birkhoff normal form for splitting methods applied to semilinear hamiltonian pdes part ii abstract splitting we consider hamiltonian pdes that can be split into a linear unbounded operator and a regular non linear part we consider abstract splitting methods associated with this decomposition where no discretization in space is made we prove a normal form result for the corresponding discrete flow under generic non resonance conditions on the frequencies of the linear operator and on the step size and under a condition of zero momentum on the nonlinearity this result implies the conservation of the regularity of the numerical solution associated with the splitting method over arbitrary long time provided the initial data is small enough this result holds for rounded numerical schemes avoiding at each step possible high frequency energy drift we apply these results to nonlinear schrodinger equations as well as the nonlinear wave equation

a kam theorem for one dimensional schrodinger equation with periodic boundary conditions abstract in this paper one dimensional 1 d nonlinear schrodinger equation iu t u xx mu g u u u 0 with periodic boundary conditions is considered m 1 12 z is a real parameter and the nonlinearity g u u j l j l 4 a jl u j u l a jl a lj r a 22 0 is a real analytic function in a neighborhood of the origin the kam machinery is adapted to fit the above equation so as to construct small amplitude periodic or quasi periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system

a kam theorem for hamiltonian partial differential equations in higher dimensional spaces in this paper we give a kam theorem for a class of infinite dimensional nearly integrable hamiltonian systems the theorem can be applied to some hamiltonian partial differential equations in higher dimensional spaces with periodic boundary conditions to construct linearly stable quasi periodic solutions and its local birkhoff normal form the applications to the higher dimensional beam equations and the higher dimensional schrodinger equations with nonlocal smooth nonlinearity are also given in this paper

lagrangians of physics and the game of fisher information transfer the lagrangians of physics arise out of a mathematical game between a smart measurer and nature personified by a demon each contestant wants to maximize his level of fisher information i the game is zero sum by conservation of information in the closed system the payoff of the game introduces a variational principle extreme physical information epi which fixes both the lagrangian and the physical constant of each scenario the epi approach provides an understanding of the relationship between measurement and physical law epi also defines a prescription for constructing lagrangians the prior knowledge required for this purpose is a rule of symmetry or conservation that implies a unitary transformation for which i remains invariant as an example when applied to the smart measurement of the space time coordinate of a particle the symmetry used is that between position time space and momentum energy space then the unitary transformation is the fourier one and epi derives the following the equivalence of energy momentum and mass the constancy of planck s parameter h and the lagrangian that implies both the klein gordon equation and the dirac equation of quantum mechanics

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

birkhoff normal form for some nonlinear pdes we consider the problem of extending to pdes birkhoff normal form theorem on hamiltonian systems close to nonresonant elliptic equilibria as a model problem we take the nonlinear wave equation with dirichlet boundary conditions on 0 g is an analytic skewsymmetric function which vanishes for u 0 and is periodic with period 2 in the x variable we prove under a nonresonance condition which is fulfilled for most g s that for any integer m there exists a canonical transformation that puts the hamiltonian in birkhoff normal form up to a reminder of order m the canonical transformation is well defined in a neighbourhood of the origin of a sobolev type phase space of sufficiently high order some dynamical consequences are obtained the technique of proof is applicable to quite general semilinear equations in one space dimension

joint data and channel estimation using blind trellis search techniques a novel method of maximum likelihood sequence estimation is proposed for data that are transmitted over unknown linear channels this procedure does not require a startup sequence for estimating the channel impulse response rather the data and the channel are simultaneously estimated it is implemented without any loss of optimality by a trellis search algorithm which searches for the best data sequence from among a number of hypothesized trellises which are constructed from the observed sequence the number of states in each trellis and the number of trellises grow exponentially with the channel memory a suboptimal trellis search algorithm is proposed whose complexity at best is slightly higher than that of the adaptive viterbi algorithm operating with a known channel response a simplified channel estimation algorithm when the number of data alphabets is greater than 2 is also proposed fast convergence of the algorithm in estimating the channel is demonstrated for binary pulse amplitude modulation over a variety of channels convergence over a wide range of snr occurs within 100 symbols the channel estimation algorithm for multi level signals converges within 500 1000 symbols we finally present an application of this algorithm to the problem of sequence estimation in the presence of rapidly time varying intersymbol interference the algorithm provides reliable zero delay decisions which results in a better channel tracking algorithm when compared to previously proposed schemes

