Abstract: We study stochastic differential equations of the type:

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 .

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Abstract: The propagation of dispersion-managed solitons in optical fibers with randomly perturbed dispersion maps is considered. The interplay between the periodic dispersion management and the random dispersive fluctuations is precisely analyzed. Analytic expressions are derived for the moments of the pulse widths as well as for the probability density functions. It is shown that a strong dispersion management stabilizes the soliton, while a small anomalous residual dispersion is necessary for preventing from a stochastic resonance phenomenon. Analytical results are confirmed by direct numerical simulations.

Abstract: The linear propagation of pulses driven by random polarization-mode dispersion is considered. Analytical expressions are derived for the probability-density functions of the pulse width, timing displacement, and degree of polarization. The study is performed in Stokes space, and frequency correlation between modes is shown to play an important role in it.

Abstract: This paper addresses the soliton dynamics in a Toda lattice with a randomly distributed chain of masses. Applying the inverse scattering transform we derive effective equations for the decay of the soliton amplitude that take into account radiative losses. Another important feature is the generation of a soliton gas consisting of a large collection of small solitons. The soliton gas plays an important role in that the changes in the conservation equations cannot be correctly understood if the soliton production is neglected.

Abstract: A weakly nonlinear model is proposed for the Rayleigh-Taylor instability in presence of ablation and thermal transport. The second harmonic generation efficiency of a single-mode disturbance is computed, as well as the nonlinear correction to the exponential growth of the fundamental modulation. Mode coupling in the spectrum of a multi-mode disturbance is thoroughly analyzed. The ablative stabilization can be clearly discussed because the derived formulas for evanescent ablation rate are in agreement with previously known results for incompressible, inviscid, irrotational, and immiscible fluids [S. W. Haan, Phys. Fluids B 3, 2349 (1991), M. Berning and A. M. Rubenchik, Phys. Fluids 10, 1564 (1998)].

Abstract: This paper deals with the propagation of broadband Schell-model sources in nonlinear media with finite relaxation time. The approach is based on the study of the Wigner distribution function and on a separation of scales technique between the microscopic random fluctuations of the field and the macroscopic intensity profile. The regime where the nonlinearity is strong and slow is considered. Precise results are obtained for the small- and large-scale characteristics of the pulse: optical intensity profile, speckle radius and typical intensity profile of the speckle spots.

Abstract: This paper deals with the transmission of a kink in a random medium described by a randomly perturbed sine-Gordon equation. Different kinds of perturbations are addressed, with time or spatial random fluctuations, with or without damping. We derive effective evolution equations for the kink velocity and width by applying a perturbation theory of the inverse scattering transform and limit theorems of stochastic calculus. Results are very different compared to a randomly perturbed Nonlinear Schrödinger equation. The effect of a random perturbation is shown to depend strongly on the interplay of the correlation length of the perturbation and the kink width.

Abstract: A weakly nonlinear model is proposed for the Rayleigh-Taylor instability in presence of surface tension. The dynamics of a multi-mode perturbation of the interface between two incompressible, inviscid, irrotational, and immiscible fluids is analyzed. The quadratic and cubic nonlinear effects are taken into account. They include the nonlinear corrections to the exponential growths of the fundamental modulations. The role of the initial modulation spectrum is discussed. A saturation criterion in terms of the product of a local rms and a particular wavenumber is exhibited. It gives theoretical foundations for numerical conjectures and allows one to analyze the effects of fundamental parameters of the problem such as the dimension or the Atwood number.

Abstract: We present new results for the time reversal of nonlinear pulses traveling in a random medium, in particular for solitary waves. We consider long water waves propagating in the presence of a spatially random depth. Both hyperbolic and dispersive regimes are considered. We demonstrate that in the presence of properly scaled stochastic forcing the solution to the nonlinear (shallow water) conservation law is regularized leading to a viscous shock profile. This enables time-reversal experiments beyond the critical time for shock formation. Furthermore we present numerical experiments for the time reversed refocusing of solitary waves in a regime where theory is not yet available. Solitary wave refocusing simulations are performed with a Boussinesq model, both in transmission and in reflection.

Abstract: The dynamics of a metastable attractive Bose-Einstein condensate trapped by a system of laser beams is analyzed in the presence of small fluctuations of the laser intensity. It is shown that the condensate will eventually collapse. The expected collapse time is inversely proportional to the integrated covariance of the time autocorrelation function of the laser intensity and it decays logarithmically with the number of atoms. Numerical simulations of the stochastic 3D Gross-Pitaevskii equation confirms analytical predictions for small and moderate values of mean field interaction.

Abstract: This paper addresses the impact of time fluctuations of a random medium on refocusing during a time-reversal experiment. Even in the presence of moderate time-perturbations a coherent refocused pulse is observed. The theory predicts the level of recompression observed as well as the conditions for the loss of statistical stabilization. It is shown that the statistical properties of the refocused pulse depend on a simple set of parameters that describe the correlation degree of the medium. The refocused pulse has in general a random shape that can be described in terms of a system of stochastic transport equations driven by a single Brownian motion. Pulse stabilization is also demonstrated for some particular configurations, and the convolution kernel that describes the pulse reshaping is explicitly computed. Numerical simulations are presented and show a very good agreement with the theoretical predictions, thus providing a clear illustration of the robustness of time reversal.

Abstract: This paper is concerned with the study of the deformation of a nonlinear pulse traveling in a random medium. We consider shallow water waves with a spatially random depth. We demonstrate that in the presence of properly scaled stochastic forcing the solution to the nonlinear conservation law is regularized leading to a viscous shock profile. This enables us to perform time-reversal experiments beyond the critical time for shock formation.

Abstract: We present new results for the time-reversal of weakly nonlinear pulses traveling in a random dissipative environment. Motivated by time-reversal experiments we describe a new theory for calculating the {\it eddy viscosity} for weakly nonlinear waves propagating over a random surface. The turbulent viscosity is calculated from first principles, namely without imposing any stress-strain hypothesis. We consider long water waves propagating in the presence of a spatially random depth. A viscous shallow water model is considered and its effective viscosity characterized. We also show that weakly nonlinear waves can still be time-reversed under weak dissipation. Incoherently scattered signals are recompressed, both, for time-reversal in transmission as well as in reflection. Under the weakly nonlinear, weakly dissipative regime dissipation only affects the refocused pulse profile regarding its amplitude, but its shape is not corrupted. Numerical experiments are presented.

Abstract: Refocusing for time reversed waves propagating in disordered media has recently been observed experimentally and studied mathematically. This surprising effect has a great potential of applications in domains such as medical imaging, underwater acoustics, wireless communications. Time refocusing for one-dimensional acoustic waves is now mathematically well understood. In this paper the important case of one-dimensional dispersive waves is addressed. Time reversal is studied in reflection and in transmission. In both cases we derive the self-averaging properties of time-reversed refocused pulses. An asymptotic analysis allows us to derive a precise description of the combined effects of randomness and dispersion. In particular we study an important regime in transmission where the coherent front wave is destroyed while time reversing the incoherent transmitted wave still enables refocusing.

Abstract: The collective oscillations of 1D repulsive Bose gas with external harmonic confinement in two different regimes are studied. The first regime is the mean field regime and the second one is the Tonks-Girardeau regime. We investigate the resonances under periodic modulations of the trap potential and the effective nonlinearity. Modulations of the effective nonlinear coefficient result from modulations of the atomic scattering length by the Feschbach resonance method or periodic variations of the transverse trap frequency. In the mean field regime we predict the bistability in the nonlinear oscillations of the condensate under periodic variations of the trap potential or scattering length. In the Tonks-Girardeau regime the resonance has the character of a linear parametric resonance. For the case of rapid strong modulations of the nonlinear coefficient we find analytical expressions for the nonlinearity managed soliton width and the frequency of the slow secondary oscillations near the fixed point. We confirm the analytical predictions by direct numerical simulations of 1D Gross-Pitaevskii equation and the effective nonlinear Schrodinger equation with quintic nonlinearity and trap potential.

