On one dimensional quantum Zakharov system

Jin-Cheng  Jiang; Chi-Kun  Lin; Shuanglin  Shao

doi:10.3934/dcds.2016040

Article Contents

2016, Volume 36, Issue 10: 5445-5475. Doi: 10.3934/dcds.2016040

This issue Previous Article Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type Next Article Zero-one law of Hausdorff dimensions of the recurrent sets

On one dimensional quantum Zakharov system

1.
Department of Mathematics, National Tsing Hua University, Hsinchu 30013
2.
Department of Mathematical Sciences, Xi'an Jiaotong-Liverpool University, SIP. Suzhou, Jiangsu, 215123

Received: April 2015

Revised: April 2016

Published: May 2017

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Abstract

In this paper, we discuss the properties of one dimensional quantum Zakharov system which describes the nonlinear interaction between the quantum Langmuir and quantum ion-acoustic waves. The system (1a)-(1b) with initial data $(E(0),n(0),\partial_t n(0))\in H^k\bigoplus H^l\bigoplus H^{l-2}$ is local well-posedness in low regularity spaces (see Theorem 1.1 and Figure 1). Especially, the low regularity result for $k$ satisfies $-3/4 < k \leq -1/4$ is obtained by using the key observation that the convoluted phase function is convex and careful bilinear analysis. The result can not be obtained by using only Strichartz inequalities for ``Schrödinger" waves.

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Mathematics Subject Classification: Primary: 35Q40, 35L56; Secondary: 81Q99.

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