asymptotic preserving schemes for the klein gordon equation in the non relativistic limit regime we consider the klein gordon equation in the non relativistic limit regime i e the speed of light c c tending to infinity we construct an asymptotic expansion for the solution with respect to the small parameter depending on the inverse of the square of the speed of light as the first terms of this asymptotic can easily be simulated our approach allows us to construct numerical algorithms that are robust with respect to the large parameter c c producing high oscillations in the exact solution

linear prediction methods for blind fractionally spaced equalization we describe adaptive methods for estimating fir zero forcing blind equalizers with arbitrary delay directly from the linear predictions of the observations while most current methods require inversion or singular value decomposition svd of the correlation matrix our methods need only to solve two linear prediction problems they can be implemented as rls or lms algorithms to recursively update the equalizer estimation they are computationally efficient the computational complexity in each recursion can be less than 15 ln sup 2 in the rls case where ln equals the equalizer length and 3l ln in the lms case where l is the number of subchannels the performance of the proposed methods and comparisons with existing approaches are shown by simulation to demonstrate their effectiveness

transfer of energy to high frequencies in the cubic defocusing nonlinear schrodinger equation we consider the cubic defocusing nonlinear schrodinger equa tion on the two dimensional torus we exhibit smooth solutions for which the support of the conserved energy moves to higher fourier modes this behavior is quantified by the growth of higher sobolev norms given any 1 k 1 s 1 we construct smooth initial data u0 withu0 h s k at some time t this growth occurs despite the hamiltonian s bound onu t h 1 and despite the conservation of the quantityu t l2

renyi entropies and fisher informations as measures of nonextensivity in a tsallis setting abstract we study nonextensive statistical scenarios a la tsallis with reference to fisher s information and renyi s entropy a new way of evaluating tsallis generalized expectation values is examined within such a context and is shown to lead to a much better cramer rao bound than the customary procedure a connection between the information measures of fisher s and renyi s is found we show that fisher s measure for translation families remains additive even in a non extensive tsallis setting

invariant cantor manifolds of quasi periodic oscillations for a nonlinear schrodinger equation this paper was written while both authors were guests of the forschungsinstitut fur mathematik at the eth zurich and we thank the institute for its hospitality pleasant working atmosphere and helpful staff in particular we thank jiirgen moser for numerous stimulating discussions on the subject we also benefitted from a remark by sigurd angenent concerning the analyticity of the solutions the second author also thanks the deutsche forschungsgemeinschaft for their financial support through a heisenberg grant

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

an abstract birkhoff normal form theorem and exponential type stability of the 1d nls we study stability times for a family of parameter dependent nonlinear schrodinger equations on the circle close to the origin imposing a suitable diophantine condition first introduced by bourgain we prove a rather flexible birkhoff normal form theorem which implies e g exponential and sub exponential time estimates in the sobolev and gevrey class respectively

a birkhoff lewis type theorem for the nonlinear wave equation we give an extension of the celebrated birkhoff lewis theorem to the nonlinear wave equation accordingly we find infinitely many periodic orbits with longer and longer minimal periods accumulating at the origin which is an elliptic equilibrium of the associated infinite dimensional hamiltonian system

a kam theorem with applications to partial differential equations of higher dimensions the existence of lower dimensional kam tori is shown for a class of nearly integrable hamiltonian systems of infinite dimensions where the second melnikov s conditions are completely eliminated and the algebraic structure of the normal frequencies are not needed as a consequence it is proved that there exist many invariant tori and thus quasi periodic solutions for nonlinear wave equations schrodinger equations and other equations of any spatial dimensions

the stability of full dimensional kam tori for nonlinear schrodinger equation abstract in this paper it is proved that the full dimensional invariant tori obtained by bourgain j funct anal 229 2005 no 1 62 94 is stable in a very long time for 1d nonlinear schrodinger equation with periodic boundary conditions