Abstract: This paper is devoted to the study of the deceleration phase of inertial confinement capsules. First the self-similar flow exhibited by Betti et al. [Phys. Plasmas, 8 (2001) 5257] is proved to be an attractor in the sense that arbitrary initial conditions converge towards this solution. The convergence rate depends on the ablation process and heat conductivity and it is shown to be a power law of the increase rate of the hotspot mass. Second the thin layer that separates the hotspot from the cold shell is described and it is shown that it also converges to a locally self-similar profile. By using and generalizing a shell model introduced by Betti et al. [Phys. Plasmas 9 (2002) 2277] a closed system of ordinary differential equations for the main hydrodynamic variables is derived. Finally the linear growth rates of the deceleration phase Rayleigh-Taylor instabilities are computed taking into account ablation and spherical convergence. Significant differences are exhibited between directly and indirectly-driven capsules.

Abstract: This paper is devoted to the study of the dynamics of two weakly-coupled Bose-Einstein condensates confined in a double-well trap and perturbed by random external forces. Energy diffusion due to random forcing allows the system to visit symmetry-breaking states when the number of atoms exceeds a threshold value. The energy distribution evolves to a stationary distribution which depends on the initial state of the condensate only through the total number of atoms. This loss of memory of the initial conditions allows a simple and complete description of the stationary dynamics of the condensate which randomly visits symmetric and symmetry-breaking states.

Abstract: A weakly nonlinear model is proposed for the Rayleigh-Taylor instability in presence of ablation and thermal transport. The nonlinear effects for a single-mode disturbance are computed, included the nonlinear correction to the exponential growth of the fundamental modulation. Mode coupling in the spectrum of a multi-mode disturbance is thoroughly analyzed by a statistical approach. The exponential growth of the linear regime is shown to be reduced by the nonlinear mode coupling. The saturation amplitude is around 0.1 lambda for long wavelengths, but higher for short instable wavelengths in the ablative regime.

Abstract: This paper demonstrates the interest of a time-reversal method for the identification of source in a randomly layered medium. An active source located inside the medium emits a pulse that is recorded on a small time-reversal mirror. The wave is sent back into the medium, either numerically in a computer with the knowledge of the medium, or physically into the real medium. Our goal is to give a precise description of the refocusing of the pulse. We identify and analyze a regime where the pulse refocuses on a ring at the depth of the source and at a critical time. Our objective is to find the location of the source and we show that the time-reveresal refocusing contains information which can be used to this effect and which cannot be obtained by a direct arrival-time analysis. The time reversal technique gives a robust procedure to locate and characterize the source also in the case with ambient noise created by other sources located at the surface.

Abstract: We consider the existence of a dynamically stable soliton in the one-dimensional cubic-quintic nonlinear Schroedinger model with strong cubic nonlinearity management for periodic and random modulations. We show that the predictions of the averaged cubic-quintic nonlinear Schroedinger (NLS) equation and modified variational approach for the arrest of collapse coincide. The analytical results are confirmed by numerical simulations of a one-dimensional cubic-quintic NLS equation with a rapidly and strongly varying cubic nonlinearity coefficient.

Abstract: In the context of the French Laser-Mégajoule fusion-research program, the hydrodynamic stability of the baseline direct-drive target is investigated at the hot spot surface during the deceleration phase by means of modeling and simulations. Using the convergence of the flow towards a self-similar solution, a closed system of ordinary differential equations is derived for the main hydrodynamic variables. An exact linear stability analysis is performed to compute the Rayleigh-Taylor growths. All theoretical predictions are compared to one-dimensional and two-dimensional single-mode detailed numerical results.

Abstract: In this paper an original interacting particle system approach is developed for studying Markov chains in rare event regimes. The proposed particle system is theoretically studied through a genealogical tree interpretation of Keynman-Kac path measures. The algorithmic implementation of the particle system is presented. An efficient estimator for the probability of ocurrence of a rare event is proposed and its variance is computed. Applications and numerical implementations are discussed. First, we apply the particle system technique to a toy model (a Gaussian random walk), which permits to illustrate the theoretical predictions. Second, we address a physically relevant problem consisting in the estimation of the outage probability due to polarization-mode dispersion in optical fibers.

Abstract: We consider an active source embedded in a randomly layered medium. We study the cross-correlation functions of the signals recorded at a series of points located at the surface. We show that this information can be processed to locate the source inside the medium. The analysis is based on a separation of scales technique and limit theorems for random differential equations. The statistical stability of the imaging method is proved. The analogy with the time-reversal of waves is enlightened, but the main difference is also put forward: we propose a passive way of imaging an unknown medium without the use of any active device. We finally extend these ideas for the location of a scatterer illuminated by a controlled source located at the surface or by a set of unknown sources generating random noise.

Abstract: We study the motion of bright matter wave solitons in nonlinear potentials, produced by periodic or random spatial variations of the atomic scattering length. We obtain analytical results for the soliton motion, the radiation of matter wave, and the radiative soliton decay in such configurations of the Bose-Einstein condensate. The stable regimes of propagation are analyzed. The results are in remarkable agreement with the numerical simulations of the Gross-Pitaevskii equation with periodic or random spatial variations of the mean field interactions.

Abstract: This paper considers the propagation of high-power large-aperture laser beams in a Kerr medium. A statistical approach is developed for the growths of filaments from small-amplitude small-scale initial modulations. Closed-form expressions are derived for the intensity distribution, contrast, and maximal beam intensity, which are valid up to the blow up of the most intense filament. Numerical experiments are found to be in good agreement with theoretical predictions.

Abstract: This paper investigates the propagation of a pulse in a randomly birefringent optical fiber. Using a separation of scales technique we derive an effective stochastic partial differential equation for the envelope of the field. Stochastic calculus then allows us to compute physically relevant quantities such as the pulse time displacement or the pulse width. We also deal with the dispersion management technique and the pulse propagation in nonlinear optical fibers. The most important features are that the effective propagation equation is driven by three independent Brownian motions, and that it depends on polarization-mode dispersion through a unique effective parameter. In that sense the pulse dynamics does not depend on the precise details of the microscopic model. Numerical simulations are in excellent agreement with theoretical results.

Abstract: This paper analyses (one-dimensional) nonlinear wave propagation over a disordered fluid body having a small viscosity. The lower boundary is disordered and modelled by a random process. As a pulse shaped nonlinear wave propagates over this turbulent boundary, the velocity and wave elevation are viewed as random fields. Starting from first principles the eddy viscosity is characterized and shown to depend on different scales. This is captured as the leading order pseudodifferential operator resulting from the asymptotic analysis of stochastic differential equations. A discussion is provided showing that mean-field theory would have not captured the correct attenuation rate for the large scale object. Numerical results are provided illustrating the accuracy of the eddy viscosity expression.

Abstract: The transmissions of matter wave solitons through linear and nonlinear inhomogeneities induced by the spatial variations of the trap and the scattering length in Bose-Einstein condensates are investigated. New phenomena, such as the enhanced transmission of a soliton through a linear trap by a modulation of the scattering length, are exhibited. The theory is based on the perturbed Inverse Scattering Transform for solitons, and we show that radiation effects are important. Numerical simulations of the Gross-Pitaevskii equation confirm the theoretical predictions.

Abstract: In this paper we study the robustness of linear pulses, solitons, and dispersion-managed solitons, under the influence of random perturbations. First, we address the problem of the estimation of the outage probability due to polarization-mode dispersion. Second, we compare the pulse broadening due to random fluctuations of the group-velocity dispersion. We use an original interacting particle system to estimate the tails of the probability density functions of the pulse widths. A new adaptative Monte Carlo method is applied that enforces the simulations to probe the regions of practical importance by selection and mutation steps.

Abstract: In this paper we extend the range of applications of the previous work [Fouque et al, Wave Motion 42, 238-260 (2005)]. We consider first the case of an active source embedded below the surface in a finely layered random medium. We perform time reversal with a time reversal mirror placed at the surface and we consider here the case where this mirror is larger than the carrier wavelength. In contrast with the situation addressed in our previous paper, where the size of the mirror was comparable to the wavelength, we show that multipathing dramatically enhances the effective aperture of the mirror so that super resolution at the location of the source can be obtained. In other words, the focal spot radius of the refocused field is much smaller than the spot size obtained in the case of a homogeneous medium. This super resolution effect is obtained by time-reversing the long incoherent waves generated by the multiple scattering due to the thin layers. We also give an application to the problem of focusing on a passive scatterer buried in the random medium and illuminated by a source at the surface.

