[1] J. Garnier, Stochastic invariant imbedding. Application to stochastic differential equations with boundary conditions, Prob. Th. Rel. Fields 103 (1995), pp. 249-271.
Abstract: We study stochastic differential equations of the type:

[2] J. Garnier, Homogenization in a periodic and time-dependent potential, SIAM, J. Appl. Math. 57 (1997), pp. 95-111.
Abstract: This paper contains a study of the long time behaviour of a diffusion process in a periodic potential. The first goal is to determine a suitable rescaling of time and space so that the diffusion process converges to some homogeneous limit. If the potential depends on time, then the usual diffusive scaling may not be the right one. Namely some drift may appear with the result that the asymptotic behaviour of the process is superdiffusive. This is applied to the homogenization of parabolic differential equations.

[3] J. Garnier, A multi-scaled diffusion-approximation theorem. Applications to wave propagation in random media, ESAIM Probab. Statist. 1 (1997), pp. 183-206, available on http://www.emath.fr/ps/.
Abstract: In this paper a multi-scaled diffusion-approximation theorem is proved so as to unify various applications in wave propagation in random media: transmission of optical modes through random planar waveguides; time delay in scattering for the linear wave equation; decay of the transmission coefficient for large lengths with fixed output and phase difference in weakly nonlinear random media.

[4] J. Garnier, L. Videau, C. Gouédard, and A. Migus, Statistical analysis of beam smoothing and some applications, J. Opt. Soc. Am. A 14 No 8 (1997), pp. 1928-1937.
Abstract:This paper aims at developing statistical tools for beam smoothing analysis. As applications we study the respective performances of two-dimensional smoothing by spectral dispersion and smoothing by optical fiber. The calculations are valid in the asymptotic framework of a large number of elements of the random phase plate and of excited optical modes of the fiber. Theoretical results and closed form expressions for the contrast and spatial spectrum of the integrated intensity of the speckle pattern are derived so as to put into evidence performance differences between these methods, which are essentially based on the longer time delay induced by the multimode fiber with respect to the one induced by the gratings and on the nature of the spectral broadening.

[5] J. Garnier, J.-P. Fouque, L. Videau, C. Gouédard, and A. Migus, Amplification of broadband incoherent light in homogeneously broadened media in presence of Kerr nonlinearity, J. Opt. Soc. Am. B 14 No 10 (1997), pp. 2563-2569.
Abstract:We have developed a statistical nonlinear model in order to explain an anomalous intensity saturation observed in the amplification of intense broadband incoherent pulses on neodynium-doped glass power chains. The physics behind this model is basically self-phase modulation creating new wavelengths scattered in the tail of the gain profile. The theory shows qualitative agreement with the experimental results.

[6] J. Garnier, Asymptotic transmission of solitons through random media, SIAM, J. Appl. Math. 58 (1998), pp. 1969-1995.
Abstract: This paper contains a study of the transmission of a soliton through a slab of nonlinear and random medium. A random nonlinear Schrodinger equation is considered, where the randomness holds in the potential and the nonlinear coefficient. Using the inverse scattering transform, we exhibit several asymptotic behaviors corresponding to the limit when the amplitudes of the random fluctuations go to zero and the size of the slab goes to infinity. The mass of the transmitted soliton may tend to zero exponentially (as a function of the size of the slab) or following a power law; or else the soliton may keep its mass, while its velocity decreases at a logarithmic rate or even slower. Numerical simulations are in good agreement with the theoretical results. Ref. [41] is a long version of this paper.

[7] J. Garnier, L. Videau, C. Gouédard, and A. Migus, Propagation and amplification of incoherent pulses in dispersive and nonlinear media, J. Opt. Soc. Am. B 15 (1998), pp. 2773-2781.
Abstract: A statistical model is developed so as to study all the relevant phenomena which can give rise to an anomalous intensity saturation in the propagation of incoherent pulses in a laser amplifier. The interplay between diffraction, self-focusing, group velocity dispersion, gain narrowing, and gain saturation is investigated. Changes in the temporal and spatial characteristics of the pulses are shown.

