construction of invariant whiskered tori by a parameterization method part ii quasi periodic and almost periodic breathers in coupled map lattices we construct quasi periodic and almost periodic solutions for coupled hamiltonian systems on an infinite lattice which is translation invariant the couplings can be long range provided that they decay moderately fast with respect to the distance for the solutions we construct most of the sites are moving in a neighborhood of a hyperbolic fixed point but there are oscillating sites clustered around a sequence of nodes the amplitude of these oscillations does not need to tend to zero in particular the almost periodic solutions do not decay at infinity we formulate an invariance equation solutions of this equation are embeddings of an invariant torus on which the motion is conjugate to a rotation we show that if there is an approximate solution of the invariance equation that satisfies some non degeneracy conditions there is a true solution close by the proof of this emph a posteriori theorem is based on a nash moser iteration which does not use transformation theory simpler versions of the scheme were developed in e fontich r de la llave y sire emph j differential equations bf 246 3136 2009 one technical tool important for our purposes is the use of weighted spaces that capture the idea that the maps under consideration are local interactions using these weighted spaces the estimates of iterative steps are similar to those in finite dimensional spaces in particular the estimates are independent of the number of nodes that get excited using these techniques given two breathers we can place them apart and obtain an approximate solution which leads to a true solution nearby by repeating the process infinitely often we can get solutions with infinitely many frequencies which do not tend to zero at infinity

existence of quasi periodic solutions of fast excited van der pol mathieu duffing equation the van der pol mathieu duffing equation x 02 h1cos 1t h2cos 2t x x2 x h3x3 h4 32cosxcos 3t is considered in this paper where h1 h2 h3 h4 1 2 are small parameters 0 the frequency 3 is large compared to 1 and 2 the above parameters are real for 0 we use kam kolmogorov arnold moser theory to prove that the van der pol mathieu duffing equation possesses quasi periodic solutions for most of the parameters 0 1 2 3 it verifies some phenomenon of fahsi and belhaq commun nonlinear sci 14 244 253 2009 and can be regarded as a extension of abouhazim et al nonlinear dyn 39 395 409 2005

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

quasi periodic solutions for perturbed autonomous delay differential equations abstract this work discusses the persistence of quasi periodic solutions for delay differential equations we prove that the perturbed system possesses a quasi periodic solution under appropriate hypotheses if an unperturbed linear system has quasi periodic solutions we extend some well known results on ordinary differential equations to delay differential equations

well posedness and stability of the repairable system with n failure modes and one standby unit the well posedness and stability of the repairable system with n failure modes and one standby unit were discussed by applying the c0 semigroups theory of function analysis firstly the integro differential equations described the system were transformed into some abstract cauchy problem of banach space secondly the system operator generates positive contractive c0 semigroups t t and so the well posedness of the system was obtained finally the spectral distribution of the system operator was analyzed it was proven that 0 is strictly dominant eigenvalue of the system operator and the dynamic solution of the system converges to the steady state solution the steady state solution was shown to be the eigenvector of the system operator corresponding to the eigenvalue 0 at the same time the dynamic solution exponentially converges to the steady state solution

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

small divisors with spatial structure in infinite dimensional hamiltonian systems a general perturbation theory of the kolmogorov arnold moser type is described concerning the existence of infinite dimensional invariant tori in nearly integrable hamiltonian systems the key idea is to consider hamiltonians with aspatial structure and to express all quantitative aspects of the theory in terms of rather general weight functions on such structures this approach combines great flexibility with an effective control of the vrious interactions in infinite dimensional systems

on the persistence of quasi periodic invariant tori for double hopf bifurcation of vector fields abstract we analyze the persistence of quasi periodic invariant 2 and 3 tori for the double hopf hopf hopf bifurcation by using the kam method we prove that in a sufficiently small neighborhood of the bifurcation point the full system has quasi periodic 2 tori for most of the parameter sets where its truncated normal form possesses 2 tori under appropriate conditions we obtain that the full system also has quasi periodic 3 tori for most parameters near the hopf bifurcation curve of its truncated normal form and along the direction of the bifurcation and these 3 tori bifurcate from invariant 2 tori we also give concrete formulas on the existence of quasi periodic invariant 2 and 3 tori which are based on coefficients of the truncated normal form

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

construction of quasi periodic breathers via kam technique by developing a kam theorem which involves an infinitely multiple normal frequency it is shown that there are plenty of breathers quasi periodic in time and super exponentially localized in space for the networks of weakly coupled oscillators this answers an open problem by aubry a2 in case the linearized system has no continuous spectrum

existence and stability of quasi periodic breathers in networks of ginzburg landau oscillators abstract it is shown that there is an attracting quasi periodic breather in networks of ginzburg landau oscillators

quasi periodic breathers in hamiltonian networks of long range coupling this work is concerned with hamiltonian networks of weakly and long range coupled oscillators with either variable or constant on site frequencies we derive an infinite dimensional kam like theorem by which we establish that given any n sites of the lattice there is a positive measure set of small amplitude quasi periodic breathers solutions of the hamiltonian network that are quasi periodic in time and exponentially localized in space having n frequencies which are only slightly deformed from the on site frequencies