a nekhoroshev type theorem for hamiltonian systems with infinitely many degrees of freedom we study the propagation of lattice vibrations in models of disorder ed classical anharmonic crystals using classical perturbation theory with an optimally chosen remainder term i e a nekhoroshev type scheme we are able to show that vibrations corresponding to localized initial conditions do essentially not propagate through the crystal up to times larger than any inverse power of the strength of the anharmonic couplings

quasi periodic solutions of hamiltonian perturbations of 2d linear schrodinger equations the general problem discussed here is the persistency of quasi periodic solutions of linear or integrable equations after hamiltonian perturbation this subject is closely related to the well known kam theory of invariant tori in smooth dynamical systems the main source of new difficulties is the fact that one considers here finite dimensional tori in an infinite dimensional phase space for instance already in the study of time periodic solutions the fact that the phase space is infinite dimensional leads to small divisor problems now the standard kam technology may be reworked in a form applicable to treat persistency problems in space dimension 1d in the case of dirichlet boundary conditions say where in particular the normal frequencies of the problem are well separated results along this line were obtained by s kuksin ki and c e wayne w among other works we mentioned ki as the main expository reference for the kam approach considering pde s in id with periodic boundary conditions and especially in space dimension 1 a significant new problem arises due to the presence of clusters of normal frequencies

direct estimation of blind zero forcing equalizers based on second order statistics most existing zero forcing equalization algorithms rely either on higher than second order statistics or on partial or complete channel identification we describe methods for computing fractionally spaced zero forcing blind equalizers with arbitrary delay directly from second order statistics of the observations without channel identification we first develop a batch type algorithm then adaptive algorithms are obtained by linear prediction and gradient descent optimization our adaptive algorithms do not require channel order estimation nor rank estimation compared with other second order statistics based approaches ours do not require channel identification at all on the other hand compared with the cma type algorithms ours use only second order statistics thus no local convergence problem exists and faster convergence can be achieved simulations show that our algorithms outperform most typical existing algorithms

conservation of energy momentum and actions in numerical discretizations of non linear wave equations for classes of symplectic and symmetric time stepping methods trigonometric integrators and the stormer verlet or leapfrog method applied to spectral semi discretizations of semilinear wave equations in a weakly non linear setting it is shown that energy momentum and all harmonic actions are approximately preserved over long times for the case of interest where the cfl number is not a small parameter such results are outside the reach of standard backward error analysis here they are instead obtained via a modulated fourier expansion in time

estimation of distribution laws and physical laws by a principle of extremized physical information abstract a new method of estimating physical probability laws is given this arises from a new form of information which in turn follows from four physically based axioms extremization of this physical information gives rise to the required probability laws as verifications of the approach the following statistical and non statistical laws of physics are derived the complex schrodinger and helmholtz wave equations relativistic quantum mechanics in the klein gordon dirac and weyl pauli formulations the boltzmann energy and maxwell boltzmann velocity distributions the lorentz transformation group of special relativity and maxwell s equations also derived are inequalities defining a class of uncertainty principles e g heisenberg s finally the einstein equations of motion but not the fields of general relativity are derived along with an equivalence d i d mc 2 between information flow rate and matter energy

long time existence for semi linear klein gordon equations with small cauchy data on zoll manifolds we prove a long time existence result for semi linear klein gordon equations with small cauchy data on zoll manifolds this generalizes a preceding result concerning the case of spheres obtained in an earlier paper by the authors the proof relies on almost orthogonality properties of products of eigenfunctions of positive elliptic selfadjoint operators on a compact manifold and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on zoll manifolds

on the reducibility of linear differential equations with quasiperiodic coefficients the system x a eq t x in rd is considered where a is a constant matrix and q a quasiperiodic analytic matrix with r basic frequencies the eigenvalues of a are arbitrary including the purely imaginary case suppose that the set formed by the eigenvalues of a and the basic frequencies of q satisfies a nonresonant condition then there is a positive measure cantorian set e such that for e e the system is reducible to constant coefficients by means of a quasiperiodic change of variables provided a nondegeneracy condition holds this condition prevents locking at resonance

interpretation of the extreme physical information principle in terms of shift information it is shown that fisher information fi can be considered as a limiting case of a related form of kullback information char22 a shift information si the compatibility of the use of si with a basic physical principle of uncertainty is demonstrated the scope of fi based theory is extended to the nonlinear klein gordon equation

