radiative transfer theory for high frequency power flows in fluid saturated poro visco elastic media some recent mathematical results are used to study the propagation features of the high frequency vibrational energy density in three dimensional fluid saturated isotropic poro visco elastic media the theory of high frequency asymptotics of the solutions of hyperbolic partial differential equations shows that their energy satisfies liouville type transport or radiative transfer equations for randomly heterogeneous materials for long propagation times these equations can be approached by diffusion equations the corresponding diffusion parameters mean free paths and diffusion constants associated with biot s linear model for such media are derived the analysis accounts for the thermal and viscous memory effects of the fluid phase and the viscous memory effect of the solid phase through time convolution operators in this respect it also extends the existing mathematical results

homogenization limits and wigner transforms we present a theory for carrying out homogenization limits for quadratic functions called energy densities of solutions of initial value problems ivps with anti self adjoint spatial pseudo differential operators pdos the approach is based on the introduction of phase space wigner matrix measures that are calculated by solving kinetic equations involving the spectral properties of the pdo the weak limits of the energy densities are then obtained by taking moments of the wigner measure the very general theory is illustrated by typical examples like semi classical limits of schrodinger equations with or without a periodic potential the homogenization limit of the acoustic equation in a periodic medium and the classical limit of the dirac equation 1997 john wiley sons inc

diffractive geometric optics for bloch wave packets we study for times of order 1 varepsilon solutions of wave equations which are fancyscript o varepsilon 2 modulations of an varepsilon periodic wave equation the solutions are of slowly varying amplitude type built on bloch plane waves with wavelength of order varepsilon we construct accurate approximate solutions of the three scale wkb type the leading profile is both transported at the group velocity and dispersed by a schrodinger equation given by the quadratic approximation of the bloch dispersion relation at the plane wave a ray average hypothesis of the small divisor type guarantees stability we introduce techniques related to those developed in nonlinear geometric optics which lead to new results even on time scales t fancyscript o 1 a pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems the leading term yields h1 asymptotics when the envelopes are only h1

analysis of dynamics in magnetotelluric data by using the fisher shannon method abstract the fisher shannon information fs plane defined by the fisher information measure and the shannon entropy power is used to investigate the complex dynamics of magnetotelluric data of three stations in taiwan in the fs plane the electric and magnetic components are significantly separated characterized by different degrees of order further investigation shows that signals measured in areas with very high level of seismic activity are well discriminated

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice i homogeneous problems we present a computational approach for the wkb approximation of the wave function of an electron moving in a periodic one dimensional crystal lattice we derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the bloch spectrum of the schrodinger operator relying on the framework of the multibranch entropy solutions introduced by brenier and corrias we compute efficiently multiphase solutions using well adapted and simple numerical schemes in this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method

semiclassical and spectral analysis of oceanic waves in this work we prove that the shallow water flow subject to strong wind forcing and linearized around an adequate stationary profile develops for large times closed trajectories due to the propagation of rossby waves while poincare waves are shown to disperse the methods used in this paper involve semi classical analysis and dynamical systems for the study of rossby waves while some refined spectral analysis is required for the study of poincare waves due to the large time scale involved which is of diffractive type

wigner measures and codimension two crossings this article gives a semiclassical description of nucleonic propagation through codimension two crossings of electronic energy levels codimension two crossings are the simplest energy level crossings which affect the born oppenheimer approximation in the zeroth order term the model we study is a two level schrodinger equation with a laplacian as kinetic operator and a matrix valued linear potential whose eigenvalues cross if the two nucleonic coordinates equal zero we discuss the case of well localized initial data and obtain a description of the wavefunction s two scaled wigner measure and of the weak limit of its position density which is valid globally in time

supercritical geometric optics for nonlinear schrodinger equations we consider the small time semi classical limit for nonlinear schrodinger equations with defocusing smooth nonlinearity for a super cubic nonlinearity the limiting system is not directly hyperbolic due to the presence of vacuum to overcome this issue we introduce new unknown functions which are defined nonlinearly in terms of the wave function itself this approach provides a local version of the modulated energy functional introduced by y brenier the system we obtain is hyperbolic symmetric and the justification of wkb analysis follows

self averaging of wigner transforms in random media we establish the self averaging properties of the wigner transform of a mixture of states in the regime when the correlation length of the random medium is much longer than the wave length but much shorter than the propagation distance the main ingredients in the proof are the error estimates for the semiclassical approximation of the wigner transform by the solution of the liouville equations and the limit theorem for two particle motion along the characteristics of the liouville equations the results are applied to a mathematical model of the time reversal experiments for the acoustic waves and self averaging properties of the re transmitted wave are proved

