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- Christophe Besse, Brigitte Bidégaray-Fesquet, Stéphane Descombes
- SIAM J. Numerical Analysis
- 2002

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.

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)

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)

Bloch equations give a quantum description of the coupling between an atom and a driving electric force. In this article, we address the asymptotics of these equations for high frequency electric fields, in a weakly coupled regime. We prove the convergence towards rate equations (i.e. linear Boltzmann equations, describing the transitions between energy… (More)

- Laurent Fesquet, Brigitte Bidégaray-Fesquet
- Signal Processing
- 2010

This article describes a new kind of processing chain based on a nonuniform 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)

- Brigitte Bidégaray-Fesquet, Éric Dumas, Guillaume James
- SIAM J. Math. Analysis
- 2013

We investigate the dynamics of a chain of oscillators coupled by fullynonlinear 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 leading… (More)

- Brigitte Bidégaray-Fesquet
- SIAM J. Numerical Analysis
- 2008

The stability of five finite difference–time domain (FD–TD) schemes coupling Maxwell equations to Debye or Lorentz models have been analyzed in [1], where numerical evidence for specific media have been used. We use von Neumann analysis to give necessary and sufficient stability conditions for these schemes for any medium, in accordance with the partial… (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)