We investigate the boundary vaJue problem （BVP） of a quasi-one-dimensional Gross-Pitaevskii equation with the Kronig-Penney potential （KPP） of period d, which governs a repulsive Bose-Einstein condensate. Under th...We investigate the boundary vaJue problem （BVP） of a quasi-one-dimensional Gross-Pitaevskii equation with the Kronig-Penney potential （KPP） of period d, which governs a repulsive Bose-Einstein condensate. Under the zero and periodic boundary conditions, we show how to determine n exact stationary eigenstates {Rn} corresponding to different chemical potentials {μn} from the known solutions of the system. The n-th eigenstate P~ is the Jacobian elliptic function with period 2din for n = 1,2,…, and with zero points containing the potential barrier positions. So Rn is differentiable at any spatial point and R2 describes n complete wave-packets in each period of the KPP. It is revealed that one can use a laser pulse modeled by a 5 potential at site xi to manipulate the transitions from the states of {Rn} with zero Point x≠xi to the states of {Rn＇} with zero Point x= Xi. The results suggest an experimental scheme for applying BEC to test the BVP and to observe the macroscopic quantum transitions.展开更多
In this paper we consider averaging and finite difference methods for solving the 3-D boundary-value problem in a multilayered domain. We consider the metal concentration in the 3 layered peat blocks. Using experiment...In this paper we consider averaging and finite difference methods for solving the 3-D boundary-value problem in a multilayered domain. We consider the metal concentration in the 3 layered peat blocks. Using experimental data the mathematical model for calculating the concentration of metal at different points in peat layers is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for the partial differential equations （PDEs） of the elliptic type of second order with piece-wise diffusion coefficients in the three layer domain. We develop here a finite-difference method for solving a problem of the above type with the periodical boundary condition in x direction. This procedure allows reducing the 3-D problem to a system of 2-D problems by using a circulant matrix.展开更多
In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space whic...In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space which consists of functions with vector valued in a general Banach space, and then describe the solution of these abstract boundary value problem by the abstract linear integral operator of Volterra type. We call this process the integral operator solving process.展开更多
基金The project supported by the National Natural Science Foundation of China under Grant Nos.10575034 and 10875039
文摘We investigate the boundary vaJue problem （BVP） of a quasi-one-dimensional Gross-Pitaevskii equation with the Kronig-Penney potential （KPP） of period d, which governs a repulsive Bose-Einstein condensate. Under the zero and periodic boundary conditions, we show how to determine n exact stationary eigenstates {Rn} corresponding to different chemical potentials {μn} from the known solutions of the system. The n-th eigenstate P~ is the Jacobian elliptic function with period 2din for n = 1,2,…, and with zero points containing the potential barrier positions. So Rn is differentiable at any spatial point and R2 describes n complete wave-packets in each period of the KPP. It is revealed that one can use a laser pulse modeled by a 5 potential at site xi to manipulate the transitions from the states of {Rn} with zero Point x≠xi to the states of {Rn＇} with zero Point x= Xi. The results suggest an experimental scheme for applying BEC to test the BVP and to observe the macroscopic quantum transitions.
文摘In this paper we consider averaging and finite difference methods for solving the 3-D boundary-value problem in a multilayered domain. We consider the metal concentration in the 3 layered peat blocks. Using experimental data the mathematical model for calculating the concentration of metal at different points in peat layers is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for the partial differential equations （PDEs） of the elliptic type of second order with piece-wise diffusion coefficients in the three layer domain. We develop here a finite-difference method for solving a problem of the above type with the periodical boundary condition in x direction. This procedure allows reducing the 3-D problem to a system of 2-D problems by using a circulant matrix.
文摘In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space which consists of functions with vector valued in a general Banach space, and then describe the solution of these abstract boundary value problem by the abstract linear integral operator of Volterra type. We call this process the integral operator solving process.
