The computational uncertainty principle states that the numerical computation of nonlinear ordinary differential equations（ODEs） should use appropriately sized time steps to obtain reliable solutions.However,the int...The computational uncertainty principle states that the numerical computation of nonlinear ordinary differential equations（ODEs） should use appropriately sized time steps to obtain reliable solutions.However,the interval of effective step size（IES） has not been thoroughly explored theoretically.In this paper,by using a general estimation for the total error of the numerical solutions of ODEs,a method is proposed for determining an approximate IES by translating the functions for truncation and rounding errors.It also illustrates this process with an example.Moreover,the relationship between the IES and its approximation is found,and the relative error of the approximation with respect to the IES is given.In addition,variation in the IES with increasing integration time is studied,which can provide an explanation for the observed numerical results.The findings contribute to computational step-size choice for reliable numerical solutions.展开更多
In recent years, peculiar physical phenomena enabled by non-Hermitian systems, especially the parity-time(PT)-symmetric systems, have drawn tremendous research interests. Particularly, special spectral degeneracies kn...In recent years, peculiar physical phenomena enabled by non-Hermitian systems, especially the parity-time(PT)-symmetric systems, have drawn tremendous research interests. Particularly, special spectral degeneracies known as exceptional points(EPs) and coherent perfect absorber-laser(CPAL) points where zero and infinite large eigenvalues coexist are the most popular topics to be studied. To date, the discussions of EPs that serve as transition boundaries between broken PT-symmetry phase and exact PT-symmetry phase have been intensively presented. However, the theoretical analysis and experimental validations of CPAL points are inadequate. Different from EPs, CPAL points, as a special solution of broken PT-symmetry phase, may exhibit even further counterintuitive physical features, which may have significant implications to study non-Hermitian physics. Here, we review some recent advances of CPAL phenomena in different sub-disciplines of physics, including optics, electronics and electromagnetics, and acoustics.Additionally, we also provide an envision of future directions and applications of CPAL systems.展开更多
We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation （NLDE） in the nonrelativistic limit regime, involving a small dimensionless parameter 0...We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation （NLDE） in the nonrelativistic limit regime, involving a small dimensionless parameter 0 〈 ε〈〈1 which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e., there are propagating waves with wavelength O（ ε^2） and O（1） in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference （CNFD） method and establish rigorously its error estimate which depends explicitly on the mesh size h and time step τ- as well as the small parameter 0 〈 ε≤1 Based on the error bound, in order to obtain ＇correct＇ numerical solutions in the nonrelativistic limit regime, i.e., 0 〈 ε≤1 , the CNFD method requests the ε-scalability： τ- = O（ε3） and h = O（√ε）. Then we propose and analyze two numerical methods for the discretization of NLDE by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time- splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their ε-scalability is improved to τ = O（ε2） and h = O（1） when 0 〈 ε 〈〈 1. Extensive numerical results are reported to confirm our error estimates.展开更多
We construct exact stationaxy solutions to the one-dimensional coupled Gross-Pitaevskii equations for the two-species Bose-Einstein condensates with equal intraspecies and interspecies interaction constants. Three typ...We construct exact stationaxy solutions to the one-dimensional coupled Gross-Pitaevskii equations for the two-species Bose-Einstein condensates with equal intraspecies and interspecies interaction constants. Three types of complex solutions as well as their soliton limits axe derived. By making use of the SU（2） unitary symmetry, we further obtain analytical time-evolving solutions. These solutions exhibit spatiotemporal periodicity.展开更多
A Kramers pair of helical edge states in quantum spin Hall effect （QSHE） is robust against normal dephasing but not robust to spin dephasing. In our work, we provide an effective spin dephasing mechanism in the pudd...A Kramers pair of helical edge states in quantum spin Hall effect （QSHE） is robust against normal dephasing but not robust to spin dephasing. In our work, we provide an effective spin dephasing mechanism in the puddles of two-dimensional （2D） QSHE, which is simulated as quantum dots modeled by 2D massive Dirac Hamiltouian. We demonstrate that the spin dephasing effect can originate from the combination of the Rashba spin-orbit coupling and electron-phonon interaction, which gives rise to inelastic backscattering in edge states within the topological insulator quantum dots, although the time-reversal symmetry is preserved throughout. Finally, we discuss the tunneling between extended helical edge states and local edge states in the QSH quantum dots, which leads to backscattering in the extended edge states. These results can explain the more robust edge transport in InAs/GaSb QSH systems.展开更多
基金supported by the National Natural Science Foundation of China[grant numbers 41375110,11471244]
文摘The computational uncertainty principle states that the numerical computation of nonlinear ordinary differential equations（ODEs） should use appropriately sized time steps to obtain reliable solutions.However,the interval of effective step size（IES） has not been thoroughly explored theoretically.In this paper,by using a general estimation for the total error of the numerical solutions of ODEs,a method is proposed for determining an approximate IES by translating the functions for truncation and rounding errors.It also illustrates this process with an example.Moreover,the relationship between the IES and its approximation is found,and the relative error of the approximation with respect to the IES is given.In addition,variation in the IES with increasing integration time is studied,which can provide an explanation for the observed numerical results.The findings contribute to computational step-size choice for reliable numerical solutions.