[8] J. Garnier, Asymptotic behavior of the quantum harmonic oscillator driven by a random time-dependent electric field, J. Statist. Phys. 93 (1998), pp. 211-241.
Abstract: This paper investigates the evolution of the state vector of a charged quantum particle in a harmonic oscillator driven by a time-dependent electric field. The external field randomly oscillates and its amplitude is small, but it acts long enough so that we can solve the problem in the asymptotic framework corresponding to a field amplitude which tends to zero and a field duration which tends to infinity. We describe the effective evolution equation of the state vector which reads as a stochastic partial differential equation. We explicitly describe the transition probabilities which are characterized by a polynomial decay of the probabilities corresponding to the low-energy eigenstates and give the exact statistical distribution of the energy of the particle.

[9] J. Garnier, L. Kallel, and M. Schoenauer, Rigorous hitting times for binary mutations, Evolutionary Computation 7 (1999), pp. 173-203 (electronic version unavailable due to copyright transfert).
Abstract: In the binary evolutionary optimization framework, two mutation operators are theoretically investigated. For both the standard mutation, in which all bits are flipped independently with the same probability, and the 1-bit-flip mutation, which flips exactly one bit per bitstring, the statistical distribution of the first hitting times of the target are thoroughly computed (expectation and variance) up to terms of order l (the size of the bitstrings) in two distinct situations: without any selection, or with the deterministic (1+1)-ES selection on the OneMax problem. In both cases, the 1-bit-flip mutation convergence time is smaller by a constant (in terms of l) multiplicative factor. These results extend to the case of multiple independent optimizers.

[10] J. Garnier, Statistics of the hot spots of smoothed beams produced by random phase plates revisited, Phys. Plasmas 6 (1999), pp. 1601-1610.
Abstract: This paper revisits and corrects the statistical theory of hot spots of speckle patterns such as those produced by a random phase plate. Analytical expressions are derived which are sensitively different from the previous results of Rose and DuBois (Phys. Fluids B 5, 590 (1993)). The departure essentially originates from a careful approach which takes into account the fact that the fields are complex-valued, while the standard mathematical theory deals with the maxima of real-valued Gaussian fields. This gives rise to an enhancement of the number of the most intense hot spots. Excellent agreements between the theoretical formulae and numerical simulations are shown.

[11] J. Garnier, C. Gouédard, and A. Migus, Statistics of the hottest spot of speckle patterns generated by smoothing techniques, Journal of Modern Optics 46 (1999), pp. 1213-1232.
Abstract: This paper is concerned with the statistical distribution of the maximal hot spots of speckle patterns such as those generated by optical smoothing methods designed for inertial confinement fusion. It is proved that the maximal intensity at the first order is proportional to the logarithm of the ratio of the pulse volume over the mean hot spot volume. Nevertheless the complete description of the maximal intensity exhibits a quite important variance. Different ways for reducing either the maximal fluence or the maximal intensity are investigated, which are based upon time incoherence or polarization smoothing.

[12] L. Videau, C. Rouyer, J. Garnier, and A. Migus, The motion of hot spots in smoothed beams, J. Opt. Soc. Am. A 16 (1999), pp. 1672-1681.
Abstract: We develop a statistical model which describes the motion of a hot spot created by smoothing techniques. We define properly the transverse and longitudinal instantaneous velocities of a hot spot and quantify its life time. This relevant parameter is found to be longer than the laser coherence time defined as the inverse of the spectrum bandwidth. We apply this model to the most usual smoothing techniques, using a sinusoidal phase modulation or a random spectrum. In case of the one-dimensional Smoothing by Spectral Dispersion, the Smoothing by Longitudinal Spectral Dispersion and the Smoothing by Optical Fiber, we give asymptotic results for hot spot velocities and life time.

[13] F. Kh. Abdullaev and J. Garnier, Modulational instability in birefringent fibers with periodic and random dispersion, Phys. Rev. E. 60 (1999), pp. 1042-1050.
Abstract: Modulational instability (MI) of electromagnetic waves in a birefringent fiber with a periodic dispersion (two-step dispersion management scheme) is investigated. The properties of new sidebands are studied. The strong variation of dispersion leads to the decreasing of the main MI region and suppression of additional resonance. In the random dispersion case the MI of all frequencies of modulation in the normal dispersion region is predicted. In the anomalous dispersion case the decreasing of the main MI peak is calculated and changes in the spectral bandwidth of MI gain are found. The analytical predictions are confirmed by the numerical simulations of the full coupled nonlinear Schrodinger equations with periodic coefficients.

