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.

中文翻译：

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量子Euler-Poisson方程的调制近似

非线性薛定ding（NLS）方程用于描述缓慢调制的时空振荡波包状解的包络，其可以作为量子Euler-Poisson方程的形式近似方程导出。在本文中，我们通过采用改进的能量函数和时空共振方法来严格证明这种近似的合理性，以克服二次项，共振和准线性所带来的困难。