branching of cantor manifolds of elliptic tori and applications to pdes we consider infinite dimensional hamiltonian systems we prove the existence of cantor manifolds of elliptic tori of any finite higher dimension accumulating on a given elliptic kam torus then close to an elliptic equilibrium we show the existence of cantor manifolds of elliptic tori which are branching points of other cantor manifolds of higher dimensional tori we also answer to a conjecture of bourgain proving the existence of invariant elliptic tori with tangential frequency along a pre assigned direction the proofs are based on an improved kam theorem its main advantages are an explicit characterization of the cantor set of parameters and weaker smallness conditions on the perturbation we apply these results to the nonlinear wave equation

on quasi periodic solutions for a generalized boussinesq equation abstract in this paper a one dimensional generalized boussinesq equation u t t u x x u 3 u x x x x 0 with hinged boundary conditions is considered it is proved that the above equation admits small amplitude quasi periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional hamiltonian system the proof is based on an infinite dimensional kam theorem and the birkhoff normal form

fisher s information measure in a tsallis nonextensive setting and its application to diffusive process fisher s information measures as adapted to a nonextensive tsallis environment are discussed for systems of particles that are in a general state of motion a lower bound to these information measures is derived with the help of a recently established upper bound to the entropy increase this lower bound to the information measure is the basis for a variational principle devised to determine unknown probability distributions in the important instance of diffusion process we show that trying to ascertain which is the probability distribution that maximizes fisher s information sheds some light on the meaning of tsallis q parameter we discuss applications to cosmological models that seem to suggest that a non extensive thermostatistics with q 1 provides an adequate scenario for discussing gravitation

system identification using a linear combination of cumulant slices a linear approach to identifying the parameters of a moving average ma model from the statistics of the output is presented first it is shown that under some constraints the impulse response of the system can be expressed as a linear combination of cumulant slices then this result is used to obtain a well conditioned linear method for estimating the ma parameters of a nongaussian process the linear combination of slices used to compute the ma parameters can be constructed from different sets of cumulants of different orders provided a general framework in which all the statistics can be combined it is not necessary to use second order statistics autocorrelation slice and therefore the proposed algorithm still provides consistent estimates in the presence of colored gaussian noise another advantage of the method is that while most linear methods give totally erroneous estimates if the order is overestimated the proposed approach does not require a previous estimation of the filter order the simulation results confirm the good numerical conditioning of the algorithm and its improvement in performance in comparison to existing methods

long time existence for semi linear klein gordon equations on tori abstract we use normal forms for sobolev energy to prove that small smooth solutions of semi linear klein gordon equations on the torus exist over a larger interval than the one given by local existence theory for almost every value mass the gain on the length of the lifespan does not depend on the dimension the result relies on the fact that the difference of square of two successive distinct eigenvalues of on t d can be bounded from below by a constant

blind sequence estimation estimating the data sequence from the received signal without knowing the transmission channel is referred to as blind sequence estimation a new blind sequence estimation scheme is proposed by exploiting the second order statistical properties of the source and the algebraic structure of the data sequence an optimal source deterministic correlation estimator and the viterbi algorithm are used to achieve blind sequence estimation the proposed approach also has a special structure particularly attractive for time varying channels

blind fractionally spaced equalization of noisy fir channels direct and adaptive solutions blind fractionally spaced equalizers reduce intersymbol interference using second order statistics without the need for training sequences methods for finding fir zero forcing blind equalizers directly from the observations are described and adaptive versions are developed in contrast most current methods require channel estimation as a first step to estimating the equalizer the direct methods can be zero forcing minimum mean square error or even minimum mean square error mmse within the class of zero forcing equalizers performance of the proposed methods and comparisons with existing approaches are shown for a variety of channels including an empirically measured digital microwave channel

fisher information as the basis for maxwell s equations maxwell s equations of classical electrodynamics may be derived on the following statistical basis consider a gedanken experiment whereby the mean space time coordinate for photons in an electromagnetic field is to be determined by observation of one photon s space time coordinate an efficient i e optimum estimate obeys a condition of minimum fisher information or minimum precision according to the second law of thermodynamics the fisher information i is a simple functional of the probability law governing space time coordinates of the particles of the field this probability law is modeled as the source free poynting energy flow density i e the ordinary local intensity in the optical sense or the square of the four vector potential when the fisher information is extremized subject to an additive constraint term in the total interaction energy maxwell s equations result