propagation observation and control of waves approximated by finite difference methods this paper surveys several topics related to the observation and control of wave propagation phenomena modeled by finite difference methods the main focus is on the property of observability corresponding to the question of whether the total energy of solutions can be estimated from partial measurements on a subregion of the domain or boundary the mathematically equivalent property of controllability corresponds to the question of whether wave propagation behavior can be controlled using forcing terms on that subregion as is often desired in engineering applications observability controllability of the continuous wave equation is well understood for the scalar linear constant coefficient case that is the focus of this paper however when the wave equation is discretized by finite difference methods the control for the discretized model does not necessarily yield a good approximation to the control for the original continuous problem in other words the classical convergence consistency stability property of a numerical scheme does not suffice to guarantee its suitability for providing good approximations to the controls that might be needed in applications observability controllability may be lost under numerical discretization as the mesh size tends to zero due to the existence of high frequency spurious solutions for which the group velocity vanishes this phenomenon is analyzed and several remedies are suggested including filtering tychonoff regularization multigrid methods and mixed finite element methods we also briefly discuss these issues for the heat beam and schrodinger equations to illustrate that diffusive and dispersive effects may help to retain the observability controllability properties at the discrete level we conclude with a list of open problems and future subjects for research

self averaging scaling limits for random parabolic waves we consider several types of scaling limits for the wigner moyal equation of the parabolic waves in random media the limiting cases of which include the standard radiative transfer limit the geometrical optics limit and the white noise limit we show under fairly general assumptions on the random refractive index field that sufficient amount of medium diversity thus excluding the white noise limit leads to statistical stability or self averaging in the sense that the limiting law is deterministic and is governed by one of the 6 different types of transport boltzmann or fokker planck equations depending on the specific scaling involved we discuss the connection to the statistical stability of time reversal procedure and the decoherence effect in quantum mechanics

a uniformly accurate ua multiscale time integrator pseudospectral method for the dirac equation in the nonrelativistic limit regime we propose and rigourously analyze a multiscale time integrator fourier pseudospectral mti fp method for the dirac equation with a dimensionless parameter varepsilon in 0 1 which is inversely proportional to the speed of light in the nonrelativistic limit regime i e 0 varepsilon ll 1 the solution exhibits highly oscillatory propagating waves with wavelength o varepsilon 2 and o 1 in time and space respectively due to the rapid temporal oscillation it is quite challenging in designing and analyzing numerical methods with uniform error bounds in varepsilon in 0 1 we present the mti fp method based on properly adopting a multiscale decomposition of the solution of the dirac equation and applying the exponential wave integrator with appropriate numerical quadratures by a careful study of the error propagation and using the energy method we establish two independent error estimates via two different mathematical approaches as h m 0 frac tau 2 varepsilon 2 and h m 0 tau 2 varepsilon 2 where h is the mesh size tau is the time step and m 0 depends on the regularity of the solution these two error bounds immediately imply that the mti fp method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau for all varepsilon in 0 1 and optimally with quadratic convergence rate at o tau 2 in the regimes when either varepsilon o 1 or 0 varepsilon lesssim tau numerical results are reported to demonstrate that our error estimates are optimal and sharp finally the mti fp method is applied to study numerically the convergence rates of the solution of the dirac equation to those of its limiting models when varepsilon to0

a wigner measure analysis of the dufort frankel scheme for the schrodinger equation we apply wigner transform techniques to the analysis of the dufort frankel difference scheme for the schrodinger equation and to the continuous analogue of the scheme in the case of a small scaled planck constant semiclassical regime in this way we are able to obtain sharp conditions on the spatial temporal grid which guarantee convergence for average values of observables as the planck constant tends to zero the theory developed in this paper is not based on local and global error estimates and does not depend on whether or not caustics develop numerical test examples are presented to help interpret the theory and to compare the dufort frankel scheme to other difference schemes for the schr odinger equation

homogenization of the schrodinger equation and effective mass theorems we study the homogenization of a schrodinger equation with a large periodic potential denoting by the period the potential is scaled as 2 we obtain a rigorous derivation of so called effective mass theorems in solid state physics more precisely for well prepared initial data concentrating on a bloch eigenfunction we prove that the solution is approximately the product of a fast oscillating bloch eigenfunction and of a slowly varying solution of an homogenized schrodinger equation the homogenized coefficients depend on the chosen bloch eigenvalue and the homogenized solution may experience a large drift the homogenized limit may be a system of equations having dimension equal to the multiplicity of the bloch eigenvalue our method is based on a combination of classical homogenization techniques two scale convergence and suitable oscillating test functions and of bloch waves decomposition

white noise and geometrical optics limits of wigner moyal equation for wave beams in turbulent media starting with the wigner distribution formulation for beam wave propagation in holder continuous non gaussian random refractive index fields we show that the wave beam regime naturally leads to the white noise scaling limit and converges to a gaussian white noise model which is characterized by the martingale problem associated to a stochastic differential integral equation of the ito type in the simultaneous geometrical optics the convergence to the gaussian white noise model for the liouville equation is also established if the ultraviolet cutoff or the fresnel number vanishes sufficiently slowly the advantage of the gaussian white noise model is that its n point correlation functions are governed by closed form equations

