  Starting from the von Neumann-Maxwell equations for the Wigner
quasi-probability distribution and for the self-consistent electric field, the
quantum analog of the classical single-wave model has been derived. The linear
stability of the quantum single-wave model has been studied, and periodic in
time patterns have been found both analytically and numerically. In addition,
some features of quantum chaos have been detected in the unstable region in
parameter space. Further, a class of standing-wave solutions of the quantum
single-wave model has also been found, which have been observed to behave as
stable solitary-wave structures. The analytical results have been finally
compared to the exact system dynamics obtained by solving the corresponding
equations in Schrodinger representation numerically.