[14] F. Kh. Abdullaev and J. Garnier, Solitons in media with random dispersive perturbations, Phys. D 134 (1999), pp. 303-315.
Abstract: A statistical approach of the propagation of solitons in media with spatially random dispersive perturbations is developed. Applying the inverse scattering transform several regimes are put into evidence which are determined by the mass and the velocity of the incoming soliton and also by the correlation length of the perturbation. Namely, the mass of the soliton is almost conserved if it is initially large. If the initial mass is too small, then the mass decays with the length of the system. The decay rate is exponential in case of a white noise perturbation, but the mass will decrease as the inverse of the square root of the length if the central wavenumber of the soliton lies in the tail of the spectrum of the perturbation.

[15] J. Garnier, Energy distribution of the quantum harmonic oscillator under random time-dependent perturbations, Phys. Rev. E 60 (1999), pp. 3676-3687.
Abstract: This paper investigates the evolution of a quantum particle in a harmonic oscillator driven by time-dependent forces. The perturbations are small, but they act long enough so that we can solve the problem in the asymptotic framework corresponding to a perturbation amplitude which tends to zero and a perturbation duration which tends to infinity. We describe the effective evolution equation of the state vector which reads as a stochastic partial differential equation. We exhibit a closed-form equation for the transition probabilities, which can be interpreted in terms of a jump process. Using standard probability tools, we are then able to compute explicitly the probabilities for observing the different energy eigenstates and give the exact statistical distribution of the energy of the particle.

[16] J. Garnier, Light propagation in square law media with random imperfections, Wave Motion 31 (2000), pp 1-19.
Abstract: This paper investigates the deformation of the wavefield transmitted through a square law medium waveguide. We consider the situation where the center of the waveguide randomly oscillates around the optical axis or the radius of the waveguide randomly pulsates. The random perturbations are small, but the waveguide is long, which gives rise to a macroscopic effect of the inhomogeneities. This effect is characterized by coupling mechanisms between optical modes, which tend to strengthen high order modes. Precise expressions for the transmitted wave are derived which exhibits some remarkable regimes, where unexpected behaviors such as shift, spreading or even focusing of the wavefield can be observed. Numerical simulations are in good agreement with the theoretical results.

[17] J. Garnier, C. Gouédard, and L. Videau, Propagation of a partially coherent beam under the interaction of small and large scales, Opt. Commun. 176 (2000), pp. 281-297.
Abstract: This paper deals with the propagation of Schell-model sources. Two different and complementary approaches are developed. The first one is standard and based on the study of the Wigner distribution function. The second one follows from a generic statistical representation of the speckle pattern as the superposition of elementary and independent modes. Precise results are obtained for the macroscopic and microscopic characteristics of the beam: optical intensity profile, Rayleigh distance, speckle radius and intensity profiles of the speckle spots. These results are finally applied to the determination of the main characteristics of the focal spot generated by a Kinoform Phase Plate. We also give the complete expressions of the above quantities when the conditions of paraxial approximation are not fulfilled.

[18] L. Videau, C. Rouyer, J. Garnier, and A. Migus, Generation of a pure phase modulated pulse by cascading effect. A theoretical approach, J. Opt. Soc. Am. B 17 (2000), pp. 1008-1017.
Abstract: New techniques to produce a spatio-temporal phase modulation without using electro-optic devices are proposed and discussed. By using nonlinear second order effect in crystal, it is possible to transfer amplitude modulations of a pump wave to the phase of a signal wave. For that, we propose the use of a well-known cascading configuration for which the phase mismatch is high. Analytical results for spatial and/or temporal incoherent phase modulation are developed with the correlation functions formalism. Furthermore highly accurate expansions of the signal phase and intensity are derived. The effects of the group velocity difference, the group velocity dispersion and the diffraction on the transfer of amplitude to phase modulation are studied. Finally an experimental demonstration into a KDP crystal with a sinusoidal pump modulation that creates sinusoidal phase modulation is proposed.

