Lie symmetry analysis of the quantum Zakharov equations

The Lie point symmetries of the one-dimensional quantum Zakharov (qZ) system of equations are considered, which is a general model to describe the coupling between the Langmuir and the ion-acoustic waves in a quantum setting. It is demonstrated that the Lie symmetries of the qZ system are exactly similar to those of the classical Zakharov equations. Further, similarity reductions are conducted based on the obtained Lie symmetries. A pure general periodic similarity ion-acoustic wave solution is obtained with the presence of constant linear and time-dependent nonlinear shears and time-dependent background, where the quantum effect increases the period of the waves.