Title: The QKP limit of the quantum Euler-Poisson equation,
Abstract: In this paper, we consider the derivation of the Kadomtsev-Petviashvili (KP)
equation for cold ion-acoustic wave in the long wavelength limit of the
two-dimensional quantum Euler-Poisson system, under different scalings for
varying directions in the Gardner-Morikawa transform. It is shown that the
types of the KP equation depend on the scaled quantum parameter $H>0$. The
QKP-I is derived for $H>2$, QKP-II for $0<H<2$ and the dispersive-less KP (dKP)
equation for the critical case $H=2$. The rigorous proof for these limits is
given in the well-prepared initial data case, and the norm that is chosen to
close the proof is anisotropic in the two directions, in accordance with the
anisotropic structure of the KP equation as well as the Gardner-Morikawa
transform. The results can be generalized in several directions.