The Schrödinger-Newton (S-N) equations were proposed by Penrose [18] as a model for gravitational collapse of the wave-function. The potential in the Schrödinger equation is the gravity due to the density of |ψ| 2 , where ψ is the wave-function. As with normal Quantum Mechanics the probability, momentum and angular momentum are conserved. We first consider the spherically symmetric case, here the stationary solutions have been found numerically by Moroz et al [15] and Jones et al [3]. The ground state which has the lowest energy has no zeros. The higher states are such that the (n + 1)th state has n zeros. We consider the linear stability problem for the stationary states, which we numerically solve using spectral methods. The ground state is linearly stable since it has only imaginary eigenvalues. The higher states are linearly unstable having imaginary eigenvalues except for n quadruples of complex eigenvalues for the (n + 1)th state, where a quadruple consists of {λ,λ, −λ, −λ}. Next we consider the nonlinear evolution, using a method involving an iteration to calculate the potential at the next time step and Crank-Nicolson to evolve the Schrödinger equation. To absorb scatter we use a sponge factor which reduces the reflection back from the outer boundary condition and we show that the numerical evolution converges for different mesh sizes and time steps. Evolution of the ground state shows it is stable and added perturbations oscillate at frequencies determined by the linear perturbation theory. The higher states are shown to be unstable, emitting scatter and leaving a rescaled ground state. The rate at which they decay is controlled by the complex eigenvalues of the linear perturbation. Next we consider adding another dimension in two different ways: by considering the axisymmetric case and the 2-D equations. The stationary solutions are found. We modify the evolution method and find that the higher states are unstable. In 2-D case we consider rigidly rotationing solutions and show they exist and are unstable.

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Much has been published about potential benefits of the adoption of cyber-physical systems (CPSs) in manufacturing industry. However, less has been said about how such automation systems might be effectively configured and supported through their lifecycles and how application modeling, visualization, and reuse of such systems might be best achieved. It is vitally important to be able to incorporate support for engineering best practice while at the same time exploiting the potential that CPS has to offer in an automation systems setting. This paper considers the industrial context for the engineering of CPS. It reviews engineering approaches that have been proposed or adopted to date including Industry 4.0 and provides examples of engineering methods and tools that are currently available. The paper then focuses on the CPS engineering toolset being developed by the Automation Systems Group (ASG) in the Warwick Manufacturing Group (WMG), University of Warwick, Coventry, U.K. and explains via an industrial case study how such a component-based engineering toolset can support an integrated approach to the virtual and physical engineering of automation systems through their lifecycle via a method that enables multiple vendors' equipment to be effectively integrated and provides support for the specification, validation, and use of such systems across the supply chain, e.g., between end users and system integrators.

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