Long wavelength limit for the quantum Euler-Poisson equation

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.