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 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 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)
The aim of this paper is to derive a raw Bloch model for the interaction of light with quantum boxes in the framework of a two-electron-species (conduction and valence) description. This requires a good understanding of the one-species case and of the treatment of level degeneracy. In contrast with some existing literature we obtain a Li-ouville equation… (More)
We study the Cauchy problem for two systems of equations (Maxwell-Debye and Maxwell-Bloch) describing laser-matter interaction phenomena. We show that these problems are locally in time well-posed for initial data in different Sobolev spaces. In the case of Maxwell-Debye system, which contains some delay term, we study the limit of the solutions when this… (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)
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 propose a 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. Following some previous work where we sampled non-uniformly already existing filter transfer functions, we propose now to design directly new filters in the… (More)