The one-dimensional two-species quantum hydrodynamic model is considered in the limit of small mass ratio of the charge carriers. Closure is obtained by adopting an equation of state pertaining to a zero-temperature Fermi gas for the electrons and by disregarding pressure effects for the ions. By an appropriate rescaling of the variables, a nondimensional parameter H, proportional to quantum diffraction effects, is identified. The system is then shown to support linear waves, which in the limit of small H resemble the classical ion-acoustic waves. In the weakly nonlinear limit, the quantum plasma is shown to support waves described by a deformed Korteweg–de Vries equation which depends in a nontrivial way on the quantum parameter H. In the fully nonlinear regime, the system also admits traveling waves which can exhibit periodic patterns. The quasineutral limit of the system is also discussed.

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Quantum ion-acoustic waves

Statistical Mechanics. — A Set of Lectures

The quantum hydrodynamic model for charged particle systems is extended to the cases of nonzero magnetic fields. In this way, quantum corrections to magnetohydrodynamics are obtained starting from the quantum hydrodynamical model with magnetic fields. The importance of the quantum corrections is described by a parameter H which can be significant in dense astrophysical plasmas. The quantum magnetohydrodynamic model is analyzed in the infinite conductivity limit. The conditions for equilibrium in ideal quantum magnetohydrodynamics are established. Translationally invariant exact equilibrium solutions are obtained in the case of the ideal quantum magnetohydrodynamic model.

A magnetohydrodynamic model for quantum plasmas

In this paper, we propose a numerical scheme to solve the two-dimensional (2D) time-dependent Schrodinger equation using collocation points and approximating the solution using multiquadrics (MQ) and the Thin Plate Splines (TPS) Radial Basis Function (RBF). The scheme works in a similar fashion as finite-difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.

A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions

We present a preliminary study of a new phenom- ena associated with the Euler-Poisson equations — the so called critical threshold phenomena, where the answer to questions of global smoothness vs. finite time breakdown depends on whether the initial configuration crosses an intrinsic,ON1O critical thresh- old. We investigate a class of Euler-Poisson equations, ranging from one-dimensional problems with or without various forcing mech- anisms to multi-dimensional isotropic models with geometrical symmetry. These models are shown to admit a critical thresh- old which is reminiscent of the conditional breakdown of waves on the beach; only waves above certain initial critical threshold experience finite-time breakdown, but otherwise they propagate smoothly. At the same time, the asymptotic long time behavior of the solutions remains the same, independent of crossing these initial thresholds. A case in point is the simple one-dimensional problem where the unforced inviscid Burgers' solution always forms a shock dis- continuity, except for the non-generic case of increasing initial profile,u 0 0. In contrast, we show that the corresponding one- dimensional Euler-Poisson equation with zero background has global smooth solutions as long as its initialN0;u 0O-configuration satisfiesu 0 p 2k0 - see (2.11) below, allowing a finite, crit- ical negative velocity gradient. As is typical for such nonlinear convection problems, one is led to a Ricatti equation which is balanced here by a forcing acting as a 'nonlinear resonance', and which in turn is responsible for this critical threshold phenom- ena.

Critical thresholds in euler-poisson equations

In this review paper we present the most important mathematical properties of dispersive limits of (non)linear Schr¨odinger type equations. Different formulations are used to study these singular limits, e.g., the kinetic formulation of the linear Schr¨odinger equation based on the Wigner transform is well suited for global-in-time analysis without using WKB-(expansion) techniques, while the modified Madelung transformation reformulating Schr¨odinger equations in terms of a dispersive perturbation of a quasilinear symmetric hyperbolic system usually only gives local-in-time results due to the hyperbolic nature of the limit equations. Deterministic analogues of turbulence are also discussed. There, turbulent diffusion appears naturally in the zero dispersion limit.

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Chi-Kun Lin

Papers

A review of dispersive limits of (non)linear schr¨odinger-type equations