[19] J. Garnier and L. Kallel, Statistical distribution of the convergence time of evolutionary algorithms for longpath problems, IEEE Transactions on Evolutionary Computation 4 (2000), pp. 16-30.
Abstract: The asymptotical behavior of a (1+1)-ES process on Rudolph's long k-paths is investigated extensively in this paper. First, in the case of , we prove that the long k-path is a longpath for the (1+1)-ES, in the sense that the entire path has to be followed before convergence occurs. For , expected convergence time is still exponential but some shortcuts will occur meanwhile which speeds up the process.
Second, in the case of constant k, the statistical distribution of convergence time is calculated, and the influence of population size is investigated for different . Besides, the histogram of the first hitting time of the solution shows an anomalous peak close to zero, which corresponds to an exceptional set of events that speed up the expected convergence time with a factor of l². A direct consequence of this exceptional set is that performing independent (1+1)-ES processes proves to be more advantageous than any population based .

[20] J. Garnier and F. Kh. Abdullaev, Modulational instability induced by randomly varying coefficients for the nonlinear Schrodinger equation, Phys. D 145 (2000), pp. 65-83.
Abstract: We introduce the theory of modulational instability (MI) of electromagnetic waves in optical fibers. The model at hand is the one-dimensional nonlinear Schrodinger equation with random group velocity dispersion and random nonlinear coefficient. We compute the MI gain which reads as the Lyapunov exponent of a random linear system. The sample and moment MI gains appear to be very different. In the anomalous dispersion regime, random fluctuations of the nonlinear coefficient reduces the sample MI gain peak, although the moment MI peak is enhanced, and the unstable bandwidth is widened. Still in the anomalous dispersion regime, random fluctuations of the group velocity dispersion reduces both the sample MI gain peakand the moment MI peak. Finally, in the normal dispersion regime, randomness extends the MI domain to the whole spectrum of modulations, and increases the MI gain peak. The linear stability analysis is confirmed by numerical simulations of the full stochastic nonlinear Schrodinger equation.

[21] J. Garnier, Propagation of solitons in a randomly perturbed Ablowitz-Ladik chain, Phys. Rev. E 63 (2001), 026608.
Abstract: This paper deals with the transmission of a soliton in a discrete, nonlinear and random medium. A random lattice nonlinear Schrödinger equation is considered, where the randomness holds in the on-site potential or in the coupling coefficients. We study the interplay of nonlinearity, randomness and discreteness. We derive effective evolution equations for the soliton parameters by applying a perturbation theory of the inverse scattering transform and limit theorems of stochastic calculus.

[22] J. Garnier, High-frequency asymptotics for Maxwell's equations in anisotropic media. Part I: Linear geometric and diffractive optics, J. Math. Phys. 42 (2001), pp. 1612-1635.
Abstract: This paper is devoted to the derivations of the equations that govern the propagation of pulses in noncentrosymmetric crystals. The method is based upon high-frequency expansions techniques for Maxwell equations. By suitable choices of the scalings we are able to derive two classical models: geometric optics and diffractive optics (Schrodinger-like equations). In the so-called geometric regime we recover the standard results on the propagation of pulses in crystals (dispersion equation, polarization states, group velocity). In the diffractive regime we exhibit original results and give a closed-form expression for the diffraction operator which reads as an anisotropic operator. Given this expression we identify a critical configuration where the diffraction reduces to a one-dimensional second-order operator instead of the standard transverse Laplacian.

[23] J. Garnier, High-frequency asymptotics for Maxwell's equations in anisotropic media. Part II: Nonlinear propagation and frequency conversion, J. Math. Phys. 42 (2001), pp. 1636-1654.
Abstract: This paper is devoted to the derivations of the equations that govern the propagation and frequency conversion of pulses in noncentrosymmetric crystals. The method is based upon high frequency expansions techniques for hyperbolic quasi-linear and semi-linear equations. In the so-called geometric regime we recover the standard results on the frequency conversion of pulses in nonlinear crystals. In the diffractive regime we show that the anisotropy of the diffraction operator involves remarkable phenomena. In particular the phase matching angle of a divergent pulse depends on the distance between the waist and the crystal plate. Finally we detect a configuration where the beam propagation in a biaxial crystal involves the generation of spatial solitons thanks to an anomalous one-dimensional diffraction.

