The purpose of this paper is to present a comparison between the modified nonlinear SchrSdinger （MNLS） equation and the focusing and defocusing variants of the （unmodified） nonlinear SchrSdinger （NLS） equation i...The purpose of this paper is to present a comparison between the modified nonlinear SchrSdinger （MNLS） equation and the focusing and defocusing variants of the （unmodified） nonlinear SchrSdinger （NLS） equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation （in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data）, or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.展开更多
Semiclassical limit to the solution of transient bipolar quantum drift-diffusion model in semiconductor simulation is discussed. It is proved that the semiclassical limit of this solution satisfies the classical bipol...Semiclassical limit to the solution of transient bipolar quantum drift-diffusion model in semiconductor simulation is discussed. It is proved that the semiclassical limit of this solution satisfies the classical bipolar drift-diffusion model. In addition, the authors also prove the existence of weak solution.展开更多
The weak solutions to the stationary quantum drift-diffusion equations （QDD） for semiconductor devices are investigated in one space dimension. The proofs are based on a reformulation of the system as a fourth-order...The weak solutions to the stationary quantum drift-diffusion equations （QDD） for semiconductor devices are investigated in one space dimension. The proofs are based on a reformulation of the system as a fourth-order elliptic boundary value problem by using an exponential variable transformation. The techniques of a priori estimates and Leray-Schauder＇s fixed-point theorem are employed to prove the existence. Furthermore, the uniqueness of solutions and the semiclassical limit δ→0 from QDD to the classical drift-diffusion （DD） model are studied.展开更多
The semiclassical limit in the transient quantum drift-diffusion equations with isentropic pressure in one space dimension is rigorously proved. The equations are supplemented with homogeneous Neumann boundary conditi...The semiclassical limit in the transient quantum drift-diffusion equations with isentropic pressure in one space dimension is rigorously proved. The equations are supplemented with homogeneous Neumann boundary conditions. It is shown that the semiclassical limit of this solution solves the classical drift-diffusion model. In the meanwhile, the global existence of weak solutions is proved.展开更多
Semiclassical limit to the solution of isentropic quantum drift-diffusion model in semicon- ductor simulation is discussed. It is proved that the semiclassical limit of this solution satisfies the classical drift-diff...Semiclassical limit to the solution of isentropic quantum drift-diffusion model in semicon- ductor simulation is discussed. It is proved that the semiclassical limit of this solution satisfies the classical drift-diffusion model. In addition, we also proved the global existence of weak solutions.展开更多
A fourth order parabolic system, the bipolar quantum drift-diffusion model in semiconductor simulation, with physically motivated Dirichlet-Neumann boundary condition is studied in this paper. By semidiscretization in...A fourth order parabolic system, the bipolar quantum drift-diffusion model in semiconductor simulation, with physically motivated Dirichlet-Neumann boundary condition is studied in this paper. By semidiscretization in time and compactness argument, the global existence and semiclassical limit are obtained, in which semiclassieal limit describes the relation between quantum and classical drift-diffusion models, Furthermore, in the case of constant doping, we prove the weak solution exponentially approaches its constant steady state as time increases to infinity.展开更多
We study the l^1-stability of a Haxniltonian-preserving scheme, developed in [Jin and Wen, Comm. Math. Sci., 3 （2005）, 285-315], for the Liouville equation with a discontinuous potential in one space dimension. We p...We study the l^1-stability of a Haxniltonian-preserving scheme, developed in [Jin and Wen, Comm. Math. Sci., 3 （2005）, 285-315], for the Liouville equation with a discontinuous potential in one space dimension. We prove that, for suitable initial data, the scheme is stable in the l^1-norm under a hyperbolic CFL condition which is in consistent with the l^1-convergence results established in [Wen and Jin, SIAM J. Numer. Anal., 46 （2008）, 2688-2714] for the same scheme. The stability constant is shown to be independent of the computational time. We also provide a counter example to show that for other initial data, in particular, the measure-valued initial data, the numerical solution may become l^1-unstable.展开更多
We study the Ll-error of a Hamiltonian-preserving scheme, developed in [19], for the Liouville equation with a piecewise constant potential in one space dimension when the initial data is given with perturbation error...We study the Ll-error of a Hamiltonian-preserving scheme, developed in [19], for the Liouville equation with a piecewise constant potential in one space dimension when the initial data is given with perturbation errors. We extend the l1-stability analysis in [46] and apply the Ll-error estimates with exact initial data established in [45] for the same scheme. We prove that the scheme with the Dirichlet incoming boundary conditions and for a class of bounded initial data is Ll-convergent when the initial data is given with a wide class of perturbation errors, and derive the Ll-error bounds with explicit coefficients. The convergence rate of the scheme is shown to be less than the order of the initial perturbation error, matching with the fact that the perturbation solution can be l1-unstable.展开更多