Abstract: In this paper we consider the hydrodynamic equations with nonlinear conduction in the case where the internal energy and the opacity have power law dependences in the density and in the temperature. This system models the situation where a dense solid is brought into contact with a thermal bath. We address regimes which support locally or globally self-similar solutions. The self-similar solutions are smooth if the surface temperature does not increase too fast in time, but it can exhibit an isothermal shock otherwise. These flows are carefully studied which allows us to clarify the role of the initial solid density in the energy absorption and the ablation process. Comparisons with numerical simulations show excellent agreement.

Abstract: The bright matter wave soliton propagation through a barrier with a rapidly oscillating position is investigated. The averaged over rapid oscillations Gross-Pitaevskii equation is derived, where the effective potential has the form of a finite well. Dynamical trapping and quantum tunneling of the soliton in the effective finite well are investigated. The analytical predictions for the effective soliton dynamics is confirmed by numerical simulations of the full Gross-Pitaevskii equation.

Abstract: Discrete solitons in the Ablowitz-Ladik (AL) and discrete nonlinear Schr\"odinger (DNLS) equations with damping and strong rapid drive are investigated. The averaged equations have the forms of the parametric AL and DNLS equations. A new type of parametric bright discrete soliton and cnoidal waves are found and the stability properties are analyzed. The analytical predictions of the perturbed inverse scattering transform are confirmed by the numerical simulations of the AL and DNLS equations with rapidly varying drive and damping.

Abstract: Mode coupling in a random waveguide can be analyzed with asymptotic analysis based on separation of scales when the propagation distance is large compared to the size of the random inhomogeneities, which have small variance. The wavelength is comparable to the scale of the inhomogeneities. In this paper we study the asymptotic form of the joint distribution of the mode amplitudes at different frequencies. We derive a deterministic system of time-frequency transport equations that describe the evolution of mode powers. This result is applied to the computations of pulse spreading in a random waveguide. It is also applied to the analysis of time-reversal in a random waveguide. We show that randomness enhances spatial refocusing and that diffraction-limited focal spots can be obtained even with small-size time reversal mirrors. However, statistical stability for narrowband refocused fields is achieved only for large-size time reversal mirrors.

Abstract: Pulse propagation in randomly perturbed single-mode waveguides is considered. By an asymptotic analysis the front pulse propagation is reduced to an effective equation with diffusion and dispersion. Apart a random time shift due to a random total travel time, two main phenomena can be distinguished. First, coupling and energy conversion between forward and backward propagating modes is responsible for an effective diffusion of the front pulse. This attenuation and spreading is somewhat similar to the one-dimensional case addressed by the O'Doherty-Anstey theory. Second, coupling between the forward propagating mode and the evanescent modes generate an effective dispersion. In the case of small-scale random fluctuations we show that the second mechanism is dominant.

Abstract: The deformation of a nonlinear pulse traveling in a dispersive random medium can be studied with asymptotic analysis based on separation of scales when the propagation distance is large compared to the correlation length of the random medium. We consider shallow water waves with a spatially random depth. We use a formulation in terms of a terrain-following Boussinesq system. We compute the effective evolution equation for the front pulse which can be written as a dissipative Kortweg-de Vries equation. We study the soliton dynamics driven by this system. We show, both theoretically and numerically, that a solitary wave is more robust than a linear wave in the early steps of the propagation. However, it eventually decays much faster after a critical distance corresponding to the loss of about half of its initial amplitude. We also perform an asymptotic analysis for a class of random bottom topographies. A universal behavior is captured through the asymptotic analysis of the metric term for the corresponding change to terrain-following coordinates. Within this class we characterize the effective height for highly disordered topographies. The probabilistic asymptotic results are illustrated by performing Monte Carlo simulations with a Schwarz-Christoffel Toolbox.

Abstract: In this paper we consider the propagation of water waves in a long-wave asymptotic regime, when the bottom topography is periodic on a short length scale. We perform a multiscale asymptotic analysis of the full potential theory model and of a family of reduced Boussinesq systems parameterized by a free parameter that is the depth at which the velocity is evaluated. We obtain explicit expressions for the coefficients of the resulting effective KdV equations. We show that it is possible to choose the free parameter of the reduced model so as to match the KdV limits of the full and reduced models. Hence the reduced model is optimal regarding the embedded linear weakly dispersive and weakly nonlinear characteristics of the underlying physical problem, which has a microstructure. We also discuss the impact of the rough bottom on the effective wave propagation. In particular nonlinearity is enhanced and we can distinguish two regimes depending on the period of the bottom where the dispersion is either enhanced or reduced compared to the flat bottom case.

Abstract: We present a simple method for all-optical measurements of background, amplitude, and timing jitters of ultra high speed pulse trains or PRBS sequences using the jitter dependence of the intercorrelation-peak shape. This method is numerically and experimentally demonstrated on a 42.66-Gbit/s PRBS sequence and then applied to measure the jitter growths occurring during the propagation of a 160-GHz pulse train in a classical SMF/DCF dispersion map.

Abstract: In this paper we consider the problem of estimating the singular support of the Green's function of the wave equation in a bounded region by cross correlating noisy signals. A collection of sources with unknown spatial distribution emit stationary random signals into the medium, which are recorded at two observation points. We show that the cross correlation of these signals has enough information to identify the singular component of the Green's function, which provides an estimate of the travel time between the two observation points. As in the recent work of Y. Colin de Verdière [math-ph/0610043], we use semiclassical arguments to approximate the wave dynamics by classical dynamics. Next we use the ergodicity of the ray dynamics to obtain an explicit expression of the cross correlation of the noisy signals. We also show that this approach is statistically stable when the averaging time is long enough, and that the accuracy of the travel time estimation is directly related to the spatial correlation function of the sources.

Abstract: We show that the propagation of pulses in an one-dimensional random medium can be characterized using a new complex martingale representation of the transmission coefficient. This representation holds in the frequency domain and in a certain asymptotic limit where diffusion approximations can be used.

Abstract: This paper concerns the homogenization of a one-dimensional elliptic equation with oscillatory random coefficients. It is well-known that the random solution to the elliptic equation converges to the solution of an effective medium elliptic equation in the limit of a vanishing correlation length in the random medium. It is also well-known that the corrector to homogenization, i.e., the difference between the random solution and the homogenized solution, converges in distribution to a Gaussian process when the correlations in the random medium are sufficiently short-range. Moreover, the limiting process may be written as a stochastic integral with respect to standard Brownian motion. We generalize the result to a large class of processes with long-range correlations. In this setting, the corrector also converges to a Gaussian random process, which has an interpretation as a stochastic integral with respect to fractional Brownian motion. Moreover, we show that the longer the range of the correlations, the larger is the amplitude of the corrector. Derivations are based on a careful analysis of random oscillatory integrals of processes with long-range correlations. We also make use of the explicit expressions for the solutions to the one-dimensional elliptic equation.

Abstract: In this paper we derive a general system of transport equations for the moments of reflected and transmitted mode amplitudes in a randomly perturbed waveguide, in a regime where backscattering is significant. The derivation is based on a limit theorem for the system of coupled differential equations for the mode amplitudes, in the limit where the amplitude of the random fluctuations of the medium is small, the correlation lengths in the transverse and longitudinal directions are of the same order of the wavelength, and the waveguide is long. Using this system we exhibit several results in some specific regimes, including the enhanced backscattering phenomenon for the reflected wave: when an incoming monochromatic wave is applied with a given incidence angle, the mean reflected power has a local maximum in the backward direction, twice as large as the mean reflected power in the other directions.

Abstract: This paper shows that the coherent interferometric imaging strategy originally proposed in the context of passive or active arrays of antennas can be implemented for synthetic aperture radar, in which a single antenna is used as an emitter and as a receiver at successive positions along a trajectory. The idea is to backpropagate the crosscorrelations of the recorded signals over selected frequency-spatial windows rather than the signals themselves. The theoretical analysis shows that the signal-to-noise ratio can be enhanced dramatically compared to the standard matched filter processing, without any loss of resolution. This holds true when the fluctuations of the recorded signals have a spatial correlation (along the antenna trajectory) that is larger than the distance between two successive positions of the antenna and smaller than the length of the antenna trajectory. As a result a good compromise between resolution and deblurring can be achieved by an appropriate choice of the spatial window size.

