Modulation approximation for the non-isentropic Euler-Poisson system

As a formal approximation, the nonlinear Schr\"{o}dinger (NLS) equation can be derived to describe the evolution of the envelopes of small oscillating wave packets-like solutions to the Euler-Poisson system. In this paper we rigorously justify that the wave packets for the non-isentropic Euler-Poisson system can be approximated by solutions of the NLS equation over a physically relevant $\mathcal{O}(\epsilon^{-2})$ time scale. Besides the difficulties such as resonances at $k=0$ and $k=\pm k_0$ and loss of derivatives arising in the modulation approximation problem in the isentropic Euler-Poisson system, new difficulties arise in the non-isentropic case. In the non-isentropic Euler-Poisson system, new resonances at wave number $k=\pm 2k_0$ appear which necessitate rescaling the correction to the modulation approximation differently for different wave numbers. In addition, it is more difficult to obtain the uniform estimates for the error $(R_{0},R_{1},R_{-1})$ between the real solutions and the approximate solutions, due to the extra interactions with the temperature. To overcome the difficulties aroused by resonances and loss of derivatives, we find several important structural identities between the diagonalized unknowns and apply a series of normal-form transforms, to obtain uniform estimates for the error over the desired $\mathcal{O}(\epsilon^{-2})$ long time scale.