normalisation holomorphe d algebres de type cartan de champs de vecteurs holomorphes singuliers nous montrons un resultat de normalisation holomorphe d une famille commutative de champs de vecteurs holomorphes au voisinage de leur point singulier commun pour ce faire nous supposons qu une condition diophanti enne portant sur une algebre de lie commutative de champs lineaires associee est satisfaite d autre part nous imposons certaines conditions algebriques sur leur forme normale formelle les champs d une telle famille sauf un peuvent ne pas avoir de partie lineaire a l origine abstract we consider a commutative family of holomorphic vector fields in a neigh bourhood of a common singular point say 0 e cn let g be a commuta tive complex lie algebra of dimension 1 let a1 an e and let us set s g l ai g xi we assume that this lie morphism is diophantine in some sense let x1 be a holomorphic vector fields in a neighbourhood of 0 e cn we assume that its linear part s is regular relative to s that it belongs to s g and has the same formal centralizer as s let x2 xi be holomorphic vector fields vanishing at 0 and commuting with x1 then there exists a formal diffeomorphism of cn o such that the family of vector fields are in normal form in these formal coordinates this means that each element of the family commutes with s we assume that the normal forms of the xi s belong to 0s 0 s g os is the ring of formal first integral of s we also assume that the family of lowest order parts of the normal forms is free over 9s then we show that there exists a holomorphic diffeomorphism of cn 0 which transforms the family into a normal form the elements of the family

spectral statistics for unitary transfer matrices of binary graphs quantum graphs have recently been introduced as model systems to study the spectral statistics of linear wave problems with chaotic classical limits it is proposed here to generalize this approach by considering arbitrary directed graphs with unitary transfer matrices an exponentially increasing contribution to the form factor is identified when performing a diagonal summation over periodic orbit degeneracy classes a special class of graphs so called binary graphs is studied in more detail for these the conditions for periodic orbit pairs to be correlated including correlations due to the unitarity of the transfer matrix can be given explicitly using combinatorial techniques it is possible to perform the summation over correlated periodic orbit pair contributions to the form factor for some low dimensional cases gradual convergence towards random matrix results is observed when increasing the number of vertices of the binary graphs

birkhoff coordinates for the focusing nls equation in this paper we construct birkhoff coordinates for the focusing nonlinear schrodinger equation near the zero solution

on linearization of third order ordinary differential equations a new algorithm for linearization of a third order ordinary differential equation is presented the algorithm consists of composition of two operations reducing order of an ordinary differential equation and using the lie linearization test for the obtained second order ordinary differential equation the application of the algorithm to several ordinary differential equations is given

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

symmetry algebras of third order ordinary differential equations the main result of this paper is the complete classification of the third order ordinary differential equations according to their symmetries the same classification was done for second order by tresse determination des invariants ponctuels de l equation differentielle ordinaire du second ordre y x y y gekronte preisschrift hirzel leipzig 1896 and recently for arbitrary order linear ordinary differential equations the sections preceding the classification consist of a brief description of the concepts and methods along the lines of krause and michel lecture notes phys 382 251 1991 these sections also contain some definitions and a table listing the prolongations of a few vector fields finally two appendices give additional information relevant to equations in real variables and describe how some of the results can be easily generalized to higher orders

quantum graphs a simple model for chaotic scattering we connect quantum graphs with infinite leads and turn them into scattering systems we show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics typical poles delay time and conductance distributions ericson fluctuations and when considered statistically the ensemble of scattering matrices reproduces quite well the predictions of the appropriately defined random matrix ensembles the underlying classical dynamics can be defined and it provides important parameters which are needed for the quantum theory in particular we derive exact expressions for the scattering matrix and an exact trace formula for the density of resonances in terms of classical orbits analogous to the semiclassical theory of chaotic scattering we use this in order to investigate the origin of the connection between random matrix theory and the underlying classical chaotic dynamics being an exact theory and due to its relative simplicity it offers new insights into this problem which is at the forefront of the research in chaotic scattering and related fields

