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2016

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In this paper, we consider the nonlinear Schrödinger equation ut + i∆u − F (u) = 0 in two dimensions. We show, by an operator-theoretic proof, that the well-known Lie and Strang formulae (which are splitting methods) are approximations of the exact solution of order 1 and 2 in time.

This article presents the derivation of a semi-classical model of electromagnetic-wave propagation in a non centro-symmetric crystal. It consists of Maxwell's equations for the wave field coupled with a version of Bloch's equations which takes fully into account the discrete symmetry group of the crystal. The model is specialized in the case of a KDP… (More)

We propose a FIR filtering technique which takes advantage of the possibility of using a very low number of samples for both the signal and the filter transfer function thanks to non-uniform sampling. This approach leads to a summation formula which plays the role of the discrete convolution for usual FIR filters. Here the formula is much more complicated… (More)

This article describes a new kind of processing chain based on a non-uniform sampling scheme provided by a level-crossing ADC. The chain implements IIR filters which process directly the non-uniform samples without resampling in a regular scheme. The non-uniformity in the sample times leads to choose a state representation for the filters. The stability is… (More)

We consider Bloch equations which govern the evolution of the density matrix of an atom (or: a quantum system) with a discrete set of energy levels. The system is forced by a time dependent electric potential which varies on a fast scale and we address the long time evolution of the system. We show that the diagonal part of the density matrix is… (More)

Bloch equations give a quantum description of the coupling between atoms and a driving electric force. It is commonly used in optics to describe the interaction of a laser beam with a sample of atoms. In this article, we address the asymptotics of these equations for a high frequency electric field, in a weak coupling regime. The electric forcing is taken… (More)

The stability of five finite difference–time domain (FD–TD) schemes coupling Maxwell equations to Debye or Lorentz models has been analyzed in [P. Petropoulos, Stability and phase error analysis of FD–TD in dispersive dielectrics, IEEE Trans. where numerical evidence for specific media have been used. We use von Neumann analysis to give necessary and… (More)

We investigate the dynamics of a chain of oscillators coupled by fully-nonlinear interaction potentials. This class of models includes Newton's cradle with Hertzian contact interactions between neighbors. By means of multiple-scale analysis, we give a rigorous asymptotic description of small amplitude solutions over large times. The envelope equation… (More)

We propose a filtering technique which takes advantage of a specific non-uniform sampling scheme which allows the capture of a very low number of samples for both the signal and the filter transfer function. This approach leads to a summation formula which plays the same role as the discrete convolution for usual FIR filters. Here the formula is much more… (More)