NLS approximation of the Euler-Poisson system for a cold ion-acoustic plasma. (English) Zbl 1532.35428
Summary: In the previous paper [Commun. Math. Phys. 371, No. 2, 357–398 (2019; Zbl 1428.35526)], we proved the nonlinear Schrödinger (NLS) approximation for the Euler-Poisson system for a hot ion-acoustic plasma, where the appearance of resonances and the loss of derivatives of quadratic terms are the main difficulties. Note that when the ion-acoustic plasma is hot, the Euler-Poisson system is Friedrich symmetrizable, and the linear term can provide a derivative to compensate the loss of derivative induced by quadratic terms after diagonalizing the linearized system. When the ion-acoustic plasma is cold, as considered in the present paper, the situation is very different from that in the previous paper. The Euler-Poisson system becomes a pressureless system, so the linear operator has no regularity, and the quadratic terms still lose a derivative in the diagonalized system. This fact makes it more difficult to prove the NLS approximation of Euler-Poisson system for a cold ion-acoustic plasma. In this paper, we take advantage of the special structure of the pressureless Euler-Poisson system and the normal-form transformation to deal with the difficulties caused by resonances, especially the difficulties caused by derivative loss, in order to prove the NLS approximation.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35Q31 | Euler equations |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 35M10 | PDEs of mixed type |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 76X05 | Ionized gas flow in electromagnetic fields; plasmic flow |
| 78A30 | Electro- and magnetostatics |
| 82D10 | Statistical mechanics of plasmas |
Keywords:
modulation approximation; nonlinear Schrödinger equation; cold ion-acoustic plasma; Euler-Poisson systemCitations:
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