Abstract: In this paper we revisit the parabolic approximation for wave propagation in random media by taking into account backscattering. We obtain a system of transport equations for the moments of the components of reflection and transmission operators. In the regime in which forward scattering is strong and backward scattering is weak, we obtain closed form expressions for physically relevant quantities related to the reflected wave, such as the beam width, the spectral width and the mean spatial power profile. In particular, we analyze the enhanced backscattering phenomenon, that is, we show that the mean power reflected from an incident quasi-plane wave has a maximum in the backscattered direction. This enhancement can be observed in a small cone around the backscattered direction and we compute the enhancement factor as well as the shape of the enhanced backscattering cone.

Abstract: This paper is devoted to the study of the deceleration phase of inertial confinement capsules. The purpose is to obtain a zero-dimensional model that has the form of a closed system of ordinary differential equations for the main hydrodynamic quantities. The model takes into account the energy released by nuclear reactions, a nonlocal model for the alpha-particle energy deposition process, and radiation loss by electron bremsstrahlung. The asymptotic analysis is performed in the case of a strong temperature dependence of the thermal conductivity. We finally study the beginning of the expansion phase after stagnation to derive an ignition criterium.

Abstract: In this paper we propose and discuss variance reduction techniques for the estimation of quantiles of the ouput of a complex model with random input parameters. These techniques are based on the use of a reduced model, such as a metamodel or a response surface. The reduced model can be used as a control variate; or a rejection method can be implemented to sample the realizations of the input parameters in prescribed relevant strata; or the reduced model can be used to determine a good biased distribution of the input parameters for the calibration of an importance sampling strategy. The different strategies are analyzed, the asymptotic variances are computed and compared, which shows the benefit of an adaptive controlled stratification method. This method is applied to a real example (computation of the peak cladding temperature during a large-break loss of coolant accident in a nuclear reactor).

Abstract: In this paper the reflection and transmission of waves by a three-dimensional random medium are studied in a white-noise and paraxial regime. The limit system derives from the acoustic wave equations and is described by a coupled system of random Schroedinger equations driven by a Brownian field whose covariance is determined by the two-point statistics of the fluctuations of the random medium. For the reflected and transmitted fields the associated Wigner distributions and the autocorrelation functions are determined by a closed system of transport equations. The Wigner distribution is then used to describe the enhanced backscattering phenomenon for the reflected field.

Abstract: The random paraxial wave equation is revisited to take into account not only random forward scattering, but also random backscattering. In this paper we are interested in the wave fronts transmitted and reflected by a strong interface buried in a random medium. In the weakly heterogeneous regime the reflected and transmitted wave fields are characterized by reflection and transmission operators that are the solutions of Ito-Schroedinger diffusion models. These models allow for the computations of the Wigner distributions and the autocorrelation functions of the reflected and transmitted waves. They also fully take into account the fact that the waves travel through the same medium during the propagation to and from the interface, which induces an increase of the beam radius and of the correlation radius, and also predicts that enhanced backscattering effect in the backscattered direction.

Abstract: This paper analyses wave propagation in a one-dimensional random medium with long-range correlation. The asymptotic regime where the fluctuations of the medium parameters are small and the propagation distance is large is studied. In this regime effective propagation equations are obtained where the effect of randomness is captured by a pseudo-differential operator. This operator is characterized by a frequency-dependent attenuation that obeys a power law with an exponent ranging from 1 to 2 that is related to the power decay rate of the autocorrelation function of the fluctuations of the medium parameters. This frequency-dependent attenuation is associated with a frequency-dependent phase, which ensures causality of the filter that realizes the approximation. A discussion is provided showing that the mean-field theory cannot capture the correct attenuation rate, because it averages a random time delay. Numerical results are given to illustrate the accuracy of the asymptotic theory.

Abstract: One of the key problems in chance constrained programming for nonlinear optimization problems is the evaluation of derivatives of joint probability functions of the form . Here is the vector of physical parameters, is a random vector describing the uncertainty of the model, is the constraints mapping, and is the vector of constraint levels. In this paper specific Monte Carlo tools for the estimations of the gradient and Hessian of are proposed when the input random vector has a multivariate normal distribution and small variances. Using the small-variance hypothesis, approximate expressions for the first- and second-order derivatives are obtained, whose Monte Carlo estimations have low computational costs. The number of calls of the constraints mapping for the proposed estimators of the gradient and Hessian of is only . These tools are implemented in penalized optimization routines adapted to stochastic optimization, and are shown to reduce the computational cost of chance constrained programming substantially.

Abstract: Inertial confinement fusion targets must be carefully designed to ignite their central hot spots and burn. Changes in the optimal implosion could reduce the fusion energy or even prevent ignition. Since there are unavoidable uncertainties due to technological defects and not perfect reproducibility from shot to shot, the fusion energy will remain uncertain. The degree with which a target can tolerate larger specifications than specified, and the probability with which a particular yield is exceeded, are possible measures of the robustness of that design. This robustness must be assessed in a very high-dimensional parameter space whose variables include every characteristics of the given target and of the associated laser pulse shape, using high-fidelity simulations. Therefore, these studies would remain computationally very intensive. In this paper we propose an approach which consist first of constructing an accurate metamodel of the yield on the whole parameter space with a reasonable data set of simulations. Then the robustness is very quickly assessed for any set of specifications with this surrogate. The yield is approximated by a neural network, and an iterative method adds new points in the data set by means of D-optimal experimental designs. The robustness study of the baseline Laser Megajoule target against one-dimensional defects illustrates this approach. A set of 2000 simulations is sufficient to metamodel the fusion energy on a large 22-dimensional parameter space around the nominal point. Furthermore, a metamodel of the robustness margin against all specifications has been obtained, providing guidance for target fabrication research and development.

Abstract: It is well known that the travel time or even the full Green's function between two passive sensors can be estimated from the cross correlation of recorded signal amplitudes generated by ambient noise sources. It is also known that the direction of the energy flux from the noise sources affects the estimation of the travel time. Using the stationary phase method we show here that the travel time can be effectively estimated when the ray joining the two sensors continues into the noise source region. We extend this analysis to passive sensor imaging of reflectors with different ambient noise source configurations by suitably migrating the cross correlations. If in addition there is multiple scattering in the medium then reflectors can be imaged with passive sensor networks or arrays by migrating suitable fourth-order cross correlations. Fourth-order cross correlations can also be used with auxiliary passive sensors in order to enhance travel time estimation in a scattering medium.

Abstract: In this paper the incoherent waves reflected by a random medium in the parabolic regime are considered. The case in which the medium has three-dimensional rapid random fluctuations and one-dimensional slow variations is analyzed. First, it is shown how the second-order statistics of the reflected wave is determined by the slow spatial variations of the background velocity, the scattering coefficient and the absorption coefficient of the medium via a system of transport equations. Next, it is shown how observations of the intensity, the spatial radius and the spectral radius of the reflected wave, can be used to invert this system in order to reconstruct the parameters of the medium. Finally, it is shown that the analytic framework set forth can also be used to analyze the time dynamics of weak localization.

Abstract: This paper presents a probabilistic analysis of an iterative two-way paraxial scheme for the simulation of wave propagation in random media. This scheme has the computational cost of the standard one-way paraxial wave equation but has the accuracy of the full wave equation in a regime beyond the classical paraxial regime. More precisely, it accurately predicts the statistics of the reflected wave field. The accuracy is determined by the order of the iterative scheme and the ratio of the random backscattering intensity over the random forward-scattering intensity.

Abstract: In this paper we carry out an asymptotic analysis of the elastic wave equations in random media in the parabolic white-noise regime. In this regime, the propagation distance is much larger than the initial beam width, which is itself much larger than the typical wavelength; moreover, the correlation length of the random medium is of the same order as the initial beam width, and the amplitude of the random fluctuations is small. In this distinguished limit we show that wave propagation is governed by a system of random paraxial wave equations. The equations for the shear waves and pressure waves have the form of Schroedinger equations driven by two correlated Brownian fields. The diffraction operators can be expressed in terms of transverse Laplacians. The covariance structure of the Brownian fields is determined by the two point-statistics of the density and Lam\'e parameters of the random medium.

