We study the computation of ground states and time dependent solutions of the Schr¨odinger-Poisson system(SPS)on a bounded domain in 2D(i.e.in two space dimensions).On a disc-shaped domain,we derive exact artific...We study the computation of ground states and time dependent solutions of the Schr¨odinger-Poisson system(SPS)on a bounded domain in 2D(i.e.in two space dimensions).On a disc-shaped domain,we derive exact artificial boundary conditions for the Poisson potential based on truncated Fourier series expansion inθ,and propose a second order finite difference scheme to solve the r-variable ODEs of the Fourier coefficients.The Poisson potential can be solved within O(M NlogN)arithmetic operations where M,N are the number of grid points in r-direction and the Fourier bases.Combined with the Poisson solver,a backward Euler and a semi-implicit/leap-frog method are proposed to compute the ground state and dynamics respectively.Numerical results are shown to confirm the accuracy and efficiency.Also we make it clear that backward Euler sine pseudospectral(BESP)method in[33]can not be applied to 2D SPS simulation.展开更多
We investigate blow-up of the focusing nonlinear Schr¨odinger equation,in the critical and supercritical cases.Numerical simulations are performed to examine the dependence of the time at which blow-up occurs on ...We investigate blow-up of the focusing nonlinear Schr¨odinger equation,in the critical and supercritical cases.Numerical simulations are performed to examine the dependence of the time at which blow-up occurs on properties of the data or the equation.Three cases are considered:dependence on the scale of the nonlinearity when the initial data are fixed;dependence upon the strength of a quadratic oscillation in the initial data when the equation and the initial profile are fixed;and dependence upon a damping factor when the initial data are fixed.In most of these situations,monotonicity in the evolution of the blow-up time does not occur.展开更多
基金Singapore A*STAR SERC PSF-Grant No.1321202067National Natural Science Foundation of China Grant NSFC41390452the Doctoral Programme Foundation of Institution of Higher Education of China as well as by the Austrian Science Foundation(FWF)under grant No.F41(project VICOM)and grant No.I830(project LODIQUAS)and grant No.W1245 and the Austrian Ministry of Science and Research via its grant for the WPI.
文摘We study the computation of ground states and time dependent solutions of the Schr¨odinger-Poisson system(SPS)on a bounded domain in 2D(i.e.in two space dimensions).On a disc-shaped domain,we derive exact artificial boundary conditions for the Poisson potential based on truncated Fourier series expansion inθ,and propose a second order finite difference scheme to solve the r-variable ODEs of the Fourier coefficients.The Poisson potential can be solved within O(M NlogN)arithmetic operations where M,N are the number of grid points in r-direction and the Fourier bases.Combined with the Poisson solver,a backward Euler and a semi-implicit/leap-frog method are proposed to compute the ground state and dynamics respectively.Numerical results are shown to confirm the accuracy and efficiency.Also we make it clear that backward Euler sine pseudospectral(BESP)method in[33]can not be applied to 2D SPS simulation.
基金This work was supported by the Austrian Science Foundation(FWF)via the START project(Y-137-TEC)by theWWTF(Viennese Science Fund,project MA-45).
文摘We investigate blow-up of the focusing nonlinear Schr¨odinger equation,in the critical and supercritical cases.Numerical simulations are performed to examine the dependence of the time at which blow-up occurs on properties of the data or the equation.Three cases are considered:dependence on the scale of the nonlinearity when the initial data are fixed;dependence upon the strength of a quadratic oscillation in the initial data when the equation and the initial profile are fixed;and dependence upon a damping factor when the initial data are fixed.In most of these situations,monotonicity in the evolution of the blow-up time does not occur.
