  The Schr\"{o}dinger-Newton model is a nonlinear system obtained by coupling
the linear Schr\"{o}dinger equation of canonical quantum mechanics with the
Poisson equation of Newtonian mechanics. In this paper we investigate the
effects of dark energy on the time-dependent Schr\"{o}dinger-Newton equations
by including a new source term with energy density $\rho_{\Lambda} = \Lambda
c^2/(8\pi G)$, where $\Lambda$ is the cosmological constant, in addition to the
particle-mass source term $\rho_m = m|\psi|^2$. The resulting
Schr\"{o}dinger-Newton-$\Lambda$ (S-N-$\Lambda$) system cannot be solved
exactly, in closed form, and one must resort to either numerical or
semianalytical (i.e., series) solution methods. We apply the Adomian
Decomposition Method, a very powerful method for solving a large class of
nonlinear ordinary and partial differential equations, to obtain accurate
series solutions of the S-N-$\Lambda$ system, for the first time. The dark
energy dominated regime is also investigated in detail. We then compare our
results to existing numerical solutions and analytical estimates, and show that
they are consistent with previous findings. Finally, we outline the advantages
of using the Adomian Decomposition Method, which allows accurate solutions of
the S-N-$\Lambda$ system to be obtained quickly, even with minimal
computational resources.
