Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner

DOI10.1016/S0167-2789(03)00124-6zbMath1073.81041MaRDI QIDQ1396643

Xiantao Li, Shih Jin

Publication date: 6 July 2003

Published in: Physica D (Search for Journal in Brave)

zbMATH Keywords

Vlasov equationstationary phase methodlinear superposition principle

Mathematics Subject Classification ID

Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05) Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory (81Q20)

Related Items (37)

GLOBAL WEAK SOLUTIONS FOR THE RELATIVISTIC WATERBAG CONTINUUM ⋮ A field-space-based level set method for computing multi-valued solutions to 1D Euler-Poisson equations ⋮ A level set method for the semiclassical limit of the Schrödinger equation with discontinuous potentials ⋮ An improved semi-Lagrangian time splitting spectral method for the semi-classical Schrödinger equation with vector potentials using NUFFT ⋮ TWO MOMENT SYSTEMS FOR COMPUTING MULTIPHASE SEMICLASSICAL LIMITS OF THE SCHRÖDINGER EQUATION ⋮ Numerical approximation of the Schrödinger equation with the electromagnetic field by the Hagedorn wave packets ⋮ A level set framework for capturing multi-valued solutions of nonlinear first-order equations ⋮ Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. I: Homogeneous problems ⋮ Computing multi-valued velocity and electric fields for 1D Euler-Poisson equations ⋮ Computation of the semiclassical limit of the Schrödinger equation with phase shift by a level set method ⋮ Superposition of multi-valued solutions in high frequency wave dynamics ⋮ A high order numerical method for computing physical observables in the semiclassical limit of the one-dimensional linear Schrödinger equation with discontinuous potentials ⋮ Exact equations for smoothed Wigner transforms and homogenization of wave propagation ⋮ Multivariate Gaussian Extended Quadrature Method of Moments for Turbulent Disperse Multiphase Flow ⋮ Efficient numerical methods for multiscale crowd dynamics with emotional contagion ⋮ Nonlinear geometric optics method-based multi-scale numerical schemes for a class of highly oscillatory transport equations ⋮ Gaussian beam formulations and interface conditions for the one-dimensional linear Schrödinger equation ⋮ Conditional quadrature method of moments for kinetic equations ⋮ High order numerical methods to a type of delta function integrals ⋮ Semiclassical regularization of Vlasov equations and wavepackets for nonlinear Schrödinger equations ⋮ Computational high frequency waves through curved interfaces via the Liouville equation and geometric theory of diffraction ⋮ A quadrature-based third-order moment method for dilute gas-particle flows ⋮ The Gaussian Wave Packet Transform for the Semi-Classical Schrödinger Equation with Vector Potentials ⋮ A class of quadrature-based moment-closure methods with application to the Vlasov-Poisson-Fokker-Planck system in the high-field limit ⋮ Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. II: Impurities, confinement and Bloch oscillations ⋮ Existence of multi-valued solutions with asymptotic behavior of parabolic Monge-Ampère equation ⋮ Realizable high-order finite-volume schemes for quadrature-based moment methods ⋮ Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation ⋮ Conservation equations for the semiclassical Schrödinger equation near caustics ⋮ Gaussian beam methods for the Schrödinger equation with discontinuous potentials ⋮ Mathematical and computational methods for semiclassical Schrödinger equations ⋮ Geometry of nonadiabatic quantum hydrodynamics ⋮ Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems ⋮ Incompressible and compressible limits of two-component Gross–Pitaevskii equations with rotating fields and trap potentials ⋮ THE VLASOV–POISSON EQUATIONS AS THE SEMICLASSICAL LIMIT OF THE SCHRÖDINGER–POISSON EQUATIONS: A NUMERICAL STUDY ⋮ Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. III: From ab initio models to WKB for Schrödinger-Poisson ⋮ Multifluid flows: a kinetic approach

Cites Work

• Geometric optics in a phase-space-based level set and Eulerian framework
• Using \(K\)-branch entropy solutions for multivalued geometric optics computations
• Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit
• On Wigner measures
• Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations
• A kinetic formulation for multi-branch entropy solutions of scalar conservation laws
• High-frequency wave propagation by the segment projection method
• On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime
• Big ray tracing: Multivalued travel time field computation using viscosity solutions of the eikonal equation
• Numerical solution of the high frequency asymptotic expansion for the scalar wave equation
• Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics
• Multi-phase computations in geometrical optics
• Some new results in multiphase geometrical optics
• The generation of modulated wavetrains in the solution of the Korteweg—de vries equation
• Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation
• The Riemann problem for the transportation equations in gas dynamics
• Sticky Particles and Scalar Conservation Laws
• The semiclassical limit of the defocusing NLS hierarchy
• A Wigner-function approach to (semi)classical limits: Electrons in a periodic potential
• Homogenization limits and Wigner transforms
• Numerical Approximations of Pressureless and Isothermal Gas Dynamics
• Direct computation of multivalued phase space solutions for Hamilton-Jacobi equations
• A Wigner-Measure Analysis of the Dufort--Frankel Scheme for the Schrödinger Equation
• TWO MOMENT SYSTEMS FOR COMPUTING MULTIPHASE SEMICLASSICAL LIMITS OF THE SCHRÖDINGER EQUATION
• Non-linear dispersive waves
• From the solution of the Tsarev system to the solution of the Whitham equations
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item

This page was built for publication: Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner