anderson transition in a three dimensional kicked rotor we investigate anderson localization in a three dimensional 3d kicked rotor by a finite size scaling analysis we identify a mobility edge for a certain value of the kicking strength k k c for k k c dynamical localization does not occur all eigenstates are delocalized and the spectral correlations are well described by wigner dyson statistics this can be understood by mapping the kicked rotor problem onto a 3d anderson model am where a band of metallic states exists for sufficiently weak disorder around the critical region k approximately k c we carry out a detailed study of the level statistics and quantum diffusion in agreement with the predictions of the one parameter scaling theory opt and with previous numerical simulations the number variance is linear level repulsion is still observed and quantum diffusion is anomalous with proportional t 2 3 we note that in the 3d kicked rotor the dynamics is not random but deterministic in order to estimate the differences between these two situations we have studied a 3d kicked rotor in which the kinetic term of the associated evolution matrix is random a detailed numerical comparison shows that the differences between the two cases are relatively small however in the deterministic case only a small set of irrational periods was used a qualitative analysis of a much larger set suggests that deviations between the random and the deterministic kicked rotor can be important for certain choices of periods heuristically it is expected that localization effects will be weaker in a nonrandom potential since destructive interference will be less effective to arrest quantum diffusion however we have found that certain choices of irrational periods enhance anderson localization effects

gamow vectors and borel summability in a class of quantum systems we analyze the detailed time dependence of the wave function x t for one dimensional hamiltonians h partial x 2 v x where v for example modeling barriers or wells and x 0 are compactly supported

quasi periodic solutions to nonlinear beam equations on compact lie groups with a multiplicative potential abstract the goal of this work is to study the existence of quasi periodic solutions to nonlinear beam equations with a multiplicative potential the nonlinearity is required to only finitely differentiable and the frequency is along a pre assigned direction the result holds on any compact lie group or homogeneous manifold with respect to a compact lie group which includes standard torus t d special orthogonal group s o d special unitary group s u d spheres s d and the real and complex grassmannians the proof is based on a differentiable nash moser iteration scheme

growth of sobolev norms in linear schrodinger equations with quasi periodic potential in this paper we consider the following problem let iu t u v x t u 0 be a linear schrodinger equation periodic boundary conditions where v is a real bounded real analytic potential which is periodic in x and quasi periodic in t with diophantine frequency vector denote s t the corresponding flow map thus s t preserves the l 2 norm and our aim is to study its behaviour on h s t d s 0 our main result is the growth in time is at most logarithmic thus if h s then more precisely is proven in 1d and 2d when v is small we also exhibit examples showing that a growth of higher sobolev norms may occur in this context and is thus essentially best possible

energy supercritical nonlinear schrodinger equations quasiperiodic solutions we construct time quasiperiodic solutions to the energy supercritical nonlinear schrodinger equations on the torus in arbitrary dimensions this introduces a new approach which could have general applicability

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

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

invariant tori of full dimension for a nonlinear schrodinger equation abstract in this paper we consider the one dimensional nonlinear schrodinger equation i u t u x x m u f u 2 u 0 with periodic boundary conditions or dirichlet boundary conditions where f is a real analytic function in some neighborhood of the origin satisfying f 0 0 f 0 0 we prove that for each given constant potential m when the frequencies as a function of the amplitudes can be regarded as the independent parameters the equation admits a whitney smooth family of small amplitude time almost periodic solutions with all frequencies the proof is based on a birkhoff normal form reduction and an improved version of the kam theorem

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

nonlinear wave and schrodinger equations on compact lie groups and homogeneous spaces we develop linear and nonlinear harmonic analysis on compact lie groups and homogeneous spaces relevant for the theory of evolutionary hamiltonian pdes a basic tool is the theory of the highest weight for irreducible representations of compact lie groups this theory provides an accurate description of the eigenvalues of the laplace beltrami operator as well as the multiplication rules of its eigenfunctions as an application we prove the existence of cantor families of small amplitude time periodic solutions for wave and schrodinger equations with differentiable nonlinearities we apply an abstract nash moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the laplace operator we provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on banach scales of sobolev functions

kam tori of hamiltonian perturbations of 1d linear beam equations abstract in this paper one dimensional 1d nonlinear beam equations u tt u xxxx mu f u with hinged boundary conditions are considered the nonlinearity f is an analytic odd function and f u o u3 it is proved that for all real parameters m 0 but a set of small lebesgue measure the above equation admits small amplitude quasi periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system the proof is based on infinite dimensional kam theory developed by kuksin lecture notes in mathematics vol 1556 springer berlin 1993 wayne commun math phys 127 1990 479 528 poschel ann sc norm sup pisa cl sci 23 1996 119 148

energy growth in schrodinger s equation with markovian forcing schrodinger s equation is considered on a one dimensional torus with time dependent potential v t v x t where v is an even trigonometric polynomial and x t is a stationary markov process it is shown that when the coupling constant is sufficiently small the average kinetic energy grows as the square root of time more generally the h s norm of the wave function is shown to behave as t s 4

growth of sobolev norms for time dependent periodic schrodinger equations with sublinear dispersion abstract in this paper we consider schrodinger equations with sublinear dispersion relation on the one dimensional torus t r 2 z more precisely we deal with equations of the form t u i v t u where v t is a quasi periodic in time self adjoint pseudo differential operator of the form v t v t x d m w t 0 m 1 d x x v is a smooth quasi periodic in time function and w is a quasi periodic time dependent pseudo differential operator of order strictly smaller than m under suitable assumptions on v and w we prove that if satisfies some non resonance conditions the solutions of the schrodinger equation t u i v t u grow at most as t t for any 0 the proof is based on a reduction to constant coefficients up to smoothing remainders of the vector field i v t which uses egorov type theorems and pseudo differential calculus the homological equations arising in the reduction procedure involve both time and space derivatives since the dispersion relation is sublinear such equations can be solved by imposing some melnikov non resonance conditions on the frequency vector

a local version of karamardian s theorem on lower semicontinuous strictly quasiconvex functions abstract it is known that lower semicontinuous strictly quasiconvex functions are quasiconvex it is shown that semilocally strictly quasiconvex functions are semilocally quasiconvex on a locally star shaped domain if some condition of semilocal lower semicontinuity is satisfied these things are analyzed in some detail the functions involved have a connected quasiorder as their range as a motivation it is shown that a multiobjective function is a quasiconvex and strictly quasiconvex with respect to the lexminmax order if its components have the same properties there is a discussion of some applications to multiobjective decision particularly with lexminmax as the underlying decision criterion

quasi periodic solutions of two dimensional schrodinger equations with quasi periodic forcing abstract this work focuses on two dimensional 2 d quasi periodically forced nonlinear schrodinger equations this means studying i u t u u e t u u 2 u 0 0 x t 2 t r with periodic boundary conditions where e is a small positive parameter t is a real analytic quasi periodic function in t with frequency vector 1 2 m it is shown that under suitable hypothesis on t there are many quasi periodic solutions for the above equation via kam theory

degenerate lower dimensional tori in hamiltonian systems abstract we study the persistence of lower dimensional tori in hamiltonian systems of the form h x y z y 1 2 z m z e p x y z where x y z t n r n r 2 m e is a small parameter and m can be singular we show under a weak melnikov nonresonant condition and certain singularity removing conditions on the perturbation that the majority of unperturbed n tori can still survive from the small perturbation as an application we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable properly degenerate hamiltonian system for which neither the kolmogorov nor the g nondegenerate condition is satisfied

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

quasi periodic solutions in a nonlinear schrodinger equation abstract in this paper one dimensional 1d nonlinear schrodinger equation i u t u x x m u u 4 u 0 with the periodic boundary condition is considered it is proved that for each given constant potential m and each prescribed integer n 1 the equation admits a whitney smooth family of small amplitude time quasi periodic solutions with n diophantine frequencies the proof is based on a partial birkhoff normal form reduction and an improved kam method

asymptotic behavior of the nonlinear schrodinger equation with harmonic trapping we consider the cubic nonlinear schrodinger equation with harmonic trapping on d 1 d 5 in the case when all directions but one are trapped aka cigar shaped trap we prove modified scattering and construct modified wave operators for small initial and final data respectively the asymptotic behavior turns out to be a rather vigorous departure from linear scattering and is dictated by the resonant system of the nls equation with full trapping on d 1 in the physical dimension d 3 this system turns out to be exactly the cr equation derived by faou germain and the first author as the large box limit of the resonant nls equation in the homogeneous zero potential setting the special dynamics of the latter equation combined with the above modified scattering results allow us to justify and extend some physical approximations in the theory of bose einstein condensates in cigar shaped traps 2015 wiley periodicals inc

on time dependent schrodinger equations global well posedness and growth of sobolev norms in this paper we consider time dependent schrodinger linear pdes of the form i t l t where l t is a continuous family of self adjoint operators we give conditions for well posedness and polynomial growth for the evolution in abstract sobolev spaces if l t h v t where v t is a perturbation smooth in time and h is a self adjoint positive operator whose spectrum can be enclosed in spectral clusters whose distance is increasing we prove that the sobolev norms of the solution grow at most as t when t for any 0 if v t is analytic in time we improve the bound to log t for some 0 the proof follows the strategy due to howland joye and nenciu of the adiabatic approximation of the flow we recover most of known results and obtain new estimates for several models including 1 degree of freedom schrodinger operators on r and schrodinger operators on zoll manifolds

quasi periodic solutions of nonlinear schrodinger equations of higher dimension abstract it is shown that there are plenty of quasi periodic solutions of nonlinear schrodinger equations of higher spatial dimension where the dimension of the frequency vectors of the quasi periodic solutions are equal to that of the space

generalizations of convex and related functions abstract the concept of semilocally convexity was recently introduced by ewing 2 this paper defines semilocally quasiconvex semi locally pseudoconvex and other related functions and investigates some of their properties

on injectivity of semimodules over additively idempotent division semirings and chain mv semirings abstract in this paper we give a characterization of injective semimodules over additively idempotent semirings consequently we provide a complete description of injective semimodules over the semifield of tropical integers and give an explicit construction of the injective hulls of semimodules over chain division semirings we also give a criterion for self injective mv semirings with an atomic boolean center and describe the structure of finitely generated injective semimodules over finite mv semirings as well as we show that every complete mv semiring with an atomic boolean center is an exact semiring which is defined by a hahn banach type separation property on semimodules arising in the tropical case from the phenomenon of tropical matrix duality moreover we show that complete boolean algebras are precisely the mv semirings in which every principal ideal is injective

localization for a class of one dimensional quasi periodic schrodinger operators we prove for small and satisfying a certain diophantine condition the operator h varepsilon 2 delta frac 1 2 pi cos 2 pi j alpha theta j in mathbb z has pure point spectrum for almost all a similar result is established at low energy for h frac d 2 dx 2 k 2 cos 2 pi x cos 2 pi alpha x theta providedk is sufficiently large

kam for quasi linear and fully nonlinear forced perturbations of airy equation we prove the existence of small amplitude quasi periodic solutions for quasi linear and fully nonlinear forced perturbations of the linear airy equation for hamiltonian or reversible nonlinearities we also prove their linear stability the key analysis concerns the reducibility of the linearized operator at an approximate solution which provides a sharp asymptotic expansion of its eigenvalues for quasi linear perturbations this cannot be directly obtained by a kam iteration hence we first perform a regularization procedure which conjugates the linearized operator to an operator with constant coefficients plus a bounded remainder these transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo differential operators at this point we implement a nash moser iteration with second order melnikov non resonance conditions which completes the reduction to constant coefficients

periodic solutions for completely resonant nonlinear wave equations with dirichlet boundary conditions we consider the nonlinear string equation with dirichlet boundary conditions utt uxx u with u u3 o u5 odd and analytic 0 and we construct small amplitude periodic solutions with frequency for a large lebesgue measure set of close to 1 this extends previous results where only a zero measure set of frequencies could be treated the ones for which no small divisors appear the proof is based on combining the lyapunov schmidt decomposition which leads to two separate sets of equations dealing with the resonant and non resonant fourier components respectively the q and the p equations with resummation techniques of divergent powers series allowing us to control the small divisors problem the main difficulty with respect to the nonlinear wave equations utt uxx mu u m 0 is that not only the p equation but also the q equation is infinite dimensional

pure dimension and projectivity of tropical polytopes abstract we study how geometric properties of tropical convex sets and polytopes which are of interest in many application areas manifest themselves in their algebraic structure as modules over the tropical semiring our main results establish a close connection between pure dimension of tropical convex sets and projectivity in the sense of ring theory these results lead to a geometric understanding of idempotency for tropical matrices as well as their direct interest our results suggest that there is substantial scope to apply ideas and techniques from abstract algebra in particular ring theory in tropical geometry

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

a kam theorem for the defocusing nls equation in this paper we prove a kam theorem for the defocusing nls equation in one space dimension with periodic boundary conditions the novelty of our result is that it is valid not only near the zero solution but on the entire sobolev space h n t c with n is an element of z 1 in particular the invariant tori which persist under small hamiltonian perturbations might be far away from the zero potential c 2011 elsevier inc all rights reserved

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

on reducibility of schrodinger equations with quasiperiodic in time potentials we prove that a linear d dimensional schrodinger equation with an x periodic and t quasiperiodic potential reduces to an autonomous equation for most values of the frequency vector the reduction is made by means of a non autonomous linear transformation of the space of x periodic functions this transformation is a quasiperiodic function of t

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

pseudolinearity and efficiency first order and second order characterizations of pseudolinear functions are derived for a nonlinear programming problem involving pseudolinear functions only it is proved that every efficient solution is properly efficient under some mild conditions

cantor families of periodic solutions for wave equations via a variational principle abstract we prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a cantor set of asymptotically full measure via a variational principle a lyapunov schmidt decomposition reduces the problem to a finite dimensional bifurcation equation variational in nature defined on a cantor set of non resonant parameters the cantor gaps are due to small divisors phenomena to solve the bifurcation equation we develop a suitable variational method in particular we do not require the typical arnold non degeneracy condition of the known theory on the nonlinear terms as a consequence our existence results hold for new generic sets of nonlinearities

full dimension tori of schrodinger equation we explore the structure of the toplitz lipschitz operator l h eliasson and s b kuksin ann math 172 371 435 2010 and define the class of semi toplitz function we then construct many almost periodic solutions of the schrodinger equation with a weak decay operator

on pseudolinear functions abstract in this paper the structure of a smooth pseudolinear function is investigated and the general form of the gradient is given explicitly

ionization of coulomb systems in mathbb r 3 by time periodic forcings of arbitrary size we analyze the long time behavior of solutions of the schrodinger equation i psi t delta b r v t x psi x in mathbb r 3 r x describing a coulomb system subjected to a spatially compactly supported time periodic potential v t x v t 2 x with zero time average

periodic solutions of nonlinear wave equations with general nonlinearities we present a variational principle for small amplitude periodic solutions with fixed frequency of a completely resonant nonlinear wave equation existence and multiplicity results follow by min max variational arguments

kam for the derivative nonlinear schrodinger equation with periodic boundary conditions abstract this paper is concerned with the derivative nonlinear schrodinger equation with periodic boundary conditions i u t u x x i f u 2 u x 0 x t r 2 z where f is real analytic in some neighborhood of the origin in c f 0 0 and f 0 0 we show the above equation possesses cantor families of smooth quasi periodic solutions of small amplitude the proof is based on an infinite dimensional kam theorem for unbounded perturbation vector fields

moser s theorem for lower dimensional tori moser s c version of kolmogorov s theorem on the persistence of maximal quasi periodic solutions for nearly integrable hamiltonian system is extended to the persistence of non maximal quasi periodic solutions corresponding to lower dimensional elliptic tori of any dimension n between one and the number of degrees of freedom the theorem is proved for hamiltonian

bounded sobolev norms for linear schrodinger equations under resonant perturbations abstract we prove that the sobolev norms for a 1 d periodic schrodinger equation remain bounded under small resonant perturbations

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

periodic solutions for a class of nonlinear partial differential equations in higher dimension we prove the existence of periodic solutions in a class of nonlinear partial differential equations including the nonlinear schrodinger equation the nonlinear wave equation and the nonlinear beam equation in higher dimension our result covers cases of completely resonant equations where the bifurcation equation is infinite dimensional such as the nonlinear schrodinger equation with zero mass for which solutions which at leading order are wave packets are shown to exist

on growth of sobolev norms in time dependent linear schrodinger equations on spheres abstract we show that solutions of linear schrodinger equations with a time dependent gevrey potential on spheres have at most logarithmically growing sobolev norms we do not use any specific knowledge of the eigenfunctions of the laplacian beltrami operator which is the main difference with our previous paper

projective modules over polyhedral semirings abstract i classify projective modules over idempotent semirings that are free on a monoid the analysis extends to the case of the semiring of convex piecewise affine functions on a polyhedron for which projective modules correspond to convex families of weight polyhedra for the general linear group

semilocal pseudolinearity and efficiency abstract semilocally pseudolinear functions are defined and their first order characterization is obtained for a multiobjective programming problem involving semilocally pseudolinear functions conditions of efficiency and proper efficiency are discussed

quasi periodic solutions for fully nonlinear forced reversible schrodinger equations abstract in this paper we consider a class of fully nonlinear forced and reversible schrodinger equations and prove existence and stability of quasi periodic solutions we use a nash moser algorithm together with a reducibility theorem on the linearized operator in a neighborhood of zero due to the presence of the highest order derivatives in the non linearity the classic kam reducibility argument fails and one needs to use a wider class of changes of variables such as diffeomorphisms of the torus and pseudo differential operators this procedure automatically produces a change of variables well defined on the phase space of the equation which diagonalizes the operator linearized at the solution this gives the linear stability

the weakly nonlinear large box limit of the 2d cubic nonlinear schrodinger equation we consider the cubic nonlinear schrodinger nls equation set on a two dimensional box of size l with periodic boundary conditions by taking the large box limit l to infty in the weakly nonlinear regime characterized by smallness in the critical space we derive a new equation set on r 2 that approximates the dynamics of the frequency modes this nonlinear equation turns out to be hamiltonian and enjoys interesting symmetries such as its invariance under fourier transform as well as several families of explicit solutions a large part of this work is devoted to a rigorous approximation result that allows to project the long time dynamics of the limit equation into that of the cubic nls equation on a box of finite size

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

resonances radiation damping and instabilitym in hamiltonian nonlinear wave equations we consider a class of nonlinear klein gordon equations which are hamiltonian and are perturbations of linear dispersive equations the unperturbed dynamical system has a bound state a spatially localized and time periodic solution we show that for ge neric nonlinear hamiltonian perturbations all small amplitude so lutions decay to zero as time tends to infinity at an anomalously slow rate in particular spatially localized and time periodic solutions of the linear problem are destroyed by generic nonlinear hamiltonian perturbations via slow radiation of energy to infinity these solutions can therefore be thought of as metastable states the main mecha nism is a nonlinear resonant interaction of bound states eigenfunc tions and radiation continuous spectral modes leading to energy transfer from the discrete to continuum modes this is in contrast to the kam theory in which appropriate nonresonance conditions im ply the persistence of invariant tori a hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the fermi golden rule arising in the theory of resonances in quantum mechanics the techniques used involve i a time dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues ii a generalization of the hamiltonian normal form appropriate for infinite dimensional dis persive systems and iii ideas from scattering theory the arguments are quite general and we expect them to apply to a large class of

invariant tori of nonlinear schrodinger equation abstract in this paper we consider one dimensional nonlinear schrodinger equation i u t u x x v x u f u 2 u 0 on 0 r under the boundary conditions a 1 u t 0 b 1 u x t 0 0 a 2 u t b 2 u x t 0 a i 2 b i 2 0 for i 1 2 it is proved that for a prescribed and analytic positive potential v x the above equation admits small amplitude quasi periodic solutions corresponding to d dimensional invariant tori of the associated infinite dimensional dynamical system

conservation of resonant periodic solutions for the one dimensional nonlinear schrodinger equation we consider the one dimensional nonlinear schrodinger equation with dirichlet boundary conditions in the fully resonant case absence of the mass term we investigate conservation of small amplitude periodic solutions for a large measure set of frequencies in particular we show that there are infinitely many periodic solutions which continue the linear ones involving an arbitrary number of resonant modes provided the corresponding frequencies are large enough say greater than a certain threshold value depending on the number of resonant modes if the frequencies of the latter are close enough to such a threshold then they can not be too distant from each other hence we can interpret such solutions as perturbations of wave packets with large wave number

quasi periodic solutions of a non autonomous wave equations with quasi periodic forcing abstract in this paper we prove existence of small amplitude quasi periodic solutions for a non autonomous quasi periodically forced nonlinear wave equations with periodic spatial boundary conditions via kam theory