on dispersion of small energy solutions of the nonlinear klein gordon equation with a potential in this paper we study small amplitude solutions of nonlinear klein gordon equations with a potential under suitable smoothness and decay assumptions on the potential and a genericity assumption on the nonlinearity we prove that all small energy solutions are asymptotically free in cases where the linear system has at most one bound state the result was already proved by soffer and weinstein we obtain here a result valid in the case of an arbitrary number of possibly degenerate bound states the proof is based on a combination of birkhoff normal form techniques and dispersive estimates

existence conditions and spreading properties of extreme entropy d dimensional distributions abstract the extremization of the information theoretic measures fisher information shannon entropy tsallis entropy which complementary describe the spreading of the physical states of natural systems gives rise to fundamental equations of motion and or conservation laws at times the associated extreme entropy distributions are known for some given constraints usually moments or radial expectation values in this work first we give the existence conditions of the maxent probability distributions in a d dimensional scenario where two moments not necessarily of consecutive order are known then we find general relations which involve four elements the extremized entropy the other two information theoretic measures and the variance of the extremum density in scenarios with different dimensionalities and moment constraints

floquet hamiltonians with pure point spectrum we consider floquet hamiltonians of the type k f i partial t h 0 beta v omega t whereh0 a selfadjoint operator acting in a hilbert space has simple discrete spectrume1 0 for a given 0 t v t is 2 periodic andr times strongly continuously differentiable as a bounded operator on and are real parameters and the periodic boundary condition is imposed in time we show roughly that providedr is large enough small enough and non resonant then the spectrum ofkf is pure point the method we use relies on a successive application of the adiabatic treatment due to howland and the kam type iteration settled by bellissard and extended by combescure both tools are revisited adjusted and at some points slightly simplified

on the integrability of degasperis procesi equation control of the sobolev norms and birkhoff resonances abstract we consider the dispersive degasperis procesi equation u t u x x t c u x x x 4 c u x u u x x x 3 u x u x x 4 u u x 0 with c r 0 in 15 the authors proved that this equation possesses infinitely many conserved quantities we prove that there are infinitely many of such constants of motion which control the sobolev norms and which are analytic in a neighborhood of the origin of the sobolev space h s with s 2 both on r and t by the analysis of these conserved quantities we deduce a result of global well posedness for solutions with small initial data and we show that on the circle the formal birkhoff normal form of the degasperis procesi at any order is action preserving

invariant tori for nearly integrable hamiltonian systems with degeneracy does not fall into a hyperplanethrough the origin the condition 1 3 is the sharpest one we have not seenits proof yet in this paper we will give a nondegeneracy condition by meansof the derivatives of h p which is equivalent to the condition 1 3 in analyticcase and under this nondegeneracy condition we obtain the russmann s results for analytic case furthermore the arguments of this paper are available to thenonanalytic case since our nondegeneracy conditions only involve the nite orderderivatives of h p main resultstheorem a suppose that h h p f p q is analytic in t

a nekhoroshev type theorem for the nonlinear wave equation abstract we prove a nekhoroshev type theorem for the nonlinear wave equation nlw 0 1 u t t u x x m u f u under dirichlet boundary conditions where f is an analytic function more precisely we prove the subexponential long time stability result of the origin for equation 0 1 by using birkhoff normal form technique for infinite dimensional hamiltonian systems and the so called tame property of the nonlinearity in gevrey or analytic space which is inspired by the work 6 where the authors proved the polynomial long time stability result in sobolev space for a class of hamiltonian pdes

non stochasticity of time dependent quadratic hamiltonians and the spectra of canonical transformations the authors study the quantum mechanical evolution generated by hamiltonians which are quadratic polynomials in q and p with time dependent coefficients this evolution is determined by a unitary implementation of the phase flow of the corresponding classical hamiltonian in the case of quadratic hamiltonians which are periodic in time the floquet operator is shown to have either a pure point spectrum or a purely transient absolutely continuous spectrum thus the motion is non stochastic in a simple model of a quadratic hamiltonian with random time dependence the quantum mechanical motion is shown to be non stochastic almost surely