Abstract: We consider the propagation of temporally pulsed electromagnetic waves in a three-dimensional random medium. The main objective is to derive effective white-noise paraxial equations from Maxwell's equations. We address the scaling regime in which 1) the carrier wavelength is small compared to the incident beam radius, which is itself small compared to the propagation distance; 2) the correlation length of the fluctuations of the random medium is of the same order as the beam radius, and the typical amplitude of the fluctuations is small. In this regime we prove that the wave field is characterized by a white-noise paraxial wave equation that has the form of a Schroedinger-type equation driven by a Brownian field. We identify the covariance function of the Brownian field in terms of the two-point statistics of the fluctuations of the dielectric permittivity and the magnetic permeability of the medium. We also study the case in which a strong interface is embedded in the random medium and study the reflected wave, which again is characterized by a Schroedinger-type equation. We discuss applications to enhanced backscattering, time reversal, and imaging.

Abstract: We analyze in detail directivity enhancement in imaging with small arrays of closely spaced sensors, in homogeneous media. Imaging is done with back propagation or migration of the array data after applying an inverse filter that increases the resolution of the image. In general, the construction of such a filter requires invasive measurements on a control array in the vicinity of the object to be imaged, which we assume are not available. The form of the filter is, however, universal if the control array encloses the imaging sensor array. It is the inverse of the finite Fourier transform operator, which has the sinc function as its kernel. We analyze the dependence of resolution enhancement on the signal-to-noise ratio both with narrow and broadband signals.

Abstract: Wave propagation in a one-dimensional random medium with short- or long-range correlations is analyzed. Multiple scattering is studied in the regime where the fluctuations of the medium parameters are small and the propagation distance is large. In this regime pulse propagation is characterized by a random time shift described in terms of a standard or fractional Brownian motion and a deterministic deformation described by a pseudo-differential operator. This operator is characterized by a frequency-dependent attenuation that obeys a power law with an exponent ranging from 0 to 2. The exponent is between 1 and 2 for a long-wavelength pulse and it is determined by the power decay rate at infinity of the autocorrelation function of the random medium parameters. The exponent is between 0 and 1 for a short-wavelength pulse and it is determined by the power decay rate at zero of the autocorrelation function of the random medium parameters. This frequency-dependent attenuation is associated with a frequency-dependent phase responsible for dispersion, which ensures causality and that the Kramers-Kronig relation is satisfied. In the time domain the effective wave equation has the form of a linear integro-differential equation with a fractional derivative.

Abstract: When a broadband pulse penetrates into a dissipative and dispersive medium, phase dispersion and frequency-dependent attenuation alter the pulse in a way that results in the appearance of a precursor field with an algebraic decay. We derive here the existence of precursors in non-dispersive, non-dissipative, but randomly heterogeneous and multiscale media. The shape of the precursor and its fractional power law decay with propagation distance depend on the random medium class. Three principal scattering precursor classes can be identified : (i) In exponentially decorrelating random media, and more generally in mixing random media, the precursor has a Gaussian shape and a peak amplitude that decays as the square root of the inverse of the propagation distance. (ii) In short-range correlation media, with rough multiscale medium fluctuations, the precursor has a skewed shape with a tail that exhibits an anomalous power law decay in time and a peak amplitude that exhibits an anomalous power law decay with propagation distance, both of which depend on the Hurst exponent that characterizes the roughness of the medium. (iii) In long-range correlation media with long-range memory, the situation mimics that of class (ii), but with modified power laws.

Abstract: We analyze the resolution of imaging functionals that migrate the cross correlation matrices of passive sensor arrays. These matrices are obtained by cross correlating signals emitted by ambient noise sources and recorded by the passive sensor array. They contain information about reflectors in the surrounding medium. Therefore, travel time or Kirchhoff migration of the cross correlations can, under favorable circumstances, produce images of such reflectors. However migration should be carried out appropriately depending on the type of illumination provided by the ambient noise sources. We present here a detailed resolution analysis of these functionals in a homogeneous medium. Resolution depends on the sensor array diameter, the distance from the array to the reflector and the central frequency, as is the case in active array imaging. When imaging with passive sensor arrays and ambient noise, resolution also depends on the space and time coherence of the noise sources because it determines an effective noise bandwidth.

Abstract: This article presents a unified kinetic formulation of partially coherent nonlinear optical waves propagating in a noninstantaneous response Kerr medium. We derive a kinetic equation that combines the weak Langmuir turbulence kinetic equation and a Vlasov-like equation within a general framework: it describes the evolution of the spectrum of a random field that exhibits a quasistationary statistics in the presence of a noninstantaneous nonlinear response. The kinetic equation sheds new light on the dynamics of partially coherent nonlinear waves and allows for a qualitative interpretation of the interplay between the noninstantaneous nonlinearity and the nonstationary statistics of the optical field. It is shown that the incoherent modulational instability of a random nonlinear wave can be suppressed by the noninstantaneous nonlinear response. Moreover, incoherent modulational instability can prevent the generation of spectral incoherent solitons.

Abstract: This paper considers the statistical properties of the waves generated by a point source in the subsurface and transmitted towards the surface through a randomly layered medium. The problem is analyzed in a regime of separation of scales and with pressure release boundary conditions at the surface. Using a probabilistic representation of the spectral density of the waves received at the surface, the transmitted intensity is analyzed. Conserved quantities specific to the pressure release boundary conditions are evaluated and the power delay spread is computed. In particular a waveguiding effect that is produced by the reflections from the surface and from the random medium is identified and analyzed. This effect is different from the propagation in a standard waveguide from the point of view of the decay of the intensity.

Abstract: In this contribution we describe and discuss a series of challenges and questions relating to understanding extreme wave phenomena in optics. Many aspects of these questions are being studied in the framework of the MANUREVA project: a multidisciplinary consortium aiming to carry out mathematical, numerical and experimental studies in this field. The central motivation of this work is the 2007 results from optical physics [D. Solli et al., Nature 450, 1054 (2007)] that showed how a fibre-optical system can generate large amplitude extreme wave events with similar statistical properties to the infamous hydrodynamic rogue waves on the surface of the ocean. We review our recent work in this area, and discuss how this observation may open the possibility for an optical system to be used to directly study both the dynamics and statistics of extreme-value processes, a potential advance comparable to the introduction of optical systems to study chaos in the 1970s.

Abstract:

Abstract: It is known that cross correlation of waves generated by noise sources, propagating in an unknown medium, and recorded by a sensor array, can provide information about the medium. In this paper the medium is a three-dimensional small-scale randomly layered medium with slow macroscopic variations. The main objective is here to set forth a framework for analysis of cross correlations of waves generated by noise sources and propagating in such a medium, moreover, use this framework to design estimators for macroscale medium features. The noise sources are located at the bottom of a random medium slab and generate a random wave field that is scattered by the rapid random fluctuations of the medium and then recorded at the surface. Taking into account the pressure release boundary conditions at the surface, this situation corresponds to the so-called daylight configuration. The analysis is carried out in the asymptotic framework where the typical wavelength is small compared to the scale of the macroscopic variations of the background medium and large compared to the decoherence length of the random fluctuations of the medium. It is shown that the cross correlation of the waves recorded at the surface contains statistically stable information about the background medium.

Abstract: We demonstrate that increased power transmission through a random single-mode or multi-mode channel can be obtained in the localization regime by optimizing the spatial wave front or the time pulse profile of the source. The idea is to select and excite the few modes or the few frequencies whose transmission coefficients are anomalously large compared to the typical exponentially small value. We prove that time reversal is optimal for maximizing the transmitted intensity at a given time or space, while iterated time reversal is optimal for maximizing the total transmitted energy. The statistical stability of the optimal transmitted intensity and energy is also obtained.

Abstract: We consider the problem of locating defects and estimating their geometric features from multi-static response matrix measurements at single or multiple frequencies. A main objective is to design specific defect detection rules and to analyze their receiver operating characteristics and the associated resolution and signal-to-noise ratios. In this paper we introduce a unified analytic framework that uses high-frequency asymptotic methods in combination with a hypothesis test based formulation to construct specific procedures for detection and characterization of cracks and inclusions. A central ingredient in our approach is the use of random matrix theory to characterize the signal space associated with the multi-static response matrix measurements. We present numerical experiments to illustrate some of our main findings.

Abstract: In this paper we study passive sensor imaging with ambient noise sources by suitably migrating cross correlations of the recorded signals. We propose and study different imaging functionals. A new functional is introduced that is an inverse Radon transform applied to a special function of the cross correlation matrix. We analyze the properties of the new imaging functional in the high-frequency regime which shows that it produces sharper images than the usual Kirchhoff migration functional. Numerical simulations confirm the theoretical predictions.

