Long wavelength limit for the quantum Euler-Poisson equation

TL;DR

This paper investigates the long wavelength behavior of the quantum Euler-Poisson equation, deriving the quantum KdV equation and analyzing its dynamics, including special cases like the inviscid Burgers equation.

Contribution

It introduces a derivation of the quantum KdV equation from the quantum Euler-Poisson system using a singular perturbation approach.

Findings

Quantum KdV equation derived under long wavelength limit

KdV dynamics observed over time scale O(ε^{-3/2})

Special case reduces to inviscid Burgers equation when H=2

Abstract

In this paper, we consider the long wavelength limit for the quantum Euler-Poisson 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 $O(\epsilon^{-3/2})$. When the nondimensional quantum parameter $H=2$, it reduces to the inviscid Burgers equation.