differential invariants for quasi linear and semi linear wave type equations abstract in this paper we consider the class of wave type equations u tt f x t u u xx g x t u u x u t and two special cases of it we derive the equivalence transformations for these equations and using these transformations we obtain differential invariants and invariant equations we employ these invariants or and invariant equations to classify all equations of this general class that can be mapped into a simple linear hyperbolic or into an elliptic equation additional applications are presented

exact solutions via equivalence transformations of variable coefficient fifth order kdv equations abstract in this paper a family of variable coefficient fifth order kdv equations has been considered by using an infinitesimal method based on the determination of the equivalence group differential invariants and invariant equations are obtained invariants provide an alternative way to find equations from the family which may be equivalent to a specific subclass of the same family and the invertible transformation which maps both equivalent equations here differential invariants are applied to obtain exact solutions

on linearization of hyperbolic equations using differential invariants abstract in this paper we consider the general class of hyperbolic equations u x t f x t u u x u t we use equivalence transformations to derive differential invariants for this class and for two subclasses then we employ these invariants to construct equations that can be linearized via local mappings further applications of the differential invariants are given

scattering reflection and impedance of waves in chaotic and disordered systems with absorption we review recent progress in analysing wave scattering in systems with both intrinsic chaos and or disorder and internal losses when the scattering matrix is no longer unitary by mapping the problem onto a nonlinear supersymmetric model we are able to derive closed form analytic expressions for the distribution of reflection probability in a generic disordered system one of the most important properties resulting from such an analysis is statistical independence between the phase and the modulus of the reflection amplitude in every perfectly open channel the developed theory has far reaching consequences for many quantities of interest including local green functions and time delays in particular we point out the role played by absorption as a sensitive indicator of mechanisms behind the anderson localization transition we also provide a random matrix based analysis of s matrix and impedance correlations for various symmetry classes as well as the distribution of transmitted power for systems with broken time reversal invariance completing previous works on the subject the results can be applied in particular to the experimentally accessible impedance and reflection in a microwave or an ultrasonic cavity attached to a system of antennas

non weyl microwave graphs one of the most important characteristics of a quantum graph is the average density of resonances l where l denotes the length of the graph this is a very robust measure it does not depend on the number of vertices in a graph and holds also for most of the boundary conditions at the vertices graphs obeying this characteristic are called weyl graphs using microwave networks that simulate quantum graphs we show that there exist graphs that do not adhere to this characteristic such graphs are called non weyl graphs for standard coupling conditions we demonstrate that the transition from a weyl graph to a non weyl graph occurs if we introduce a balanced vertex a vertex of a graph is called balanced if the numbers of infinite leads and internal edges meeting at a vertex are the same our experimental results confirm the theoretical predictions of e b davies and a pushnitski analysis and pde 4 729 2011 and are in excellent agreement with the numerical calculations yielding the resonances of the networks

on the complete integrability and linearization of nonlinear ordinary differential equations ii third order equations we introduce a method for finding general solutions of third order nonlinear differential equations by extending the modified prelle singer method we describe a procedure to deduce all the integrals of motion associated with the given equation so that the general solution follows straightforwardly from these integrals the method is illustrated with several examples further we propose a powerful method of identifying linearizing transformations the proposed method not only unifies all the known linearizing transformations systematically but also introduces a new and generalized linearizing transformation in addition to the above we provide an algorithm to invert the non local linearizing transformation through this procedure the general solution for the original nonlinear equation can be obtained from the solution of the linear ordinary differential equation

convergence versus integrability in birkhoff normal form we show that any analytically integrable hamiltonian system near an equilibrium point admits a convergent birkhoff normalization the proof is based on a new geometric approach to the topic

linearization criteria for systems of two second order stochastic ordinary differential equations we provide the necessary and sufficient conditions for the linearization of systems of two second order stochastic ordinary differential equations the linearization criteria are given in terms of coefficients of the system followed by some illustrations this paper gives a new treatment for the linearization of two second order stochastic ordinary differential equations and with some examples

long time stability of small finite gap solutions of the cubic nonlinear schrodinger equation on t2 abstract in this paper we study long time stability of a class of nontrivial quasi periodic solutions depending on one spacial variable of the cubic defocusing non linear schrodinger equation on the two dimensional torus we prove that these quasi periodic solutions are orbitally stable for finite but long times provided that their fourier support and their frequency vector satisfy some complicated but explicit condition which we show holds true for most solutions the proof is based on a normal form result more precisely we expand the hamiltonian in a neighborhood of a quasi periodic solution we reduce its quadratic part to diagonal constant coefficients through a kam scheme and finally we remove its cubic terms with a step of nonlinear birkhoff normal form the main difficulty is to impose second and third order melnikov conditions this is done by combining the techniques of reduction in order of pseudo differential operators with the algebraic analysis of resonant quadratic hamiltonians

on the level spacing distribution in quantum graphs we derive a formula for the level spacing probability distribution in quantum graphs we apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases moreover we derive an exact and explicit formula for the level spacing distribution of integrable quantum graphs

experimental investigation of wigner s reaction matrix for irregular graphs with absorption we use tetrahedral microwave networks consisting of coaxial cables and attenuators connected by t joints to make an experimental study of wigner s reaction k matrix for irregular graphs in the presence of absorption from measurements of the scattering matrix s for each realization of the microwave network we obtain distributions of the imaginary and real parts of k our experimental results are in good agreement with theoretical predictions

experimental and numerical investigation of the reflection coefficient and the distributions of wigner s reaction matrix for irregular graphs with absorption we present the results of an experimental and numerical study of the distribution of the reflection coefficient p r and the distributions of the imaginary p v and the real p u parts of the wigner reaction k matrix for irregular fully connected hexagon networks graphs in the presence of strong absorption in the experiment we used microwave networks which were built of coaxial cables and attenuators connected by joints in the numerical calculations experimental networks were described by quantum fully connected hexagon graphs the presence of absorption introduced by attenuators was modeled by optical potentials the distribution of the reflection coefficient p r and the distributions of the reaction k matrix were obtained from measurements and numerical calculations of the scattering matrix s of the networks and graphs respectively we show that the experimental and numerical results are in good agreement with the exact analytic ones obtained within the framework of random matrix theory

a unification in the theory of linearization of second order nonlinear ordinary differential equations in this letter we introduce a new generalized linearizing transformation glt for second order nonlinear ordinary differential equations snodes the well known invertible point ipt and non point transformations npt can be derived as sub cases of the glt a wider class of nonlinear odes that cannot be linearized through npt and ipt can be linearized by this glt we also illustrate how to construct glts and to identify the form of the linearizable equations and propose a procedure to derive the general solution from this glt for the snodes we demonstrate the theory with two examples which are of contemporary interest

on the linearization of some second order odes via contact transformations only those second order ordinary differential equations odes that admit an eight dimensional lie algebra of lie point symmetries are linearizable via a point transformation however all second order odes are linearizable via a contact transformation unfortunately the latter result is not useful as one cannot usually find the contact symmetries of second order odes by analysing realizations of three dimensional lie algebras we show how to explicitly construct the linearization of some second order odes via contact transformations

equivalence classes and linearization of the riccati and abel chain abstract the problem of linearization by point transformations is solved for equations in the generalized riccati and abel chain of order not exceeding the fourth it is shown in particular that nonlinear third order and fourth order equations from the chain are not linearizable by any point transformations the lie pseudo group of equivalence transformations for equations of arbitrary orders from the chain are then found together with expressions for the transformed parameter functions an important subgroup of the group of equivalence transformations found is considered and some associated equivalence classes are exhibited