文摘In recent years, peculiar physical phenomena enabled by non-Hermitian systems, especially the parity-time(PT)-symmetric systems, have drawn tremendous research interests. Particularly, special spectral degeneracies known as exceptional points(EPs) and coherent perfect absorber-laser(CPAL) points where zero and infinite large eigenvalues coexist are the most popular topics to be studied. To date, the discussions of EPs that serve as transition boundaries between broken PT-symmetry phase and exact PT-symmetry phase have been intensively presented. However, the theoretical analysis and experimental validations of CPAL points are inadequate. Different from EPs, CPAL points, as a special solution of broken PT-symmetry phase, may exhibit even further counterintuitive physical features, which may have significant implications to study non-Hermitian physics. Here, we review some recent advances of CPAL phenomena in different sub-disciplines of physics, including optics, electronics and electromagnetics, and acoustics.Additionally, we also provide an envision of future directions and applications of CPAL systems.
基金supported by the Ministry of Education of Singapore(Grant No.R146-000-196-112)National Natural Science Foundation of China(Grant No.91430103)
文摘We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation （NLDE） in the nonrelativistic limit regime, involving a small dimensionless parameter 0 〈 ε〈〈1 which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e., there are propagating waves with wavelength O（ ε^2） and O（1） in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference （CNFD） method and establish rigorously its error estimate which depends explicitly on the mesh size h and time step τ- as well as the small parameter 0 〈 ε≤1 Based on the error bound, in order to obtain ＇correct＇ numerical solutions in the nonrelativistic limit regime, i.e., 0 〈 ε≤1 , the CNFD method requests the ε-scalability： τ- = O（ε3） and h = O（√ε）. Then we propose and analyze two numerical methods for the discretization of NLDE by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time- splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their ε-scalability is improved to τ = O（ε2） and h = O（1） when 0 〈 ε 〈〈 1. Extensive numerical results are reported to confirm our error estimates.
基金Supported by funds from the Ministry of Science and Technology of China under Grant No.2012CB821403by National Natural Science Foundation of China under Grant No.11374036
文摘We construct exact stationaxy solutions to the one-dimensional coupled Gross-Pitaevskii equations for the two-species Bose-Einstein condensates with equal intraspecies and interspecies interaction constants. Three types of complex solutions as well as their soliton limits axe derived. By making use of the SU（2） unitary symmetry, we further obtain analytical time-evolving solutions. These solutions exhibit spatiotemporal periodicity.
基金supported by the National Basic Research Program of China(Grant Nos.2015CB921102,2012CB821402 and 2012CB921303)the National Natural Science Foundation of China(Grant Nos.11534001 and11274364)
文摘A Kramers pair of helical edge states in quantum spin Hall effect （QSHE） is robust against normal dephasing but not robust to spin dephasing. In our work, we provide an effective spin dephasing mechanism in the puddles of two-dimensional （2D） QSHE, which is simulated as quantum dots modeled by 2D massive Dirac Hamiltouian. We demonstrate that the spin dephasing effect can originate from the combination of the Rashba spin-orbit coupling and electron-phonon interaction, which gives rise to inelastic backscattering in edge states within the topological insulator quantum dots, although the time-reversal symmetry is preserved throughout. Finally, we discuss the tunneling between extended helical edge states and local edge states in the QSH quantum dots, which leads to backscattering in the extended edge states. These results can explain the more robust edge transport in InAs/GaSb QSH systems.
