Quantum corrections to nonlinear ion acoustic wave with Landau damping

Quantum corrections to nonlinear ion acoustic wave with Landau damping have been computed using Wigner equation approach. The dynamical equation governing the time development of nonlinear ion acoustic wave with semiclassical quantum corrections is shown to have the form of higher KdV equation which has higher order nonlinear terms coming from quantum corrections, with the usual classical and quantum corrected Landau damping integral terms. The conservation of total number of ions is shown from the evolution equation. The decay rate of KdV solitary wave amplitude due to presence of Landau damping terms has been calculated assuming the Landau damping parameter $\alpha_1 = \sqrt{{m_e}/{m_i}}$ to be of the same order of the quantum parameter $Q = {\hbar^2}/({24 m^2 c^2_{s} L^2})$. The amplitude is shown to decay very slowly with time as determined by the quantum factor $Q$.