on the role of quadratic oscillations in nonlinear schrodinger equations we consider a nonlinear semi classical schrodinger equation for which it is known that quadratic oscillations lead to focusing at one point described by a nonlinear scattering operator if the initial data is an energy bounded sequence we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations the proof relies on a linearizability condition which can be expressed in terms of wigner measures when the initial data is a sum of such quadratic oscillations we prove that the associate solution is the superposition of the nonlinear evolution of each of them up to a small remainder term in an appendix we transpose those results to the case of the nonlinear schrodinger equation with harmonic potential

kinetic limit for wave propagation in a random medium we study crystal dynamics in the harmonic approximation the atomic masses are weakly disordered in the sense that their deviation from uniformity is of the order e the dispersion relation is assumed to be a morse function and to suppress crossed recollisions we then prove that in the limit e 0 the disorder averaged wigner function on the kinetic scale time and space of order e 1 is governed by a linear boltzmann equation

propagation of quantum expectations with husimi functions we analyze the dynamics of expectation values of quantum observables for the time dependent semiclassical schrodinger equation to benefit from the positivity of husimi functions we switch between observables obtained from weyl and anti wick quantization we develop and prove a second order egorov type propagation theorem with husimi functions by establishing transition and commutator rules for weyl and anti wick operators we provide a discretized version of our theorem and present numerical experiments for schrodinger equations in dimensions two and six that validate our results

wigner measures and observability for the schrodinger equation on the disk we analyse the structure of semiclassical and microlocal wigner measures for solutions to the linear schrodinger equation on the disk with dirichlet boundary conditions our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk we prove a structure theorem expressing the restriction of the wigner measures on each invariant torus in terms of second microlocal measures they are obtained by performing a finer localization in phase space around each of these tori at the limit of the uncertainty principle and are shown to propagate according to heisenberg equations on the circle our construction yields as corollaries a that the disintegration of the wigner measures is absolutely continuous in the angular variable which is an expression of the dispersive properties of the equation b an observability inequality saying that the l 2 norm of a solution on any open subset intersecting the boundary resp the l 2 norm of the neumann trace on any nonempty open set of the boundary controls its full l 2 norm resp h 1 norm these results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary

dispersion and strichartz estimates for the liouville equation we consider the liouville equation associated with a metric g of class c2 and we prove dispersion and strichartz estimates for the solution of this equation in terms of geodesics associated with g we introduce the notion of focusing and dispersive metric to characterize metrics such that the same dispersion estimate as in the euclidean case holds to deal with the case of non trapped long range perturbation of the euclidean metric we prove a global velocity moments effect on the solution in particular we obtain global in time strichartz estimates for metrics such that the dispersion estimate is not satisfied

the phase flow method this paper introduces the phase flow method a novel accurate and fast approach for constructing phase maps for nonlinear autonomous ordinary differential equations the method operates by initially constructing the phase map for small times using a standard ode integration rule and builds up the phase map for larger times with the help of a local interpolation scheme together with the group property of the phase flow the computational complexity of building up the complete phase map is usually that of tracing a few rays in addition the phase flow method is provably and empirically very accurate once the phase map is available integrating the ode for initial conditions on the invariant manifold only makes use of local interpolation thus having constant complexity the paper develops applications in the field of high frequency wave propagation and shows how to use the phase flow method to 1 rapidly propagate wave fronts 2 rapidly calculate wave amplitudes along these wave fronts and 3 rapidly evaluate multiple wave arrival times at arbitrary locations

finiteness of prescribed fibers of local biholomorphisms a geometric approach let x be a stein manifold of complex dimension at least two f x rightarrow mathbb c n a local biholomorphism and q in f x in this paper we formulate sufficient conditions involving only objects naturally associated to q in order for the fiber over q to be finite assume that f 1 l is 1 connected for the generic complex line l containing q and f 1 l has finitely many components whenever l is an exceptional line through q using arguments from topology and differential geometry we establish a sharp estimate on the size of f 1 q it follows that for n ge 2 a local biholomorphism of x onto mathbb c n is invertible if and only if the pull back of every complex line is 1 connected

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice iii from ab initio models to wkb for schrodinger poisson this work is concerned with the semiclassical approximation of the schrodinger poisson equation modeling ballistic transport in a 1d periodic potential by means of wkb techniques it is derived by considering the mean field limit of a n body quantum problem then k multivalued solutions are adapted to the treatment of this weakly nonlinear system obtained after homogenization without taking into account for pauli s exclusion principle numerical experiments display the behaviour of self consistent wave packets and screening effects