On one dimensional Quantum Zakharov system
  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 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. 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 "Schr\"{o}dinger" waves. 
Jin-Cheng Jiang, Chi-Kun Lin and Shuanglin Shao
math.AP