tsallis nonextensive thermostatistics and fisher s information measure abstract the connection between fisher s ideas concerning information measures and nonextensive thermostatistics net is investigated the cramer rao bound is generalized to a net environment a relationship between fisher s information and tsallis entropy is established

superexponential stability of kam tori we study the dynamics in the neighborhood of an invariant torus of a nearly integrable system we provide an upper bound to the diffusion speed which turns out to be of superexponentially small size exp exp 1 being the distance from the invariant torus we also discuss the connection of this result with the existence of many invariant tori close to the considered one

long time stability of kam tori for nonlinear wave equation abstract it is proved that the kam tori thus quasi periodic solutions are long time stable for infinite dimensional hamiltonian systems generated by nonlinear wave equation by constructing a partial normal form of higher order around the kam torus and showing that p tame property persists under kam iterative procedure and normal form iterative procedure

kam tori for 1d nonlinear wave equations with periodic boundary conditions in this paper one dimensional 1d nonlinear wave equations with periodic boundary conditions are considered v is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u 0 it is proved that for most potentials v x the above equation admits small amplitude periodic or quasi periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system the proof is based on an infinite dimensional kam theorem which allows for multiple normal frequencies

convergence to normal forms of integrable pdes in an infinite dimensional hilbert space we consider a family of commuting analytic vector fields vanishing at the origin and which are nonlinear perturbations of some fundamental linear vector fields we prove that one can construct by the method of poincar e normal form a local analytic coordinate transformation near the origin transforming the family into a normal form the result applies to the kdv and nls equations and to the toda lattice with periodic boundary conditions one gets existence of birkhoff coordinates in a neighbourhood of the origin the proof is obtained by directly estimating in an iterative way the terms of the poincar e normal form and of the transformation to it through a rapid convergence algorithm

almost global existence for hamiltonian semilinear klein gordon equations with small cauchy data on zoll manifolds this paper is devoted to the proof of almost global existence results for klein gordon equations on zoll manifolds e g spheres of arbitrary dimension with hamiltonian nonlinearities when the cauchy data are smooth and small the proof relies on birkhoff normal form methods and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on zoll manifolds 2007 wiley periodicals inc

blind sequence detection without channel estimation this paper proposes a new blind sequence estimation method for single input single output siso systems utilizing an optimal trellis search which is performed by a channel independent viterbi algorithm civa in contrast to the traditional viterbi algorithm that requires accurate channel estimation civa does not require channel coefficients instead the metrics are calculated from a bank of test vectors designed off line the proposed algorithm has outstanding performance under most of the channel conditions specifically it does not suffer from ill conditioned channels in addition it does not depend on channel correlation estimation and therefore has fast convergence simulations demonstrate its superior performance over even most training based equalization algorithms

birkhoff normal form for partial differential equations with tame modulus we prove an abstract birkhoff normal form theorem for hamiltonian partial differential equations pdes the theorem applies to semilinear equations with nonlinearity satisfying a property that we call tame modulus such a property is related to the classical tame inequality by moser in the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times we also develop a general scheme to apply the abstract theory to pdes in one space dimensions and we use it to study some concrete equations nonlinear wave nlw equation nonlinear schrodinger nls equation with different boundary conditions an application to an nls equation on the d dimensional torus is also given in all cases we deduce bounds on the growth of high sobolev norms in particular we get lower bounds on the existence time of solutions

birkhoff normal form for splitting methods applied to semilinear hamiltonian pdes part i finite dimensional discretization we consider discretized hamiltonian pdes associated with a hamiltonian function that can be split into a linear unbounded operator and a regular nonlinear part we consider splitting methods associated with this decomposition using a finite dimensional birkhoff normal form result we show the almost preservation of the actions of the numerical solution associated with the splitting method over arbitrary long time and for asymptotically large level of space approximation provided the sobolev norm of the initial data is small enough this result holds under generic non resonance conditions on the frequencies of the linear operator and on the step size we apply these results to nonlinear schrodinger equations as well as the nonlinear wave equation

