguides of complex cross-section. Analytical solutions of nonlinear Rayleigh-Lamb waves and rod waves have been known for some time. These solutions explain the phenomenon of cumulative double harmonic generation of guided waves. These solutions, however, are only applicable to simple geometries. This paper combines the general approach of the analytical solutions with semi-analytical finite element models to generalize the method to more complex geometries, specifically waveguides with arbitrary cross-sections. Supporting comparisons with analytical solutions are presented for simple cases. This is followed by the study of the case of a rail track. One reason for studying nonlinear guided waves in rails is the potential measurement of thermal stresses in welded rail.     

Squeezed Number State Solutions of Generalized Two-Mode Harmonic Oscillators Model： an Algebraic Approach

Some analytical solutions of generalized two-mode harmonic oscillators model are obtained by utilizing an algebraic diagonalization method. We find two types of eigenstates which are formulated as extended SU(1,1), SU(2) squeezed number states respectively. Some statistical properties of these states are also discussed.     

Massive nonlinear sigma models and BPS domain walls in harmonic superspace

Four-dimensional massive nonlinear sigma models and BPS wall solutions are studied in the off-shell harmonic superspace approach in which supersymmetry is manifest. The general nonlinear sigma model can be described by an analytic harmonic potential which is the hyper-Kähler analog of the Kähler potential in theory. We examine the massive nonlinear sigma model with multi-center four-dimensional target hyper-Kähler metrics and derive the corresponding BPS equation. We study in some detail two particular cases with the Taub-NUT and double Taub-NUT metrics. The latter embodies, as its two separate limits, both Taub-NUT and Eguchi–Hanson metrics. We find that domain wall solutions exist only in the double Taub-NUT case including its Eguchi–Hanson limit.     

Unitary quantum three-body problem in a harmonic trap

We consider either 3 spinless bosons or 3 equal mass spin-1/2 fermions, interacting via a short-range potential of infinite scattering length and trapped in an isotropic harmonic potential. For a zero-range model, we obtain analytically the exact spectrum and eigenfunctions: for fermions all the states are universal; for bosons there is a coexistence of decoupled universal and efimovian states. All the universal states, even the bosonic ones, have a tiny 3-body loss rate. For a finite range model, we numerically find for bosons a coupling between zero angular momentum universal and efimovian states; the coupling is so weak that, for realistic values of the interaction range, these bosonic universal states remain long-lived and observable.     

Squeezed Number State Solutions of Generalized Two-Mode Harmonic Oscillators Model:an Algebraic Approach

Some analytical solutions of generalized two-mode harmonic oscillators model are obtained by utilizing an algebraic diagonalization method. We find two types of eigenstates which are formulated as extended SU(1,1), SU(2)squeezed number states respectively. Some statistical properties of these states are also discussed.     

Application of a modified rational harmonic balance method for a class of strongly nonlinear oscillators

An analytical approximate technique for conservative nonlinear oscillators is proposed. This method is a modification of the rational harmonic balance method in which analytical approximate solutions have rational form. This approach gives us the frequency of the motion as a function of the amplitude of oscillation. We find that this method works very well for the whole range of parameters, and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed. The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values and the results reveal that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems with complex nonlinearities.     

Approximate analytical solutions of the generalized Woods-Saxon potentials including the spin-orbit coupling term and spin symmetry

We study the approximate analytical solutions of the Dirac equation for the generalized Woods-Saxon potential with the pseudo-centrifugal term. We apply the Nikiforov-Uvarov method (which solves a second-order linear differential equation by reducing it to a generalized hypergeometric form) to spin- and pseudospin-symmetry to obtain, in closed form, the approximately analytical bound state energy eigenvalues and the corresponding upper- and lower-spinor components of two Dirac particles. The special cases κ = ±1 (s = $ \tilde l $ = 0, s-wave) and the non-relativistic limit can be reached easily and directly for the generalized and standard Woods-Saxon potentials. We compare the non-relativistic results with those obtained by others.     

Electromagnetic interpretation of the massless spin-1 field equation in curved space-time

The source-free Maxwell equations associated to the massless spin-1 free field equation are considered in curved space-time. The gauge invariance of the theory is discussed by using as starting point the notion of the spinor potential. The structure of the electromagnetic field in the case of the Robertson-Walker space-time is discussed by using the solutions of the massless spin-1 equations previously determined. The flat-universe case of the standard cosmology is studied exactly and considered in some limiting physical situations.     

Mixed spin transverse Ising model with longitudinal random crystal field interactions

The effect of a longitudinal random crystal field interaction on the phase diagrams of the mixed spin transverse Ising model consisting of spin-1/2 and spin-1 is investigated within the finite cluster approximation based on a single-site cluster theory. In order to expand a cluster identity of spin-1, we transform the spin-1 to spin-1/2 representation containing Pauli operators. We derive the state equations applicable to structures with arbitrary coordination number N. The phase diagrams obtained in the case of a honeycomb lattice (N=3) and a simple-cubic lattice (N=6), are qualitatively different and examined in detail. We find that both systems exhibit a variety of interesting features resulting from the fluctuation of the crystal field interactions. Received: 13 February 1998 / Accepted: 17 March 1998     

Wave functions and mass spectrum for a system of two relativistic spinor quarks coupled by a chromodynamical interaction

Exact solutions of relativistic quasipotential equations in the configuration representation were found for a system of two spin-1/2 quarks interacting via a Coulomb-like chromodynamical potential. Quantization conditions were found for the pseudoscalar, pseudovector, and vector cases. The present analysis was performed within the Hamiltonian formulation of quantum field theory via a transition to the relativistic configuration representation for the case of two relativistic spin-1/2 quarks of equal mass.     

Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales

We study the attenuation, caused by weak damping, of harmonic waves through a discrete, periodic structure with frequency nominally within the Propagation Zone (i.e., propagation occurs in the absence of the damping). The period of the structure consists of a linear stiffness and a weak linear/nonlinear damping. Adapting the transfer matrix method and using harmonic balance for the nonlinear terms, a four-dimensional linear/nonlinear map governing the dynamics is obtained. We analyze this map by applying the method of multiple scales upto first order. The resulting slow evolution equations give the amplitude decay rate in the structure. The approximations are validated by comparing with other analytical solutions for the linear case and full numerics for the nonlinear case. Good agreement is obtained. The method of analysis presented here can be extended to more complex structures.     

Matrix form and certain solutions of the Feynman-Gell-Mann type equations for a spin-3/2 particle

We consider a P-odd variant of a spin-3/2 particle theory in the matrix formulation. We determine a matrix algebra of a first-order equation. We find solutions of the free equation for a massive particle in the form of projection matrices-dyads. We obtain exact solutions of the equation in the case of an external plane wave field with linear polarization and in the case of an axially symmetric magnetic field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 3–8, October, 1987.The authors wish to thank A. A. Bogush for his interest in this work and useful discussions.