Long Wavelength Limit for the Quantum Euler--Poisson Equation

Abstract

Abstract. In this paper, we consider the long wavelength limit for the quantum EulerPoisson equation. Under the Gardner-Morikawa transform, we derive the quantum Korteweg-de Vries (KdV) equation by a singular perturbation method. We show that the KdV dynamics can be seen at time interval of order O(ǫ −3/2 ). When the nondimensional quantum parameter H = 2, it reduces to the inviscid Burgers equation.

Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force

“…Xi et al [36] provided the time decay rates for the high-order spatial derivatives. Moreover, Pu and his coauthors studied the global existence of smooths solutions for the full compressible quantum Navier-Stokes equations [24] and derived the famous KdV equations from the quantum Euler-Poisson equations [17]. However, the optimal L q − L 2 time decay for smooth solutions of the vQMHD system when there is the potential force is not known yet, which is the main topic in the present paper.…”

“…The quantum terms data back to Wigner [30] and one may would like to refer to Haas [7,8] for quantum plasma. Furthermore, for the derivation of the quantum KdV equations from the quantum Euler-Poisson equations is studied [17].…”

Justification of the NLS Approximation for the Euler–Poisson Equation

“…Almost at the same time, [27] also established the Zakharov-Kuznetsov equation in 3D from the Euler-Poisson system. Recently, the authors in the present paper [28,29] obtained rigorously the quantum KdV limit in 1D and the KP-I and KP-II equations in 2D for the Euler-Poisson system for cold as well as hot plasma taking quantum effect into account, where the electron equilibrium is given by a Fermi-Dirac distribution. Han-Kwan [15] also introduced a long wave scaling for the Vlasov-Poisson equation and derived the KdV equation in 1D, the KP-II equation in 2D and the Zakharov-Kuznetsov equation in 3D using the modulated energy method.…”

The KP limit of a reduced quantum Euler–Poisson equation

“…Such an approximation by the KdV equation was justified recently. 21 As a first step toward a justification of the KP equation as an envelope equation, we consider in this paper the following RQEP equation:…”