[24] J. Garnier, F. Kh. Abdullaev, E. Seve, and S. Wabnitz, Role of polarization mode dispersion on modulational instability in optical fibers, Phys. Rev. E 63 (2001), 066616.
Abstract: We introduce the theory of modulational instability (MI) of electromagnetic waves in fibers with random polarization mode dispersion. Applying a linear stability analysis and stochastic calculus we show that the MI gain spectrum reads as the maximal eigenvalue of a constant effective matrix. In the limit of small or large fluctuations, we give explicit expressions for the MI gain spectra. In the general configurations we give the explicit form of the effective matrix and compute numerically the maximal eigenvalue. In the anomalous dispersion regime, polarization dispersion widens the unstable bandwidth. Depending on the type of variations of the birefringence parameters, polarization dispersion reduces or enhances the MI gain peak. In the normal dispersion regime, random effects may extend the instability domain to the whole spectrum of modulations. The linear stability analysis is confirmed by numerical simulation of the full stochastic coupled nonlinear Schrödinger equations.

[25] J. Garnier, Solitons in random media with long-range correlation, Waves Random Media 11 (2001), pp. 149-162.
Abstract: A statistical approach of the propagation of solitons in media with spatially random potential is developed. Applying the inverse scattering transform several regimes are put into evidence which are determined by the mass and the velocity of the incoming soliton as well as by the correlation length of the random potential. Namely, the mass of the soliton is conserved if its initial amplitude is large enough. If the initial mass is small, then the mass decays with the length of the system. The decay rate is exponential in case of a white noise perturbation, but it obeys a power law if the carrier wavenumber of the soliton lies in the tail of the spectrum of the potential. Furthermore, the scattered radiation propagates in backward direction in case of a white noise perturbation, while it propagates in forward direction (with the same carrier wavenumber as the soliton) in case of a colored noise with long range correlation.

[26] M.-O. Bernard, J. Garnier, and J.-F. Gouyet, Laplacian growth of parallel needles. A Fokker-Planck equation approach, Phys. Rev. E 64 (2001) 041401.
Abstract: Using a conformal transformation to set up the iterative nonlinear equations, we study analytically the kinetics of growth of parallel needles. We establish a discrete Fokker-Planck equation for the probability of finding at time t a given distribution of needle lengths. In the linear regime, it shows a short-wavelength Laplacian instability which we investigate in detail. From the crossover of the solutions to the nonlinear regime, we deduce analytically the general scale invariance of the two-dimensional models.

[27] J. Garnier and L. Videau, Statistical analysis of the sizes and velocities of laser hot spots of smoothed beams, Phys. Plasmas 8 (2001), pp. 4914-4924.
Abstract: This paper presents a precise description of the characteristics of the hot spots of a partially coherent pulse. The average values of the sizes and velocities of the hot spots are computed, as well as the corresponding probability density functions. Applications to the speckle patterns generated by optical smoothing techniques for uniform irradiation in plasma physics are discussed.

[28] J. Garnier, Long-time dynamics of Korteweg-de Vries solitons driven by random perturbations, J. Statist. Phys 105 (2001), pp. 789-833.
Abstract: This paper deals with the transmission of a soliton in a random medium described by a randomly perturbed Korteweg-de Vries equation. Different kinds of perturbations are addressed, depending on their specific time or position dependences, with or without damping. We derive effective evolution equations for the soliton parameter by applying a perturbation theory of the inverse scattering transform and limit theorems of stochastic calculus. Original results are derived that are very different compared to a randomly perturbed Nonlinear Schrödinger equation. First the emission of a soliton gas is proved to be a very general feature. Second some perturbations are shown to involve a speeding-up of the soliton, instead of the decay that is usually observed in random media.

[29] J. Garnier and J.-P. Fouque, Amplification of incoherent light with wide spectrum, Proceedings of the Third International Conference on Mathematical and Numeriacl Aspects of Wave Propagation Phenomena, G. Cohen, ed., SIAM-INRIA, 1995, pp. 584-593.
Abstract: We consider a large-band incoherent pulse and its propagation in an amplifier. We show how the intensity grows and how the correlation function behaves in the medium. We see how a small nonlinearity may greatly affect the amplification.

