A formal approach for correct-by-construction system substitution

The substitution of a system with another one may occur in several situations like system adaptation, system failure management, system resilience, system reconfiguration, etc. It consists in replacing a running system by another one when given conditions hold. This contribution summarizes our proposal to define a formal setting for proving the correctness of system substitution. It relies on refinement and on the Event-B method.Comment: EDCC-2014, Student-Forum, System Substitution, state rRecovery, correct-bycorrection, Event-B, refinemen

Nonlinear Photonic Crystals: IV. Nonlinear Schrodinger Equation Regime

We study here the nonlinear Schrodinger Equation (NLS) as the first term in a sequence of approximations for an electromagnetic (EM) wave propagating according to the nonlinear Maxwell equations (NLM). The dielectric medium is assumed to be periodic, with a cubic nonlinearity, and with its linear background possessing inversion symmetric dispersion relations. The medium is excited by a current $\mathbf{J}$ producing an EM wave. The wave nonlinear evolution is analyzed based on the modal decomposition and an expansion of the exact solution to the NLM into an asymptotic series with respect to some three small parameters $\alpha$, $\beta$ and $\varrho$. These parameters are introduced through the excitation current $\mathbf{J}$ to scale respectively (i) its amplitude and consequently the magnitude of the nonlinearity; (ii) the range of wavevectors involved in its modal composition, with $\beta ^{-1}$ scaling its spatial extension; (iii) its frequency bandwidth, with $\varrho ^{-1}$ scaling its time extension. We develop a consistent theory of approximations of increasing accuracy for the NLM with its first term governed by the NLS. We show that such NLS regime is the medium response to an almost monochromatic excitation current $\mathbf{J}$. The developed approach not only provides rigorous estimates of the approximation accuracy of the NLM with the NLS in terms of powers of $\alpha$, $\beta$ and $\varrho$, but it also produces new extended NLS (ENLS) equations providing better approximations. Remarkably, quantitative estimates show that properly tailored ENLS can significantly improve the approximation accuracy of the NLM compare with the classical NLS