Abstract: This article is composed of two parts. The first part is aimed at providing an overview on the kinetic description of random nonlinear waves considering the one-dimensional nonlinear Schroedinger (NLS) equation as a representative model of optical wave propagation. We expose, in particular, the key problem of achieving a closure of the infinite hierarchy of moment equations for the random field. The hierarchy is closed at the first order when the statistics of the random wave is non-stationary or when the response time of the nonlinearity is non-instantaneous, which, respectively, leads to the Vlasov kinetic equation and the weak-Langmuir turbulence equation. When the amount of nonstationary statistics is comparable to the amount of non-instantaneous nonlinearity, we derive a generalized Vlasov-Langmuir equation that provides a unified formulation of the Vlasov and Langmuir approaches. On the other hand, when the statistics of the random wave is stationary and the nonlinear response instantaneous, the closure of the hierarchy of moment equations requires a second-order perturbation expansion procedure, which leads to the Hasselmann (or wave turbulence) kinetic equation. Contrarily to the Vlasov and Langmuir equations, the Hasselmann equation is irreversible, a feature which is expressed by a H-theorem of entropy growth that describes wave thermalization toward the thermodynamic equilibrium distribution, i.e. the Rayleigh-Jeans (RJ) spectrum. In the second part of the paper, we discuss a process of anomalous thermalization by considering the example of the scalar NLS equation whose integrability is broken by the presence of third-order dispersion. The anomalous thermalization is characterized by an irreversible evolution of the wave toward an equilibrium state of a fundamental different nature than the conventional RJ equilibrium state. The wave turbulence kinetic equation reveals that the anomalous thermalization is due to the existence of a local invariant in frequency space, which originates in degenerate resonances of the system. In contrast to integral invariants that lead to a generalized RJ distribution, here, it is the local nature of the invariant that makes the new equilibrium states fundamentally different than the usual RJ equilibrium states. We study in detail the anomalous thermalization by means of numerical simulations of the NLS equation and of the wave turbulence equation by using an improved criterion of applicability of the kinetic theory. The spectrum of the field is shown to exhibit an intriguing asymmetric deformation, which is characterized by the unexpected emergence of a constant spectral pedestal in the long-term evolution of the field. It turns out that the local invariant explains all the essential properties of the anomalous thermalization of the wave.

Abstract: We consider the long-term evolution of a random nonlinear wave that propagates in a multimode optical waveguide. The optical wave exhibits a thermalization process characterized by an irreversible evolution toward an equilibrium state. The tails of the equilibrium distribution satisfy the property of energy equipartition among the modes of the waveguide. As a consequence of this thermalization, the optical field undergoes a process of classical wave condensation, which is characterized by a macroscopic occupation of the fundamental mode of the waveguide. Considering the nonlinear Schr¬odinger equation with a confining potential, we formulate a wave turbulence description of the random wave into the basis of the eigenmodes of the waveguide. The condensate amplitude is calculated analytically as a function of the wave energy, and it is found in quantitative agreement with the numerical simulations. The analysis reveals that the waveguide configuration introduces an effective physical frequency cutoff, which regularizes the ultraviolet catastrophe inherent to the ensemble of classical nonlinear waves. The numerical simulations have been performed in the framework of a readily accessible nonlinear fiber optics experiment.

Abstract: The cross correlations of the wave signals emitted by ambient noise sources can be used to estimate the Green's function of the wave equation in an inhomogeneous medium. In this paper we clarify the role of random scattering in the Green's function estimation in the radiative transport regime and we show how this insight can be used to estimate the velocity of propagation of a smooth background medium.

Abstract: Coherent interferometric imaging is based on the backpropagation of local space-time cross correlations of array data and was introduced in order to improve images when the medium between the array and the object to be imaged is inhomogeneous and unknown [Borcea et al., Inverse Problems, 21 (2005), 1419]. Although this method has been shown to be effective and is well founded theoretically, the coherent interferometric imaging function is computationally expensive and therefore difficult to use. In this paper we show that this function is equivalent to a windowed beamformer energy function, that is, a quadratic function that involves only time gating and time delaying signals in emission and in reception. In this form the coherent interferometric imaging can be implemented efficiently both in hardware and software, that is, at a computational cost that is comparable to the usual beamforming and migration imaging methods. We also revisit the trade-off between enhanced image stability and loss of resolution in coherent interferometry from the point of view of its equivalence to a windowed beamformer energy imaging function.

Abstract: This paper deals with the narrow escape problem when there are two gates which open alternatively in a random way. We set up the problem and carry out a rigorous asymptotic analysis to derive the mean escape time (MET) for a Brownian particle inside a domain to exit the domain through the switching gates. We show that the MET decreases as the switching rate between the gates increases and we give upper and lower bounds for the decay rate. We then consider the case when there are multiple switching gates and derive the leading order term of the asymptotic expansion of the MET.

Abstract: We analyze the resolution and statistical fluctuations of images when the ambient medium is random and scattering can be modeled primarily by wavefront distortion. We compare the coherent interferometric imaging method to the widely used Kirchhoff migration and show how the latter loses statistical stability at an exponential rate with the distance of propagation. In Kirchhoff migration we form images by superposing the array data back propagated to the image domain. In coherent interferometry we back propagate local cross correlations of the array data. This is a denoising process that enhances the signal-to-noise ratio of images but also reduces the resolution. We quantify analytically the trade-off between enhanced stability and reduced resolution in coherent interferometric imaging.

Abstract: Travel time estimation and reflector imaging can be carried out using the cross correlations of signals generated by ambient noise sources and recorded at sensor arrays. We study here the mean and variance of the estimated quantities both with respect to the distribution of the noise sources and with respect to the distribution of the randomly scattering medium. In particular, we discuss the trade-off between resolution enhancement due to illumination diversification by scattering and the associated signal-to-noise ratio reduction, also due to scattering.

Abstract: We analyze the notion offield-field cross correlations associated with scattered coda waves or clutter, observed at pairwise distinct receivers, to obtain an empirical Green's function (EGF) with an emphasis on high-frequency body waves. The scattered waves are generated in a slab with random medium fluctuations by an incident wave packet below. Following the dyadic parabolic scaling of wave packets, and scaling the random fluctuations appropriately, we arrive at a description in terms of a system of Ito-Schroedinger diffusion models. Studying the Wigner distributions of the fields generated by these models, leads to a blurring transformation providing a complete characterization of the mentioned cross correlations.

Abstract: The long-term behavior of a modulationally unstable nonintegrable system is known to be characterized by the soliton turbulence self-organization process: It is thermodynamically advantageous for the system to generate a large-scale coherent soliton in order to reach the (``most disordered") equilibrium state. We show that this universal process of self-organization breaks down in the presence of a highly nonlocal nonlinear response. A wave turbulence approach based on a Vlasov-like kinetic equation reveals the existence of an incoherent soliton turbulence process: It is advantageous for the system to self-organize into a large-scale, spatially localized, incoherent soliton structure.

Abstract: Within the structural reliability context, the aim of this paper is to present a new accelerated Monte-Carlo simulation method, named ADS - Adaptive Directional Stratification -, and designed to overcome the following industrial constraints: robustness of the estimation of a low structural failure probability (less than 10^-3), limited computational resources and complex (albeit often monotonic) physical model. This new stochastic technique is an original variant of adaptive accelerated simulation method, combining stratified sampling and directional simulation and including two steps in the adaptation stage (ADS-2). First, we theoretically study the properties of two possible failure probability estimators and get the asymptotic and non-asymptotic expressions of their variances. Then, we propose some improvements for our new method. To begin with, we focus on the root-finding algorithm required for the directional approach: we present a stop criterion for the dichotomic method and a strategy to reduce the required number of calls to the costly physical model under monotonic hypothesis. Lastly, to overcome the limit involved by high dimensional inputs, we introduce the ADS-2+ method, which has the same ground as the ADS-2 method, but additionnally uses a statistical test to detect the most significant inputs and carries out the stratification only along them. To conclude, we test the ADS-2 and ADS-2+ methods on academic examples in order to compare them with the classical structural reliability methods and to make a numerical sensitivity analysis over some parameters. We also apply the methods to a flood model and a nuclear reactor pressurized vessel model, to practically demonstrate their interest on real industrial examples.

