Series Solution of the Time-Dependent Schrödinger–Newton Equations in the Presence of Dark Energy via the Adomian Decomposition Method

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