[30] J.-P. Fouque and J. Garnier, On waves in random media in the diffusion-approximation regime, Proceedings of the meeting Waves in Random and other Complex Media, R. Burridge, G. Papanicolaou, and L. Pastur, eds., IMA Vol. 96, Springer Verlag, New York, 1997, pp. 31-48.
Abstract: The aim of this contribution is to present recent results obtained at the "Centre de Mathématiques Appliquées de l'Ecole Polytechnique" by the group working on waves in random media (F. Bailly, J. Chillan, J.F. Clouet, J.P. Fouque and J. Garnier). These results are based on various generalizations of classical diffusion-approximation results. In the first section we study the spreading of an acoustic pulse travelling through a randomly layered medium In the second section we present a justification of the parabolic and white noise approximation for waves in random media in the high frequency regime leading to a stochastic Schrodinger equation The third section is devoted to the effect of a weak nonlinearity on a wave equation with a random potential. In the last section we study the amplification of an incoherent optical pulse propagating in a nonlinear Kerr medium.

[31] J. Garnier, L. Videau, C. Gouédard, and A. Migus, Which optical smoothing for LMJ and NIF ?, Proceedings of the meeting Solid state lasers for applications to ICF 1996, M. André and H.T. Powell, eds., SPIE, Vol. 3047, 1997, pp. 260-271.
Abstract: This paper uses statistical theory to investigate the respective performances of two-dimensional smoothing by spectral dispersion and smoothing by optical fiber, both techniques being proposed and implemented for uniform irradiation in plasma physics. The calculations are valid in the asymptotic framework of a large number of elements of the random phase plate or the excited optical modes of the fiber. Theoretical results and closed-form expressions for the contrast and spatial spectrum of the integrated intensity of the speckle pattern are derived so as to put into evidence performance differences between these methods. These differences essentially originate from the much longer time delay induced by the multimode fiber with respect to the one induced by the gratings and from the interplay between the nature of the delay line vs. the nature of the spectral broadening.

[32] L. Videau, A. Boscheron, J. Garnier, C. Gouédard, C. Feral, M. Laurent, J. Paye, C. Sauteret, and A. Migus, Recent results of optical smoothing on the Phebus Laser, Proceedings of the meeting Solid state lasers for applications to ICF 1996, M. André and H.T. Powell, eds., SPIE, Vol. 3047, 1997, pp. 757-762.
Abstract:

[33] L. Videau, J. Garnier, C. Feral, C. Gouédard, C. Sauteret, and A. Migus, Spectral broadening and nonlinear limitation of partially incoherent pulses in high power amplifiers, Proceedings of the meeting CLEO'97, OSA Technical Digest Series, Vol. 11, 1997, pp. 353-354.
Abstract: Spatial and temporal incoherent pulses have been amplified in a high power glass laser up to 1.5 kJ. Performance has been studied as a function of initial bandwidth and energy input, and are compared to a statistical model of amplification.

[34] L. Videau, E. Bar, C. Rouyer, C. Gouédard, J. Garnier, and A. Migus, Control of the amplification of large band amplitude modulated pulses in Nd-glass amplifier chain, Proceedings of the meeting Solid state lasers for applications to ICF 1998, W. Howard Lowdermilk, ed., SPIE, Vol. 3492, 1999, pp. ???-???.
Abstract: The development of the coming generation of the Megajoule-class laser requires optical smoothing to obtain a focal spot large enough with a good uniformity. Different optical smoothing techniques have been proposed and experimented, such as Smoothing by Optical Fiber (SOF) and Smoothing by Spectral Dispersion (SSD). SOF seems to be an efficient method but experiments have shown a limitation in term of the amplification performance. In this paper we present recent results obtained on the Phebus facility and compare them with a theoretical model which takes into account the interaction between time-space incoherence and nonlinear effects. We also discuss different techniques that can be applied to circumvent the anomalous intensity saturation.

[35] L. Videau, J. Garnier, C. Rouyer, and A. Migus, Speckle movement description in case of 1D-SSD and longitudinal-SSD for a temporal sinusoidal phase modulation, Proceedings of the meeting Solid state lasers for applications to ICF 1998, W. Howard Lowdermilk, ed., SPIE, Vol. 3492, 1999, pp. 277-284.
Abstract: Smoothing techniques are important for Ignition Confinement Fusion in order to reduce instabilities in the plasma interaction. The future ICF configurations (French LMJ and US NIF) are designed for the indirect drive scheme so that high laser intensitites are likely to induce parametric instabilities in the extended window and in the hohlraum gas. A lot of work have been concerned with the effects of smoothing techniques for reducing parametric instabilities. Very often theoretical papers consider speckle patterns as a collection of hot spots moving in the forward direction. We have developed a statistical formalism which is based on the study of the time-space autocorrelation function of the field. We are then able to compute the motions of the hot spots. We apply this method in different 1D-types of SSD techniques with sinusoidal phase modulation. Results show that the motions may be backward and/or with speed larger than light velocity and that the hot spot lifetime may be longer than the coherence time of the laser. The relevant parameters are the modulation frequency f and depth b. For a given spectrum (equal to the product f b ) different speeds and lifetimes are possible, so the choice of the couple (f,b) is crucial for reducing the interaction length between the laser and parametric instabilitites.

[36] F. Abdullaev and J. Garnier, Modulational instability in birefringent fibers with strong dispersion management, to appear in the proceedings of the conference SCT'99 (Solitons, Collapses, and Turbulence, Chernogolovka, Russia, 1999).
Abstract: Strong dispersion management in birefringent fibers with periodic dispersion is shown to reduce modulational instability domain and suppress additional resonances.

[37] F. Abdullaev, J. Garnier, E. Seve, and S. Wabnitz, Modulational instability in optical fibers with polarization mode dispersion, proceedings of the conference NLGW'99 (Nonlinear Guided Waves, Dijon, 1999), OSA Technical Digest Series, paper WB3.
Abstract: Random polarization mode dispersion leads to a substantial extension of the modulational instability domain in both the normal and anomalous dispersion regime of fibers.

[38] J. Garnier and L. Kallel, How to detect all maxima of a function ?, proceedings of the Second EVONET Summer School on Theoretical Aspects of Evolutionary Computing (Anvers, 1999), Springer, Berlin, 2001, pp. 343-370.
Abstract: This paper provides a new methodology allowing one to estimate the number and the sizes of the attraction basins of a landscape specified in relation to some modification operator.

[39] J. Garnier and F. Kh. Abdullaev, Long-range transmission of solitons in random media, proceedings of the conference Photonics West (San José, 2001), SPIE Proceedings Series, Vol. 4271 (2001), pp. 32-42.
Abstract: A statistical approach of the propagation of solitons in media with spatially random perturbations is developed. Applying the inverse scattering transform several regimes are put into evidence which are determined by the mass and the velocity of the incoming soliton and also by the correlation length of the perturbation. The mass of the transmitted soliton may tend to zero exponentially (as a function of the size of the slab) or following a power law; or else the soliton may keep its mass if it is initially large enough, while its velocity decreases at a logarithmic rate or even slower. Numerical simulations are in good agreement with the theoretical results.

[40] J. Garnier, Some applications of the anisotropic diffraction in biaxial crystals, proceedings of the conference Photonics West (San José, 2001), SPIE Proceedings Series, Vol. 4271 (2001), pp. 138-149.
Abstract: We analyze the propagation of pulses in noncentrosymmetric crystals by applying high-frequency expansions techniques for Maxwell equations. As a first application we give a closed-form expression for the anisotropic diffraction operator. Given this expression we identify a critical configuration in biaxial crystals where the diffraction reduces to a one-dimensional second-order operator for the ordinary wave instead of the standard transverse Laplacian. The beam propagation in such a configuration involves the generation of spatial solitons because of this anomalous one-dimensional diffraction. As a second application we present closed-form formulas for the interference patterns from biaxial crystal plates between two polarizers. These formulas agree with experimental patterns.

[41] J. Garnier, Theoretical and numerical study of solitons in random media, technical report, long version of [6].
Abstract: This paper contains a study of the transmission of a soliton through a slab of nonlinear and random medium. A random nonlinear Schrodinger equation is considered, where the randomness holds in the potential and the nonlinear coefficient. Using the inverse scattering transform, we exhibit several asymptotic behaviors corresponding to the limit when the amplitudes of the random fluctuations go to zero and the size of the slab goes to infinity. The mass of the transmitted soliton may tend to zero exponentially (as a function of the size of the slab) or following a power law; or else the soliton may keep its mass, while its velocity decreases at a logarithmic rate or even slower. Numerical simulations are in good agreement with the theoretical results. A short version of this paper has been published in SIAM, Journal on Applied Mathematics [6]. We present here more numerical simulations and give the proofs of all the technical estimates.

