Nonlinear stationary solutions of the Wigner and Wigner–Poisson equations

Exact nonlinear stationary solutions of the one-dimensional Wigner and Wigner–Poisson equations in the terms of the Wigner functions that depend not only on the energy but also on position are presented. In this way, the Bernstein–Greene–Kruskal modes of the classical plasma are adapted for the quantum formalism in the phase space. The solutions are constructed for the case of a quartic oscillator potential, as well as for the self-consistent Wigner–Poisson case. Conditions for well-behaved physically meaningful equilibrium Wigner functions are discussed.