invariants of two and three dimensional hyperbolic equations abstract we consider linear hyperbolic equations of the form u t t i 1 n u x i x i i 1 n x i x 1 x n t u x i t x 1 x n t u t u x 1 x n t u we derive equivalence transformations which are used to obtain differential invariants for the cases n 2 and n 3 motivated by these results we present the general results for the n dimensional case it appears at least for n 2 that this class of hyperbolic equations admits differential invariants of order one but not of order two we employ the derived invariants to construct interesting mappings between equivalent equations

scattering and transport properties of tight binding random networks we study numerically scattering and transport statistical properties of tight binding random networks characterized by the number of nodes n and the average connectivity we use a scattering approach to electronic transport and concentrate on the case of a small number of single channel attached leads we observe a smooth crossover from insulating to metallic behavior in the average scattering matrix elements the conductance probability distribution w t the average conductance the shot noise power p and the elastic enhancement factor f by varying from small 0 to large 1 values we also show that all these quantities are invariant for fixed n moreover we proposes a heuristic and universal relation between and p and the disorder parameter

ordinary differential equations which linearize on differentiation in this paper we discuss ordinary differential equations odes which linearize upon one or more differentiations although the subject is fairly elementary equations of this type arise naturally in the context of integrable systems

birkhoff coordinates for the toda lattice in the limit of infinitely many particles with an application to fpu abstract in this paper we study the birkhoff coordinates cartesian action angle coordinates of the toda lattice with periodic boundary condition in the limit where the number n of the particles tends to infinity we prove that the transformation introducing such coordinates maps analytically a complex ball of radius r n in discrete sobolev analytic norms into a ball of radius r n with r r 0 independent of n if and only if 2 then we consider the problem of equipartition of energy in the spirit of fermi pasta ulam we deduce that corresponding to initial data of size r n 2 0 r 1 and with only the first fourier mode excited the energy remains forever in a packet of fourier modes exponentially decreasing with the wave number finally we consider the original fpu model and prove that energy remains localized in a similar packet of fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet the proof of the theorem on birkhoff coordinates is based on a new quantitative version of a vey type theorem by kuksin and perelman which could be interesting in itself

symmetries and integrability of a fourth order euler bernoulli beam equation the complete symmetry group classification of the fourth order euler bernoulli ordinary differential equation where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement is obtained we perform the lie and noether symmetry analysis of this problem in the lie analysis the principal lie algebra which is one dimensional extends in four cases viz the linear exponential general power law and a negative fractional power law it is further shown that two cases arise in the noether classification with respect to the standard lagrangian that is the linear case for which the noether algebra dimension is one less than the lie algebra dimension as well as the negative fractional power law in the latter case the noether algebra is three dimensional and is isomorphic to the lie algebra which is sl 2 r this exceptional case although admitting the nonsolvable algebra sl 2 r remarkably allows for a two parameter family of exact solut

explicitly solvable cases of one dimensional quantum chaos we identify a set of quantum graphs with unique and precisely defined spectral properties called regular quantum graphs although chaotic in their classical limit with positive topological entropy regular quantum graphs are explicitly solvable the proof is constructive we present exact convergent periodic orbit expansions for individual energy levels thus obtaining an analytical solution for the spectrum of regular quantum graphs that is complete explicit and exact

reducing nonideal to ideal coupling in random matrix description of chaotic scattering application to the time delay problem we write explicitly a transformation of the scattering phases reducing the problem of quantum chaotic scattering for systems with m statistically equivalent channels at nonideal coupling to that for ideal coupling unfolding the phases by their local density leads to universality of their local fluctuations for large m a relation between the partial time delays and diagonal matrix elements of the wigner smith matrix is revealed for ideal coupling this helped us in deriving the joint probability distribution of partial time delays and the distribution of the wigner time delay

invariants of a family of third order ordinary differential equations we consider a family of third order ordinary differential equations odes with the cubic nonlinearity in second order derivative which is closed with respect to contact transformations using lie s infinitesimal method we construct the basis of differential invariants for this family of equations and the operators of invariant differentiation invariants provide a simple way of finding the equations which may be equivalent to a given nonlinear ode and the transformation connecting two equivalent equations some examples are given to illustrate the results obtained in the paper

on the generalizations of the kummer schwarz equation abstract we carry out complete group classifications of two non homogeneous generalizations of the kummer schwarz equation in particular we prove that the symmetry algebra for one of them is s l 2 r s l 2 r and that it can be mapped by an equivalence transformation to the homogeneous kummer schwarz equation for this purpose we calculate the continuous equivalence group for that equation we also study the linearization of the considered equations

experimental simulation of quantum graphs by microwave networks we present the results of experimental and theoretical study of irregular tetrahedral microwave networks consisting of coaxial cables annular waveguides connected by t joints the spectra of the networks were measured in the frequency range 0 0001 16 ghz in order to obtain their statistical properties such as the integrated nearest neighbor spacing distribution and the spectral rigidity d3sld the comparison of our experimental and theoretical results shows that microwave networks can simulate quantum graphs with time reversal symmetry in particular we use the spectra of the microwave networks to study the periodic orbits of the simulated quantum graphs we also present experimental study of directional microwave networks consisting of coaxial cables and faraday isolators for which the time reversal symmetry is broken in this case our experimental results indicate that spectral statistics of directional microwave networks deviate from predictions of gaussian orthogonal ensembles in random matrix theory approaching especially for small eigenfrequency spacing s results for gaussian unitary ensembles experimental results are supported by the theoretical analysis of directional graphs

parametric correlations of the energy levels of ray splitting billiards parameter dependent statistical properties of the spectra of ray splitting billiards are studied experimentally and theoretically the autocorrelation functions c x and c omega x of level velocities as well as the generalized conductance c 0 are calculated for two different classically chaotic ray splitting billiards experimentally a modified sinai ray splitting billiard is studied consisting of a thin microwave rectangular cavity with two quarter circle shaped teflon inserts the length of the cavity serves as the experimentally adjustable parameter for the theoretical estimates of the parametric correlations we compute the quantum spectrum of a scaling triangular ray splitting billiard our experimental and numerical results are compared with each other and with the predictions of random matrix theory

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

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

parameter dependent spectral statistics of chaotic quantum graphs neumann versus circular orthogonal ensemble boundary conditions the parameter dependent spectral statistics of totally connected quantum graphs with n 4 30 vertices such as the parametric velocities correlation functions and the distribution of curvatures are studied the inverse participation ratio ipr an important measure of localization effects was also numerically investigated in the calculations we successfully used two different theoretical approaches the first approach was based on the graphs eigenenergies and wave functions calculations while the second one used the eigenphases and the eigenvectors of the bond scattering matrix s k we considered graphs with neumann and circular orthogonal ensemble coe boundary conditions we show that in contrast to large neumann graphs for which the departure of many parameter dependent spectral statistics from the random matrix theory rmt predictions is observed for large coe graphs the spectral statistics and ipr are in good agreement with the rmt predictions

symmetry lie algebras of nth order ordinary differential equations abstract we show that an nth n 3 order linear ordinary differential equation has exactly one of n 1 n 2 or n 4 the maximum point symmetries the lie algebras corresponding to the respective numbers of point symmetries are obtained then it is shown that a necessary and sufficient conditon for an nth n 3 order equation to be linearizable via a point transformation is that it must admit the n dimensional abelian algebra na1 a1 a1 a1 we discuss in detail the symmetry realizations of n 1 a1 s a1 finally we prove that an nth n 3 order equation q n h t q qn 1 cannot admit exactly an n 3 dimensional algebra of point symmetries which is a subalgebra of na1 gl 2 r

linearization of third order ordinary differential equations by point and contact transformations we present here the solution of the problem on linearization of third order ordinary differential equations by means of point and contact transformations we provide in explicit forms the necessary and sufficient conditions for linearization the equations for determining the linearizing point and contact transformations as well as the coefficients of the resulting linear equations