[42] J. Garnier, Approche probabiliste du séquencage à grande échelle, technical report (in French).
Abstract: Cette première approche a pour but de présenter le type de résultats qu'on peut obtenir par une approche probabiliste du problème, qui repose essentiellement sur une analyse statistique dans le cadre asymptotique où la longueur de génome est nettement plus grande que toutes les autres échelles caractéristiques. Afin de simplifier cette étude préliminaire, on a négligé toutes les sources d'erreurs. On propose donc dans ce document des résultats théoriques qui décrivent les distributions statistiques des quantités intéressantes : les nombres et tailles des trous, des contigs de séquences et de clones, la longueur du plus grand contig de clones et plus particulièrement l'"instant" où le contig maximal recouvre entièrement le génome. On va commencer par examiner dans la Section 1 la stratégie de séquençage la plus simple, à savoir le shot gun simple. On présentera ensuite dans la Section 2 des résultats encore partiels sur le séquençage par paires d'extrémités.
Avertissement : Ce rapport est plus un document de travail utile pour la suite qu'un objet fini. C'est pourquoi la présentation est encore un peu sèche.

[43] J. Garnier and L. Kallel, Efficiency of local search with multiple local optima, submitted to SIAM Journal on Discrete Mathematics.
Abstract: The first contribution of this paper is a theoretical investigation of combinatorial optimization problems. Their landscapes are specified by the set of neighborhoods of all points of the search space. The aim of the paper consists in the estimation of the number N of local optima and the distributions of the sizes of their attraction basins. For different types of landscapes we give precise estimates of the size of the random sample that ensures that at least one point lies in each attraction basin.
A practical methodology is then proposed for identifying these quantities (N and distributions) for an unknown landscape, given a random sample of starting points and a local steepest ascent search. This methodology can be applied to any landscape specified with a modification operator and provides bounds on search complexity to detect all local optima. Experiments demonstrate the efficiency of this methodology for guiding the choice of modification operators, eventually leading to the design of problem-dependent optimization heuristics.

[44] J. Garnier, Instability of a quantum particle induced by a randomly varying spring coefficient, to appear in the proceedings of the conference Ascona'99.
Abstract: This paper investigates the evolution of a quantum particle in a harmonic oscillator whose spring coefficient randomly fluctuates around its mean value. The perturbations are small, but they act long enough so that we can solve the problem in the asymptotic framework corresponding to a perturbation amplitude which tends to zero and a perturbation duration which tends to infinity. We describe the effective evolution equation of the state vector which reads as a stochastic partial differential equation. We exhibit a closed-form equation for the transition probabilities, which can be interpreted in terms of a jump process. Using standard probability tools, we are then able to compute explicitly the probabilities for observing the different energy eigenstates and give the exact statistical distribution of the energy of the particle.

[45] J. Garnier and L. Kallel, Optimization of binary mutations for evolutionary algorithms, submitted to Evolutionary Computation.
Abstract: This paper deals with theoretical parameter design for Evolutionary Algorithms, in the case of the mutation parameter. First, we prove the explicit relationship between the convergence time of an Evolutionary Algorithm and its mutation operator, in the case of the Onemax problem and for all possible memoryless binary mutations. Second, this relationship is used for deriving the optimal mutation strategy that minimizes the convergence time of the algorithm towards the optimum. We address both the static and time dependent mutation cases, and we show that the optimal mutation strategy does not depend on the string size.

[46] J. Garnier, Exponential localization versus soliton propagation, to appear in the NATO ARW proceedings "Nonlinearity and disorder".
Abstract: The scattering of a wavepacket by a random nonlinear medium is analyzed. In the linear limit strong localization occurs, which means that the transmission coefficient decays exponentially with a characteristic localization length. In some nonlinear homogeneous media solitons propagate without changes in their shape or velocity. Solitons are therefore candidates to test the robustness of the exponential localization in random nonlinear media. Using the inverse scattering transform for the nonlinear Schrödinger equation different typical behaviors can be exhibited depending on the amplitude of the incoming soliton.