Instantaneous Formulation for Transitions Between Two Instantaneous Bound States and Its Gauge Invariance

We have precisely derived a ＂rigorous instantaneous formulation＂ for transitions between two bound states when the bound states are well-described by instantaneous Bethe-Salpeter （BS） equation （i.e. the kernel of the equation is instantaneous ＂occasionally＂）. The obtained rigorous instantaneous formulation, in fact, is expressed as an operator sandwiched by two ＂reduced BS wave functions＂ properly, while the reduced BS wave functions appearing in the formulation are the rigorous solutions of the instantaneous BS equation, and they may relate to Schroedinger wave functions straightforwardly. We also show that the rigorous instantaneous formulation is gauge-invariant with respect to the Uem（1） transformation precisely, if the concerned transitions are radiative. Some applications of the formulation are outlined.     

Instantaneous Formulation for Transitions Between Two Instantaneous Bound States and Its Gauge Invariance

We have precisely derived a “rigorous instantaneous formulation“ for transitions between two bound states when the bound states are well-described by instantaneous Bethe-Salpeter (BS) equation (i.e. the kernel of the equation is instantaneous“occasionally“). The obtained rigorous instantaneous formulation, in fact, is expressed as an operator sandwiched by two “reduced BS wave functions“ properly, while the reduced BS wave functions appearing in the formulation are the rigorous solutions of the instantaneous BS equation, and they may relate to Schr(o)dinger wave functions straightforwardly. We also show that the rigorous instantaneous formulation is gauge-invariant with respect to the Uem(1) transformation precisely, if the concerned transitions are radiative. Some applications of the formulation are outlined.     

Solutions of Two Kinds of Non-Isospectral Generalized Nonlinear Schrodinger Equation Related to Bose-Einstein Condensates

Two non-isospectral generalized nonlinear Schrodinger （ONLS） equations, which are two important models of nonlinear excitations of matter waves in Bose-Einstein condensates, are studied. Two novel transformations are constructed such that these two GNLS equations are transformed to the well-known nonlinear Schr6dinger （NLS） equation, which is an isospectral equation. Therefore, once one solution of the NLS equation is provided, we can immediately obtain one solution for two ONLS equations by these transformations. Thus it is unnecessary to solve these two non-isospectral GNLS equations directly. Soliton solutions and periodic solutions are obtained for them by two transformations from the corresponding solutions of the NLS equation, which are generated by Darboux transformation.     

Relativistic studies of the decay constants of S wave and P wave mesons

Relativistic corrections are important in hadronic physics since even for the heavy hadrons there are sizable relativistic corrections. Therefore one should use a relativistic model to describe the higher excited states. This note summarizes our predictions for the decay constants of the S wave and P wave heavy mesons by means of the instantaneous relativistic Bethe Salpeter equation （Salpeter equation）.     

Auto-Baecklund Transformation and Soliton-Type Solutions of the Generalized Variable-Coefficient Kadomtsev-Petviashvili Equation

Using the truncated Painlevé expansion, an auto-Baecklund transformation and soliton-type solutions of the generalized variable-coefficient Kadomtsev-Petviashvili （GKP） equation are obtained by symbolic computation. Since the cylindrical Korteweg-de Vries （cKdV） equation, the cylindrical KP （cKP） equation and the generalized cKP （GcKP） equation are all special cases of the GKP equation, we can also obtain the corresponding results of these equations.     

An Explicit Backlund Transformation of Burgers Equation with Applications

In this paper, an explicit Bgcklund transformation （BT） of the Burgers equation is obtained by using the further extended tanh method [Phys. Lett. A 307 （2003） 269; Chaos, Solitons ＆ Fractals 17 （2003） 669]. Based on the BT and some newly obtained seed solutions, infinite sequences of exact solutions for the Burgers equation are generated. Further more, this BT of the Burgers equation is applied to solve the variant Boussinesq equations and the approximate equations of long water wave.     

Backlund Transformations and Explicit Solutions of （2d-1）-Dimensional Barotropic and Quasi-Geostrophic Potential Vorticity Equation

By the Backlund transformation method, an important （2＋1）-dimensional nonlinear barotropie and quasigeostrophic potential vorticity （BQGPV） equation is investigated. Some simple special Backlund transformation theorems are proposed and used to get explicit solutions of the BQGPV equation. Furthermore, all solutions of a second order linear ordinary differential equation including an arbitrary function can be used to construct explicit solutions of the （2＋1）-dimensional BQGPV equation. Some figures are also given out to describe these solutions.     

New Solitary Solutions of （2＋1）-Dimensional Variable Coefficient Nonlinear Schrodinger Equation with an External Potential

By a series of transformations, the （2＋1）-dimensional variable coefficient nonlinear Schr？dinger equation can turn to the Klein-Gordon equation. Many new double travelling wave solutions of the Klein-Gordon equation are obtained. Thus, the new solitary solutions of the variable coefficient nonlinear Schr？inger equation with an external potential can be found.     

Explicit Solutions for Generalized （2＋1）-Dimensional Nonlinear Zakharov-Kuznetsov Equation

The exact solutions of the generalized （2＋1）-dimensional nonlinear Zakharov-Kuznetsov （Z-K） equation are explored by the method of the improved generalized auxiliary differential equation. Many explicit analytic solutions of the Z-K equation are obtained. The methods used to solve the Z-K equation can be employed in further work to establish new solutions for other nonlinear partial differential equations.     