nonlocal transformations and linearization of second order ordinary differential equations the class of nonlinear second order equations that are linearizable by means of generalized sundman transformations s transformations is identified as the class of equations admitting first integrals that are polynomials of first degree in the first order derivative this class is also characterized in terms of the coefficients of the equations and constructive methods to derive the linearizing s transformations are presented only the equations of a well defined subclass can also be linearized by invertible point transformations these invertible point transformations can be constructed by using the algorithms for the calculation of linearizing s transformations several examples illustrate that both types of linearization are strictly different

families of line graphs and their quantization any directed graph g with n vertices and j edges has an associated line graph l g where the j edges form the vertices of l g we show that the non zero eigenvalues of the adjacency matrices are the same for all graphs of such a family ln g we give necessary and sufficient conditions for a line graph to be quantisable and demonstrate that the spectra of associated quantum propagators follow the predictions of random matrices under very general conditions line graphs may therefore serve as models to study the semiclassical limit of large matrix size of a quantum dynamics on graphs with fixed classical behaviour

conditions for linearization of a projectable system of two second order ordinary differential equations a new method for linearizing a system of ordinary differential equations is introduced the method is applied to a system of two second order ordinary differential equations it is shown that for a particular class of equations the method gives more general linearization criteria than linearization via point transformations examples of systems of equations which are not linearizable via point transformations but linearizable by the new method are given

the characterization of third order ordinary differential equations admitting a transitive fiber preserving point symmetry group abstract third order ordinary differential equations admitting a transitive symmetry group of fiber preserving point transformations are completely characterized we give the general expressions of these equations as well as the canonical equations representing the equivalence classes the symmetry groups admitted by these equations are also characterized in some cases we give the transformation taking the canonical equation into the general one the linearization problem for third order equations is solved

new q conditional symmetries and exact solutions of some reaction diffusion convection equations arising in mathematical biology abstract a theorem giving a complete description of q conditional symmetries of a class of nonlinear reaction diffusion convection equations is proved furthermore the q conditional symmetries obtained and the method of additional generating conditions are applied for finding exact solutions of the generalized fisher fitzhugh nagumo and kolmogorov petrovskii piskunov equations the symmetries and solutions constructed are compared with those obtained by other authors in particular it was established that the known travelling wave solutions of these equations are particular cases of more general non lie solutions the relation between q conditional symmetries and generalized conditional symmetries is also shown

the third order ordinary differential equations admitting the painleve i and ii as first integrals third order ordinary differential equations admitting the painleve equations pi and pii as first integrals are completely characterized this is done by determining necessary and sufficient conditions for these equations to belong to the equivalence class of certain canonical equations this characterization also includes equations admitting first integrals which are equivalent to pi and pii the general form of these equations is given in some cases we determine the transformation taking the equation into a canonical one

linearization of fourth order ordinary differential equations by point transformations the solution of the problem on linearization of fourth order equations by means of point transformations is presented here we show that all fourth order equations that are linearizable by point transformations are contained in the class of equations which is linear in the third order derivative we provide the linearization test and describe the procedure for obtaining the linearizing transformations as well as the linearized equation for ordinary differential equations of order greater than 4 we obtain necessary conditions which separate all linearizable equations into two classes

classical 1d maps quantum graphs and ensembles of unitary matrices we study a certain class of classical one dimensional piecewise linear maps for these systems we introduce an infinite family of markov partitions in equal cells the symbolic dynamics generated by these systems is described by bi stochastic doubly stochastic matrices we analyse the structure of graphs generated from the corresponding symbolic dynamics we demonstrate that the spectra of quantized graphs corresponding to the regular classical systems have locally poissonian statistics while quantized graphs derived from classically chaotic systems display statistical properties characteristic of the circular unitary ensemble even though the corresponding unitary matrices are sparse

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