Justification of the NLS Approximation for the Euler–Poisson Equation
Justification of the NLS Approximation for the Euler–Poisson Equation
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DOI：
10.1007/s00220-019-03576-4
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发表时间：
2018-01
影响因子：
2.4
通讯作者：
Huimin Liu;Xueke Pu
中科院分区：
文献类型：
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作者：
Huimin Liu;Xueke Pu
非线性薛定谔(NLS)方程可以作为一种形式的近似方程来描述离子Euler-Poisson方程的慢调制空间和暂时振荡波包状解的包络。在这篇文章中，我们通过给出离子Euler-Poisson系统的精确解与通过NLS方程得到的形式近似之间的Soblev范数误差估计，严格证明了这种近似。由于问题的拟线性，证明由几个困难组成，例如共振和正则性损失。通过引入正规型变换和截断函数，以及精心构造方程的能量泛函，克服了这些困难。
The nonlinear Schrödinger (NLS) equation can be derived as a formal approximation equation describing the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler–Poisson equation. In this paper, we rigorously justify such approximation by giving error estimates in Sobolev norms between exact solutions of the ion Euler–Poisson system and the formal approximation obtained via the NLS equation. The justification consists of several difficulties such as the resonances and loss of regularity, due to the quasilinearity of the problem. These difficulties are overcome by introducing normal form transformation and cutoff functions and carefully constructed energy functional of the equation.