Abstract: In this paper, a perturbation theory for the nonlinear Schroedinger equation with non-vanishing boundary conditions based on the inverse scattering transform is presented. It is applied to study the stability of the soliton propagation on a continuous-wave background. It is shown that the soliton is rather robust with respect to dispersive perturbations but it can be strongly affected by damping. In particular, it is shown that adiabatic approaches can underestimate the decay of the soliton energy.

Abstract: The aim of this paper is to study a topological derivative based anomaly detection algorithm. We compare its performance with other imaging approaches such as MUltiple Signal Classification algorithm (MUSIC), backpropagation, and Kirchhoff migration. We also investigate its stability with respect to medium and measurement noises as well as its resolution. A simple postprocessing of the data set is introduced and shown to be essential in order to obtain an efficient topological based imaging functional, both in terms of resolution and stability.

Abstract: The problem addressed in this paper is the combined detection and localization of a point reflector embedded in a medium by sensor array imaging when the array response matrix is obtained in a noisy environment. We study a detection test based on reverse-time migration of the array response matrix that is the most powerful for a given false alarm rate and compare it with a test based on the singular values of the response matrix. Moreover, we show that reflector localization should be performed with reverse-time migration rather than any other form of weighted-subspace migration and we give the standard deviation of the localization error.

Abstract: Random media have material properties with such complicated spatial variations that they can only be described statistically. When looking at waves propagating in these media, we can only expect in general a statistical description of the wave. But sometimes there exists a deterministic result. In this paper we restrict ourselves to one-dimensional wave problems that arise naturally in many applications, in gravity waves in shallow channels, in layered elastic media such as the earth's mantle, in optical fiber transmission, etc.
In the first part of this lecture we review some asymptotic methods for stochastic differential equations with a small parameter. We apply these methods to compute the localization length of a wave traveling through a slab of random medium. Localization is characterized by an exponential decay of the transmittivity as a function of the size of the slab. It appears as a universal feature in wave propagation in one-dimensional linear random media.
In the second part we address the problem of the propagation of a soliton through a slab of nonlinear and random medium. Indeed some nonlinear dispersive systems such as the Nonlinear Schrödinger equation have special solutions called solitons that can propagate without change of form or diminution of speed. Solitons are therefore candidates to test the robustness of the exponential localization in nonlinear and random media. Using the inverse scattering transform we can exhibit several typical behaviors depending on the amplitude of the incoming soliton.

Abstract: Contents: Introduction; MI in homogeneous media; MI in periodically inhomogeneous media; Modulational instability in random media; MI in nonlinear discrete optical systems; Conclusions; Acknowledgements

Abstract: Random media have material properties with such complicated spatial variations that they can only be described statistically. When looking at waves propagating in these media, we can only expect in general a statistical description of the wave. But sometimes there exists a deterministic result: the wave dynamics only depends on the statistics of the medium, and not on the particular realization of the medium. Such a phenomenon arises when the different scales present in the problem (wavelength, correlation length, and propagation distance) can be separated. In this lecture we restrict ourselves to one-dimensional wave problems that arise naturally in acoustics and geophysics.

Abstract: Contents: Introduction; The main equations; Solitons in random single-mode fibers; Dispersion-managed solitons under random perturbations; Randomly birefringent fibers; Solitons in random quadratic media; Spatial solitons in random waveguides; Two-dimensional solitons in random media; Conclusions; Acknowledgements

Contents:
1. Introduction.
2. Bright solitons in quasi one-dimensional BEC. 2.1. The 1D Gross-Pitaevskii equation. 2.2. Adiabatic soliton compression . 2.3. Transmission through nonlinear barriers and wells. 2.4. Trapping by dynamically managed linear potentials. 2.5. Controllable soliton emission by spatial variations of the scattering length.
3. Bright solitons in nonlinear optical lattices. 3.1. Propagation through a weak nonlinear periodic potential. 3.2. Propagation through a weak random nonlinear potential.
4. Multidimensional bright solitons in BEC . 4.1. 2D bright solitons in BEC with time-varying scattering length. 4.2. 2D bright solitons in BEC with spatially-varying scattering length. 4.3. 2D bright solitons in dipolar BEC. 4.4. 3D bright solitons in anisotropic trap.
5. Future Challenges
References

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.

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.

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.

Abstract:

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.

Abstract: We study nonlinear effects in amplification of partially coherent pulses in a high power laser chain. We compare statistical models with experimental results for temporal and spatial effects. First we show the interplay between self-phase modulation which broadens spectrum bandwidth and gain narrowing which reduces output spectrum. Theoretical results are presented for spectral broadening and energy limitation in case of time-incoherent pulses. In a second part, we introduce spatial incoherence with a multimode optical fiber which provides a smoothed beam. We show with experimental result that spatial filter pinholes are responsible for additive energy losses in the amplification. We develop a statistical model which takes into account the deformation of the focused beam as a function of B integral. We estimate the energy transmission of the spatial filter pinholes and compare this model with experimental data. We find a good agreement between theory and experiments. As a conclusion, we present an analogy between temporal and spatial effects with spectral broadening and spectral filter. Finally, we propose some solutions to control energy limitations in smoothed pulses amplification.

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.

Abstract: Strong dispersion management in birefringent fibers with periodic dispersion is shown to reduce modulational instability domain and suppress additional resonances.

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.

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.

Abstract: We propose new smoothing techniques involving nonlinear cascaded processes. The scheme is based on the transfer of incoherent amplitude modulations of a pump beam to the phase of a monochromatic plane wave signal. For that, we can use nonlinear cascaded processes which create crossed phase modulation without efficient energy transfer. With this technique, we should be able to produce a temporal or/and spatial incoherent phase modulation without using electro- optic devices. We describe the mechanisms and the different schemes we propose such as a random temporal phase modulator or a temporally varying random phase plate. We present theoretical results by developing analytical calculations of the nonlinear phase. Then, we analyze the incoherent phase modulation with the correlation functions formalism. We present results which take into account temporal limitations such as the Group Velocity Dispersion (GVD) or the Group Velocity Walk-off (GVW) and spatial limitations such as the Diffraction and the Spatial Walk-off. Finally, we present an all-optical sinusoidal phase modulator setup.

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.

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.

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.

Abstract: This paper addresses the soliton dynamics in a Toda lattice with a randomly distributed chain of masses. Applying the inverse scattering transform we derive effective equations for the decay of the soliton amplitude that take into account radiative losses. The decay rate does not depend on the incoming energy for large-amplitude soliton. An important feature is the generation of a soliton gas consisting of a large collection of small solitons. The soliton gas plays an important role in that the changes in the conservation equations cannot be correctly understood if the soliton production is neglected.

Abstract: In this paper time reversal of acoustic waves in a dissipative random one-dimensional medium is analyzed. It is shown that time reversal can be used as an efficient and statistically stable method to image a dissipative layer embedded in a random scattering medium. The quantities needed to achieve this goal appear as the solutions of a system of transport equations which are solved by a Monte Carlo method.

Abstract: In this review paper we consider the problem of estimating the singular support of the Green's function of the wave equation by using passive sensors. We assume that noise sources emit stationary random signals into the medium which are recorded by sensors. We show that the cross correlation of the recorded signals has enough information to identify the singular component of the Green's function, which provides an estimate of the travel time between the sensors. We consider different situations, such as when the support of the noise distribution extends over all space or is spatially limited, the medium is open or bounded, homogeneous or inhomogeneous, dissipative or not. We discuss the limitations of the cross correlation technique and identify configurations under which travel time estimation is possible. We show that iterated cross correlations using auxiliary sensors can be efficient for travel time estimation when the support of the noise source distribution is spatially limited.

Abstract: It is possible to estimate the travel time or even the full Green's function between two passive sensors from the cross correlation of recorded signal amplitudes generated by ambient noise sources. Using the stationary phase method we show that it is possible to image reflectors buried in a smoothly varying medium by migrating the cross correlations of the noise signals.

Abstract: In this paper we consider passive sensor imaging with ambient noise sources by suitably migrating the cross correlations of the recorded signals. We use an imaging functional that relates to the classic Kirchhoff migration functional. We analyze the properties of the imaging functional in the high-frequency regime which shows that it produces sharp images. We identify the scaling assumptions that allow us to image respectively, the support of the random sources and the medium variations when these may come in the form of clutter. Numerical simulations confirm the theoretical predictions.