Instantaneous Bethe-Salpeter Equation and Its Exact Solution

We present an approach to solve Bethe-Salpeter (BS) equations exactly without any approximation if the kernel of the BS equations exactly is instantaneous, and take positronium as an example to illustrate the general features of the exact solutions. The key step for the approach is from the BS equations to derive a set of coupled and well-determined integration equations in linear eigenvalue for the components of the BS wave functions equivalently, which may be solvable numerically under a controlled accuracy, even though there is no analytic solution. For positronium, the exact solutions precisely present corrections to those of the corresponding Schrödinger equation in order v¹ (v is the relative velocity) for eigenfunctions, in order v² for eigenvalues, and the mixing between S and D components in J^PC=1^-- states etc., quantitatively. Moreover, we also point out that there is a questionable step in some existent derivations for the instantaneous BS equations if one is pursuing the exact solutions. Finally, we emphasize that one should take the O(v) corrections emerging in the exact solutions into account accordingly if one is interested in the relativistic corrections for relevant problems to the bound states.     

Ground State Mass Spectrum for Scalar Diquarks with Bethe-Salpeter Equation

In this article,we study the structures of the pseudoscalar mesons π,K and the scalar diquarks Ua,Da,Sa in the framework of the coupled rainbow Schwinger-Dyson equation and ladder Bethe-Salpeter equation with the confining effective potential.The u,d,s quarks have small current masses,and the renormalization is very large,the mass poles in the timelike region are absent which implements confinement naturally.The Bethe-Salpeter wavefunctions of the pseudoscalar mesons π,K,and the scalar diquarks Ua,Da,Sa have the same type (Gaussian type) momentum dependence,center around zero momentum and extend to the energy scale about q2 = 1 GeV2,which happens to be the energy scale for the chiral symmetry breaking,the strong interactions in the infrared region result in bound (or quasi-bound) states.The numerical results for the masses and decay constants of the π and K mesons can reproduce the experimental values,and the ground state masses of the scalar diquarks Ua,Da,Sa are consistent with the existing theoretical calculations.We suggest a new Lagrangian which may explain the uncertainty of the masses of the scalar diquarks.     

时间分数阶亚扩散方程的高阶差分方法

提出两差分格式求解时间分数阶亚扩散方程.两个格式都是绝对稳定的,收敛阶均为O(τ^q+h²),其中q(q=2-β或2)与方程解的光滑性有关,β(0 < β < 1)是分数阶导数的阶、τ和h分别是时间和空间方向步长.数值实验验证了理论结果的正确性,并与其他方法进行比较,显示了本文方法的有效性和精确性.     

二维非定常Sine-Gordon方程辛算法及其孤子数值模拟

在矩形域[-a,a]×[-a,a]内对微分算子L=(ə²)/(əx²)+(ə²)/(əy²)用5点差分格式将二维非定常Sine Gordon方程离散化为一个2×799²阶非线性Hamilton系统.对该系统使用Euler中心格式,得到一个非线性方程组.对此方程组建立迭代解法并给出了这个迭代方法的收敛条件和收敛速度.Sine Gordon方程单孤子和双孤子的数值模拟试验显示该辛算法是有效的.     

Degenerate Solutions of the Nonlinear Self-Dual Network Equation

The N-fold Darboux transformation(DT) T_n~([N]) of the nonlinear self-dual network equation is given in terms of the determinant representation. The elements in determinants are composed of the eigenvalues λ_j(j = 1, 2..., N)and the corresponding eigenfunctions of the associated Lax equation. Using this representation, the N-soliton solutions of the nonlinear self-dual network equation are given from the zero "seed" solution by the N-fold DT. A general form of the N-degenerate soliton is constructed from the determinants of N-soliton by a special limit λ_j →λ_1 and by using the higher-order Taylor expansion. For 2-degenerate and 3-degenerate solitons, approximate orbits are given analytically,which provide excellent fit of exact trajectories. These orbits have a time-dependent "phase shift", namely ln(t~2).     

Relativistic symmetries with the trigonometric Pschl-Teller potential plus Coulomb-like tensor interaction

The Dirac equation is solved to obtain its approximate bound states for a spin-1/2 particle in the presence of trigonometric Pschl-Teller(tPT) potential including a Coulomb-like tensor interaction with arbitrary spin-orbit quantum number κ using an approximation scheme to substitute the centrifugal terms κ(κ± 1)r-2.In view of spin and pseudo-spin(p-spin) symmetries,the relativistic energy eigenvalues and the corresponding two-component wave functions of a particle moving in the field of attractive and repulsive tPT potentials are obtained using the asymptotic iteration method(AIM).We present numerical results in the absence and presence of tensor coupling A and for various values of spin and p-spin constants and quantum numbers n and κ.The non-relativistic limit is also obtained.     

The Breit Equation

Contrary to the conventional view, the Breit equation can be solved. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Influence of the Renormalization Condition to the Equation of State of the Nuclear Matter

In the relativistic σ-ω model, the influences of the parameters b, c, d in the potential U(σ)=(1/2!)bσ²+(1/3!)cσ³+ (1/4!)dσ⁴ to the incompressibility, effective mass and binding energy of the nuclear matter are studied in detail. The calculation of equation of state of nuclear matter shows that the values of b, c, d depend on the renormalization condition, we also find that a soft equation of state of nuclear matter can be obtained in a suitable renormalization condition, and the experimental incompressibility coefficient can be reproduced. These results are also used to study the thermal properties of hot Δ-resonant nuclear matter.     

Solution of the Dirac Equation for Ring-Shaped Modified Kratzer Potential

We propose a new exactly solvable potential which is Formed by modified Kratzer potential plus a new ring-shaped potential η cot^2 θ/r^2 The solutions of the Dirac equation with equal scalar and vector ring-shaped modified Kratzer potential are found by using the Nikiforov-Uvarov method. The nonrelativistic limit of the energy spectrum has been discussed.