  In this paper, we introduce the notion of a noncommutative Poisson bialgebra,
and establish the equivalence between matched pairs, Manin triples and
noncommutative Poisson bialgebras. Using quasi-representations and the
corresponding cohomology theory of noncommutative Poisson algebras, we study
coboundary noncommutative Poisson bialgebras which leads to the introduction of
the Poisson Yang-Baxter equation. A skew-symmetric solution of the Poisson
Yang-Baxter equation naturally gives a (coboundary) noncommutative Poisson
bialgebra. Rota-Baxter operators, more generally O-operators on noncommutative
Poisson algebras, and noncommutative pre-Poisson algebras are introduced, by
which we construct skew-symmetric solutions of the Poisson Yang-Baxter equation
in some special noncommutative Poisson algebras obtained from these structures.
