In this paper we continue our study [DSS20] of the nonlinear Schr\"odinger equation (NLS) with bounded initial data which do not vanish at infinity. Local well-posedness on $\mathbb{R}$ was proved for real analytic data. Here prove global 1D NLS lying in $L^{p}$ any $2 < p \infty$, provided is sufficiently smooth. We use complete integrability cubic Schr{\"o}dinger equation.

The Schr\"{o}dinger-Newton system is a nonlinear obtained by coupling together the linear Schr\"{o}dinger equation of quantum mechanics with Poisson Newtonian mechanics. In present work we will investigate effects cosmological constant (dark energy or vacuum fluctuation) on system, modifying through addition new term. corresponding Schr\"{o}dinger-Newton-$\Lambda$ cannot be solved exactly, and ...