Abstract: In this paper we consider the problem of reconstructing sources in attenuating acoustic media using a time-reversal technique. We first justify the use of the adjoint of the attenuated wave operator instead of the ideal one for modifying the time-reversal process as a first order correction of the attenuation effect. Then we present a modified approach for higher order corrections. We use a thermo-viscous law model for the attenuation losses.

Abstract: The detection, localization, and characterization of a target embedded in a medium is an important problem in wave sensor imaging. The responses between each pair of source and receiver are collected so that the available information takes the form of a response matrix between the source array and the receiver array. When the data are corrupted by additive noise we show how the target can be efficiently detected, localized and characterized using recent tools of random matrix theory.

Abstract: In the context of structural reliability, a small probability to be assessed, a high computational time model and a relatively large input dimension are typical constraints which brought together lead to an interesting challenge. Indeed, in this framework many existing stochastic methods fail in estimating the failure probability with robustness. Therefore, the aim of this article is to present and prove theoretical results about the validity of an original method we have introduced to overcome the specific constraints mentioned above. This new method turns out to be very competitive compared with the existing techniques. It is a variant of accelerated Monte-Carlo simulation method, named ADS-2 - Adaptive Directional Stratification. It combines, in a two steps adaptive strategy, the stratification into quadrants and the directional simulation techniques. Two ADS-2 estimators are presented and their properties are studied

Abstract: We consider an elliptic pseudo-differential equation with a linear potential, whose coefficient contains a highly oscillating part modeled by a stationary ergodic random field, and the random field is constructed as some function of a centered Gaussian process with non-integrable de-correlation rate. We show first that homogenization is simply averaging. We then characterize the corrector: For its mean-zero part, we characterize the magnitude of the fluctuation; further, divided by this magnitude, this mean-zero corrector converges to a Gaussian random process in probability distribution and weakly in the functional space. For the deterministic part of the corrector, we determine its size. As our paper shows, depending on the de-correlation rate of the random field, and the singularity of the Green's function, either the deterministic or the random part of the corrector can dominate.

Abstract: In this paper we develop iterative approaches for imaging extended inclusions from multi-static response measurements at single or multiple frequencies. Assuming measurement noise, we perform a detailed stability and resolution analysis of the proposed algorithms in two different asymptotic regimes. We consider both the Born approximation in the nonmagnetic case and a high-frequency regime in the general case. Based on a high-frequency asymptotic analysis of the measurements, an algorithm for finding a good initial guess for the illuminated part of the inclusion is provided and its optimality is shown. We illustrate our main findings with a variety of numerical examples.

Abstract: In this paper we consider the problem of reconstructing the spatial support of noise sources from boundary measurements using cross correlation techniques. We consider media with and without attenuation and provide efficient imaging functionals in both cases. We also discuss the case where the noise sources are spatially correlated. We present numerical results to show the viability of the different proposed imaging techniques.

Abstract: In this paper we consider resolution estimates in both the linearized conductivity problem and the wave imaging problem. Our purpose is to provide explicit formulas for the resolving power of the measurements in the presence of measurement noise. We show that the low-frequency regime in wave imaging as well as the inverse conductivity problem are very sensitive to measurement noise while high-frequencies increase stability in wave imaging.

Abstract: In this paper we analyze wave propagation in three-dimensional random media. We consider a source with limited spatial and temporal support that generates spherically diverging waves. The waves propagate in a random medium whose fluctuations have small amplitude and correlation radius larger than the typical wavelength but smaller than the propagation distance. In a regime of separation of scales we prove that the wave is modified in two ways by the interaction with the random medium: first, its time profile is affected by a deterministic diffusive and dispersive convolution; second the wave fronts are affected by random perturbations that can be described in terms of a Gaussian process whose amplitude is of the order of the wavelength and whose correlation radius is of the order of the correlation radius of the medium. Both effects depend on the two-point statistics of the random medium.

Abstract: The aim of this paper is to develop new Coherent Interferometry (CINT) algorithms to correct the effect of an unknown cluttered sound speed (random fluctuations around a known constant) on photoacoustic images. By back-projecting the correlations between the pre-processed pressure measurements, we show that we are able to provide statistically stable photo-acoustic images. The pre-processing is exactly in the same way as when we use the circular or the line Radon inversion to obtain photo-acoustic images. Moreover, we provide a detailed stability and resolution analysis of the new CINT-Radon algorithms. We also present numerical results to illustrate their performance and to compare them with Kirchhoff-Radon migration functionals.

Abstract: The aim of this chapter is to review direct (non-iterative) anomaly detection algorithms that take advantage of the smallness of the ultrasound anomalies. In particular, we numerically investigate their stability with respect to medium and measurement noises as well as their resolution.

Abstract: The detection, localization, and characterization of a collection of targets embedded in a medium is an important problem in multistatic wave imaging. The responses between each pair of source and receiver are collected and assembled in the form of a response matrix, known as the multi-static response matrix. When the data are corrupted by additive noise, we study the structure of the response matrix using random matrix theory to show how the targets can be efficiently detected, localized and characterized. We address both the case of a collection of point reflectors in which the singular vectors have all the same form and the case of small electromagnetic inclusions in which the singular vectors may have different forms depending on their magnetic or dielectric type.

Abstract: We analyze the "field-field" cross correlation associated with partly coherent scattered waves generated by a wave packet. The configuration consists of a slab in which random medium fluctuations occur; the bottom of the slab is bounded by a deterministic discontinuity (a smooth reflector). Following the dyadic parabolic scaling of wave packets, and scaling the random fluctuations appropriately, we arrive at a description in terms of a system of linear Ito-Schroedinger diusion models. Studying the Wigner distributions of the elds generated by these models, leads to a blurring transformation providing a complete characterization of the cross correlation. We obtain a description of the cross correlation in terms of this transformation applied to the eective transmission operator arising as a solution of the Ito-Schroedinger model. Most of the analysis is focussed on the interaction of the deterministic reflector with the random fluctuations through the waves in the regime of fine scales.

Abstract: It was shown in [Garnier et al., SIAM J. Imaging Sciences 2 (2009), 396] that it is possible to image reflectors by backpropagating cross correlations of signals generated by ambient noise sources and recorded at sensor arrays. The resolution of the image depends on the directional diversity of the noise signals relative to the sensor array and on the reflector location. When directional diversity is limited it is possible to enhance it by exploiting the scattering properties of the medium since scatterers will act as secondary noise sources. However, scattering increases the fluctuation level of the cross correlations and therefore blurs the image. In this paper we study the trade-off between resolution enhancement and signal-to-noise ratio reduction due to scattering.

Abstract: We give a detailed analysis of long range cumulative scattering effects from rough boundaries in waveguides. We assume small random fluctuations of the boundaries and obtain a quantitative statistical description of the wave field. The method of solution is based on coordinate changes that straighten the boundaries. The resulting problem is similar from the mathematical point of view to that of wave propagation in random waveguides with interior inhomogeneities. We quantify the net effect of scattering at the random boundaries and show how it differs from that of scattering by internal inhomogeneities.

Abstract: In this paper, we consider the problem of reconstructing sources in a homogeneous viscoelastic medium from wavefield measurements using time-reversal algorithms. Our motivation is the recent advances on hybrid methods in biomedical imaging. We first present a modified time-reversal imaging algorithm based on a weighted Helmholtz decomposition and justify mathematically that it provides a better approximation than by simply time reversing the displacement field. Then, we investigate the source inverse problem in an elastic attenuating medium. We provide a regularized time-reversal imaging which corrects the attenuation effect at the first order.

Abstract: With each domain and material parameter, an infinite number of tensors, called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain significant information on the shape of the domain. In the recent paper [Ammari et al., Math. Of Comp. 81 (2012), 367], a recursive optimal control scheme to recover fine shape details of a given domain using GPTs is proposed. In this paper, we show that the GPTs can be used for shape description. We also show that high-frequency oscillations of the boundary of a domain are only contained in its high-order GPTs. Indeed, we provide a stability and resolution analysis for the reconstruction of small shape changes from the GPTs. By developing a level set version of the recursive optimization scheme, we make the change of topology possible and show that the GPTs can capture the topology of the domain. We provide numerical evidence that GPTs can capture topology and high-frequency shape oscillations. Both the analytical and numerical results of this paper clearly show that the concept of GPTs is a very promising new tool for shape description.