This work is concerned with e1-error estimates on a Hamiltonian-preserving scheme for the Liouville equation with pieeewise constant potentials in one space dimension. We provide an analysis much simpler than these in...This work is concerned with e1-error estimates on a Hamiltonian-preserving scheme for the Liouville equation with pieeewise constant potentials in one space dimension. We provide an analysis much simpler than these in literature and obtain the same half-order convergence rate. We formulate the Liouville equation with discretized velocities into a series of linear convection equations with piecewise constant coefficients, and rewrite the numerical scheme into some immersed interface upwind schemes. The e1-error estimates are then evaluated by comparing the derived equations and schemes.展开更多
This paper studies the existence, semiclassical limit, and long-time behavior of weak solutions to the unipolar isentropic quantum drift-diffusion model, a fourth order parabolic system. Semi-discretization in time an...This paper studies the existence, semiclassical limit, and long-time behavior of weak solutions to the unipolar isentropic quantum drift-diffusion model, a fourth order parabolic system. Semi-discretization in time and entropy estimates give the global existence and semiclassical limit of nonnegative weak solutions to the one-dimensional model with a nonnegative large initial value and a Dirichlet-Neumann boundary condition. Furthermore, the weak solutions are proven to exponentially approach constant steady state as time increases to infinity.展开更多
We study the semi-classical limit of the Schro¨dinger equation in a crystal in the presence of an external potential and magnetic field. We first introduce the Bloch-Wigner transform and derive the asymptotic equ...We study the semi-classical limit of the Schro¨dinger equation in a crystal in the presence of an external potential and magnetic field. We first introduce the Bloch-Wigner transform and derive the asymptotic equations governing this transform in the semi-classical setting. For the second part, we focus on the appearance of the Berry curvature terms in the asymptotic equations. These terms play a crucial role in many important physical phenomena such as the quantum Hall effect. We give a simple derivation of these terms in different settings using asymptotic analysis.展开更多
The authors investigate the global existence and semiclassical limit of weak solutions to a sixth-order parabolic system,which is a quantum-corrected macroscopic model derived recently to simulate the quantum effects ...The authors investigate the global existence and semiclassical limit of weak solutions to a sixth-order parabolic system,which is a quantum-corrected macroscopic model derived recently to simulate the quantum effects in miniaturized semiconductor devices.展开更多
基金supported by the National Science Foundation under grant DMS-0807653
文摘The purpose of this paper is to present a comparison between the modified nonlinear SchrSdinger （MNLS） equation and the focusing and defocusing variants of the （unmodified） nonlinear SchrSdinger （NLS） equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation （in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data）, or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.
基金Supported by NSFC (10541001, 10571101, 10401019, and 10701011)by Basic Research Foundation of Tsinghua University
文摘Semiclassical limit to the solution of transient bipolar quantum drift-diffusion model in semiconductor simulation is discussed. It is proved that the semiclassical limit of this solution satisfies the classical bipolar drift-diffusion model. In addition, the authors also prove the existence of weak solution.
文摘The weak solutions to the stationary quantum drift-diffusion equations （QDD） for semiconductor devices are investigated in one space dimension. The proofs are based on a reformulation of the system as a fourth-order elliptic boundary value problem by using an exponential variable transformation. The techniques of a priori estimates and Leray-Schauder＇s fixed-point theorem are employed to prove the existence. Furthermore, the uniqueness of solutions and the semiclassical limit δ→0 from QDD to the classical drift-diffusion （DD） model are studied.
基金the National Natural Science Foundation of China(Nos.10401019,10701011,10541001)
文摘The semiclassical limit in the transient quantum drift-diffusion equations with isentropic pressure in one space dimension is rigorously proved. The equations are supplemented with homogeneous Neumann boundary conditions. It is shown that the semiclassical limit of this solution solves the classical drift-diffusion model. In the meanwhile, the global existence of weak solutions is proved.