breathers in nonlinear lattices existence linear stability and quantization abstract it has been proved by the principle of anti integrable limit renamed here anti continuous limit and the implicit function theorem that under rather weak hypotheses there exist spatially localized and time periodic solutions breathers in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong the models can be translationally invariant or not and be of any dimension there are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions multibreathers the whole set of these solutions can be labeled by a discrete coding sequence the condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called phonobreather which correspond to special codes which can be also spatially chaotic and where no oscillator is at rest these results extend almost without change to models with coupled rotors the existence of rotobreathers which are exact solutions consisting of a single rotor rotating while the other rotors oscillate is proved there are also multirotobreather solutions and phono rotobreathers etc it is also shown that the phase of the single breathers constituting a multibreather or multirotobreather solution can be submitted to a phase torsion these new dynamical structures carry a nonzero energy flow in models with two dimensions and more the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow the dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti continuous limit the existence of two kinds of polarobreathers which are spatially exponentially localized dynamical solutions is proved for a small enough electronic transfer integral the polarobreathers of the first kind include the standard static polaron the lattice configuration is static time independent when the electron is in an excited state i e the electronic oscillator oscillates periodically they can be spatially chaotic a polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation the most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site the standard floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the newton operator involved in the implicit function theorem the krein theory of bifurcations is reinterpreted from this point of view the linear stability of the single breather and rotobreather solutions is then proved at least at small enough coupling finally it is shown that in the quantum version of translationally invariant classical models where there cannot exist in principle any strictly localized excitation there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers the bandwidth of these excitations goes exponentially to zero with the number of bound phonons this theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist these numerical applications are currently written elsewhere

analytic solutions of nonlinear elliptic equations on rectangular tori abstract in this paper we consider the nonlinear elliptic equations on rectangular tori using methods in the study of kam theory and anderson localization we prove that these equations admit many analytic solutions

solutions of einstein s equations with conformal symmetries and a contact slice abstract first we show that the initial space like slice of the generalized robertson walker grw space time is einstein if and only if the electric part of the space time weyl conformal tensor vanishes we also show that the purely spatial component of the space time weyl tensor vanishes if and only if the initial slice has constant curvature then assuming that a spatially complete synchronous space time solution of einstein s equations admits a conformal vector field whose normal component is non constant and spatial component is conformal on each space like slice we show that each slice is conformal to i a euclidean space ii a sphere or iii a hyperbolic space or iv riemannian product of an open interval and and a riemannian space finally under the preceding hypotheses if a slice has a k contact metric then it is isometric to a unit sphere

dynamics of breathers in discrete nonlinear schro dinger models abstract we review some recent results concerning the existence and stability of spatially localized and temporally quasiperiodic non stationary excitations in discrete nonlinear schrodinger dnls models in two dimensions we show the existence of linearly stable stationary and non stationary localized vortex like solutions we also show that stationary on site localized excitations can have internal breathing modes which are spatially localized and symmetric the excitation of these modes leads to slowly decaying quasiperiodic oscillations finally we show that for some generalizations of the dnls equation where bistability occurs a controlled switching between stable states is possible by exciting an internal breathing mode above a threshold value

construction of quasi periodic solutions of delay differential equations via kam techniques abstract this work focuses on the existence of quasi periodic solutions for linear autonomous delay differential equation under quasi periodic time dependent perturbation near an elliptic hyperbolic equilibrium point using the time 1 map of the solution operator newton iteration scheme space splitting and kam techniques it is shown that under appropriate hypothesis there exist quasi periodic solutions with the same frequencies as the perturbation for most parameters we show that if the delay differential equation is analytic we obtain analytic parameterizations of the solutions

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

conformal symmetries of einstein s field equations and initial data this paper examines the initial data for the evolution of the space time solution of einstein s equations admitting a conformal symmetry under certain conditions on the extrinsic curvature of the initial complete spacelike hypersurface and sectional curvature of the space time with respect to sections containing the normal vector field we have shown that the initial hypersurface is conformally diffeomorphic to a sphere or a flat space or a hyperbolic space or the product of an open real interval and a complete 2 manifold it has been further shown that if the initial hypersurface is compact then it is conformally diffeomorphic to a sphere finally the conformal symmetries of a generalized robertson walker space time have been described

conformal killing vector fields on spacetime solutions of einstein s equations and initial data abstract working within the adm formalism we have established results characterizing a conformal killing vector ckv field on a spacetime solution of einstein s field equations and the initial data on a spacelike hypersurface we have considered the cases i the ckv is tangential to spacelike slices ii the ckv is closed and the initial hypersurface is totally umbilical and has constant mean curvature cmc and iii spacetime is a perfect fluid has a non vanishing ckv transverse to a compact cmc hypersurface orthogonal to the 4 velocity finally we show that a non null ckv that is also a geodesic vector field is either homothetic or locally closed

anti integrability in dynamic and variational problems abstract we show using a series of examples that the existence of a singular limit called anti integrable can be useful for proving the existence of chaotic solutions and elucidating some of their properties leading to some new and unexpected results in particular the existence proof of chaotic trajectories in large classes of dynamical systems with discrete time becomes straightforward the method applies even for maps in infinite dimension symplectic or not similar ideas can be used for proving the existence of breathers or self localised modes and of chaotic distributions of breathers in any array of weakly coupled non linear oscillators with continuous time the existence of insulating bipolaronic structures in electron phonon systems electronic self localisation at large electron phonon coupling can also be proven on the same basis

homothetic and conformal symmetries of solutions to einstein s equations we present several results about the nonexistence of solutions of einstein s equations with homothetic or conformal symmetry we show that the only spatially compact globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature and also admitting an infinitesimal proper homothetic symmetry are everywhere locally flat this assumes that the matter fields either obey certain energy conditions or are the yang mills or massless klein gordon fields we find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions we show that if the dominant energy condition is assumed then minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation in other words with the exceptions cited homothetic or conformal killing fields are in fact killing in spatially compact or asymptotically flat spactimes in the conformal procedure for solving the initial value problem we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry