Long wavelength limit of non-isentropic Euler-Poisson system

Abstract

This paper is devoted to study the long wavelength limit of Euler-Poisson system with variable temperature for ions in plasma. By the Gardner-Morikawa transformation, we formally derive the Korteweg-de Vries (KdV) equation and the linear KdV equation from the non-isentropic Euler-Poisson system. Then, we establish the rigorous justification of the KdV equation from Euler-Poisson equation in a time interval $ [0, \tau_*\varepsilon^{-\frac{3}{2}}] $ for some $ \tau_*>0 $. The long wavelength limit of Euler-Poisson equation follows from the suitable uniform bound of the remainder terms by the nonlinear energy estimates.