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## Modulation approximation for the quantum Euler-Poisson equation

 1 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China 2 Faculty of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China 3 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

* Corresponding author: Huimin Liu

Received  April 2020 Revised  August 2020 Published  October 2020

Fund Project: The first author D. Bian is supported by NSFC under the Contract 11871005. The second author H. Liu is supported by NSFC under the Contract 12001338, the Youth Fund of Shanxi University of Finance and Economics of China under Z06180 and the Science and Technology Innovation Project of Shanxi Province of China under 2020L0256. The last author X. Pu is supported by NSFC under the contract 11871172 and the Natural Science Foundation of Guangdong Province of China under 2019A1515012000

The nonlinear Schrödinger (NLS) equation is used to describe the envelopes of slowly modulated spatially and temporally oscillating wave packet-like solutions, which can be derived as a formal approximation equation of the quantum Euler-Poisson equation. In this paper, we rigorously justify such an approximation by taking a modified energy functional and a space-time resonance method to overcome the difficulties induced by the quadratic terms, resonance and quasilinearity.

Citation: Dongfen Bian, Huimin Liu, Xueke Pu. Modulation approximation for the quantum Euler-Poisson equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020292
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