long time analysis of nonlinearly perturbed wave equations via modulated fourier expansions a modulated fourier expansion in time is used to show long time near conservation of the harmonic actions associated with spatial fourier modes along the solutions of nonlinear wave equations with small initial data the result implies the long time near preservation of the sobolev type norm that specifies the smallness condition on the initial data

lower dimensional invariant tori with prescribed frequency for the nonlinear schrodinger equation abstract in this paper one dimensional 1d nonlinear schrodinger equation i u t u x x m u u 2 u f u 2 u 0 subject to dirichlet boundary conditions is considered where the nonlinearity f is a real analytic function near u 0 with f 0 f 0 0 it is proved that for each given constant potential m and each prescribed integer b 1 the above equation admits a whitney smooth family of small amplitude time quasi periodic solutions whose b dimensional frequencies are just small dilation of a prescribed diophantine vector accordingly we obtain the existence of lower dimensional invariant kam tori with tangential frequencies constrained to a given diophantine direction in an infinite dimensional phase space setting

perturbations of lower dimensional tori in the resonant zone for reversible systems consider a nearly integrable system which is reversible with respect to an involution of n n type for a given l resonant surface 1 l n we show that there is a subset of it with positive measure such that in each resonant torus corresponding to a point of this set at least one dimensional invariant submanifold can survive any small perturbations

spectrum for quantum duffing oscillator and small divisor equation with large variable coefficient by establishing a new estimate of the solutions for the small divisor equation with large variable coefficients and using the kam technique we prove a reduction theorem that entails the pure point nature of the floquet spectrum of the quantum duffing oscillator with a small perturbation temporal quasi periodic with nonresonant frequencies 2010 wiley periodicals inc

spectral 1 f noise derived from extremized physical information we consider the evolution of a temporal signal x t that is an intrinsic random field of order zero in the sense of a certain measurement estimation experiment the state of disorder of x t should increase toward an equilibrium state the disorder of x t is measured by its physical information scri and the equilibrium state is determined by the condition that scri be an extremum the equilibrium state is shown to have a power spectrum s ensuremath omega of the form mathrm ensuremath omega mathrm ensuremath mathrm ensuremath alpha 1 ensuremath le ensuremath alpha ensuremath le 2 that of 1 f noise

analyticity of the scattering operator for semilinear dispersive equations we present a general algorithm to show that a scattering operator associated to a semilinear dispersive equation is real analytic and to compute the coefficients of its taylor series at any point we illustrate this method in the case of the schrodinger equation with power like nonlinearity or with hartree type nonlinearity and in the case of the wave and klein gordon equations with power nonlinearity finally we discuss the link of this approach with inverse scattering and with complete integrability

quasi periodic solutions of derivative nonlinear schrodinger equations with a given potential abstract it is proved that for a prescribed potential v x there are many quasi periodic solutions of derivative nonlinear schrodinger equation i u t u x x v x u u x 2 u subject to dirichlet boundary condition by means of a kam theorem to a reversible system

long time sobolev stability for small solutions of quasi linear klein gordon equations on the circle we prove that higher sobolev norms of solutions of quasi linear klein gordon equations with small cauchy data on s 1 remain small over intervals of time longer than the ones given by local existence theory this result extends previous ones obtained by several authors in the semi linear case the main new difficulty one has to cope with is the loss of one derivative coming from the quasi linear character of the problem the main tool used to overcome it is a global paradifferential calculus adapted to the sturm liouville operator with periodic boundary conditions

absolute continuity of the floquet spectrum for a nonlinearly forced harmonic oscillator we prove that the floquet spectrum of the time periodic schrodinger equation corresponding to a mildly nonlinear resonant forcing is purely absolutely continuous for suitably small

time quasi periodic unbounded perturbations of schrodinger operators and kam methods we eliminate by kam methods the time dependence in a class of linear differential equations in l2 subject to an unbounded quasi periodic forcing this entails the pure point nature of the floquet spectrum of the operator h0 ep t for e small here h0 is the one dimensional schrodinger operator p2 v v x x 2 for x the time quasi periodic perturbation p may grow as x 2 2 and the frequency vector is non resonant the proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on kuksin s estimate of solutions of homological equations with non constant coefficients