文摘Semiclassical limit to the solution of isentropic quantum drift-diffusion model in semicon- ductor simulation is discussed. It is proved that the semiclassical limit of this solution satisfies the classical drift-diffusion model. In addition, we also proved the global existence of weak solutions.
基金Supported by the Natural Science Foundation of China （No. 10571101, No. 10626030 and No. 10871112）
文摘A fourth order parabolic system, the bipolar quantum drift-diffusion model in semiconductor simulation, with physically motivated Dirichlet-Neumann boundary condition is studied in this paper. By semidiscretization in time and compactness argument, the global existence and semiclassical limit are obtained, in which semiclassieal limit describes the relation between quantum and classical drift-diffusion models, Furthermore, in the case of constant doping, we prove the weak solution exponentially approaches its constant steady state as time increases to infinity.
基金Innovation Project of the Chinese Academy of Sciences grants K5501312S1,K5502212F1,K7290312G7 and K7502712F7NSFC grant 10601062+1 种基金NSF grant DMS-0608720NSAF grant 10676017
文摘We study the l^1-stability of a Haxniltonian-preserving scheme, developed in [Jin and Wen, Comm. Math. Sci., 3 （2005）, 285-315], for the Liouville equation with a discontinuous potential in one space dimension. We prove that, for suitable initial data, the scheme is stable in the l^1-norm under a hyperbolic CFL condition which is in consistent with the l^1-convergence results established in [Wen and Jin, SIAM J. Numer. Anal., 46 （2008）, 2688-2714] for the same scheme. The stability constant is shown to be independent of the computational time. We also provide a counter example to show that for other initial data, in particular, the measure-valued initial data, the numerical solution may become l^1-unstable.
文摘We study the Ll-error of a Hamiltonian-preserving scheme, developed in [19], for the Liouville equation with a piecewise constant potential in one space dimension when the initial data is given with perturbation errors. We extend the l1-stability analysis in [46] and apply the Ll-error estimates with exact initial data established in [45] for the same scheme. We prove that the scheme with the Dirichlet incoming boundary conditions and for a class of bounded initial data is Ll-convergent when the initial data is given with a wide class of perturbation errors, and derive the Ll-error bounds with explicit coefficients. The convergence rate of the scheme is shown to be less than the order of the initial perturbation error, matching with the fact that the perturbation solution can be l1-unstable.
文摘This work is concerned with e1-error estimates on a Hamiltonian-preserving scheme for the Liouville equation with pieeewise constant potentials in one space dimension. We provide an analysis much simpler than these in literature and obtain the same half-order convergence rate. We formulate the Liouville equation with discretized velocities into a series of linear convection equations with piecewise constant coefficients, and rewrite the numerical scheme into some immersed interface upwind schemes. The e1-error estimates are then evaluated by comparing the derived equations and schemes.
基金the National Natural Science Foundation of China(No. 10401019)
文摘This paper studies the existence, semiclassical limit, and long-time behavior of weak solutions to the unipolar isentropic quantum drift-diffusion model, a fourth order parabolic system. Semi-discretization in time and entropy estimates give the global existence and semiclassical limit of nonnegative weak solutions to the one-dimensional model with a nonnegative large initial value and a Dirichlet-Neumann boundary condition. Furthermore, the weak solutions are proven to exponentially approach constant steady state as time increases to infinity.
基金supported in part by Department of Energy under Contract No.DE-FG02-03ER25587by Office of Naval Research under Contract No.N00014-01-1-0674by National Science Foundation grant DMS-0708026
文摘We study the semi-classical limit of the Schro¨dinger equation in a crystal in the presence of an external potential and magnetic field. We first introduce the Bloch-Wigner transform and derive the asymptotic equations governing this transform in the semi-classical setting. For the second part, we focus on the appearance of the Berry curvature terms in the asymptotic equations. These terms play a crucial role in many important physical phenomena such as the quantum Hall effect. We give a simple derivation of these terms in different settings using asymptotic analysis.
基金Project supported by the National Natural Science Foundation of China (Nos. 10871112, 10771008)the Research Fund for the Doctoral Program of Higher Education of China (No. 20090005120009)+1 种基金 the Fundamental Research Funds for the Central Universities (No. BUPT2009RC0702)the Talents Scheme Funds of BUPT
文摘The authors investigate the global existence and semiclassical limit of weak solutions to a sixth-order parabolic system,which is a quantum-corrected macroscopic model derived recently to simulate the quantum effects in miniaturized semiconductor devices.