on the existence of sobolev quasi periodic solutions of multidimensional nonlinear beam equation in this paper we prove the existence of quasi periodic solutions of the multidimensional nonlinear beam equation with finitely differentiable nonlinearities and quasi periodic forcing in time

quasi periodic solutions for d dimensional beam equation with derivative nonlinear perturbation in this paper we consider the d dimensional beam equation with convolution potential under periodic boundary conditions we will apply the kolmogorov arnold moser theorem in eliasson and kuksin ann math 172 371 435 2010 into this system and obtain that for sufficiently small e there is a large subset s of s such that for all s s the solution u of the unperturbed system persists as a time quasi periodic solution which has all lyapunov exponents equal to zero and whose linearized equation is reducible to constant coefficients

an a posteriori kam theorem for whiskered tori in hamiltonian partial differential equations with applications to some ill posed equations the goal of this paper is to develop a kam theory for tori with hyperbolic directions which applies to hamiltonian partial differential equations even to some ill posed ones the main result has an a posteriori format that is we show that if there is an approximate solution of an invariance equation which also satisfies some non degeneracy conditions then there is a true solution nearby this allows besides dealing with the quasi integrable case for the validation of numerical computations or formal perturbative expansions as well as for obtaining quasi periodic solutions in degenerate situations the a posteriori format also has other automatic consequences smooth dependence on parameters bootstrap of regularity etc we emphasize that the non degeneracy conditions required are just quantities evaluated on the approximate solution no global assumptions on the system such as twist hence they are readily verifiable in perturbation expansions we will pay attention to the quantitative relations between the sizes of the approximation and the non degeneracy conditions this will allow us to prove what experts call small twist theorems the non degeneracy conditions vanishes as the perturbation goes to zero but much slower than the error of the approximation the method of proof is based on an iterative method for solving a functional equation for the parameterization of the torus expressing that the range of the parameterization admits an evolution and is invariant we also solve functional equations for bundles which imply that are invariant under the linearization the iterative method does not use transformation theory nor action angle variables the main result does not assume that the system is close to integrable more surprisingly we do not need that the equations we study define an evolution for all initial conditions and are well posed even if the systems we study do not admit solutions for all initial conditions we show that there is a systematic way to choose initial conditions on which one can define an evolution which is quasi periodic we first develop an abstract theorem then we show how this abstract result applies to some concrete examples the examples considered in this paper are the scalar boussinesq equation and the boussinesq system both are pde models that aim to describe water waves in the long wave limit for these equations we construct small amplitude time quasi periodic solutions which are even in the spatial variable the strategy for the abstract theorem is inspired by that in fontich et al electron res announc math sci 16 9 22 2009 j differ equ 246 8 3136 3213 2009 the main part of the paper is to study infinite dimensional analogues of dichotomies which applies even to ill posed equations and which is stable under addition of unbounded perturbations this requires that we assume smoothing properties we also present very detailed bounds on the change of the splittings under perturbations

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

invariant tori for commuting hamiltonian pdes we generalize to some pdes a theorem by eliasson and nekhoroshev on the persistence of invariant tori in hamiltonian systems with r integrals of motion and n degrees of freedom r n the result we get ensures the persistence of an r parameter family of r dimensional invariant tori the parameters belong to a cantor like set the proof is based on the lyapunov schmidt decomposition and on the standard implicit function theorem some of the persistent tori are resonant we also give an application to the nonlinear wave equation with periodic boundary conditions on a segment and to a system of coupled beam equations in the first case we construct 2 dimensional tori while in the second case we construct 3 dimensional tori

almost periodic one dimensional systems the schrodinger equation with potential v x 2 1cos x 2 2cos x is considered its solution reduces to the problem of finding the eigenvectors of a matrix the eigenvalues of this matrix show a band structure which is very sensitive to the value of the parameter the solutions of the schrodinger equation are presented and their physical meaning is discussed the potential v x has a complex multiwell structure and quantum tunneling occurs the accuracy of all approximations is carefully studied

time decay for solutions of schrodinger equations with rough and time dependent potentials in this paper we establish dispersive estimates for solutions to the linear schrodinger equation in three dimensions frac 1 i partial t psi triangle psi v psi 0 qquad psi s f 0 1 where v t x is a time dependent potential that satisfies the conditions sup t v t cdot l frac 3 2 mathbb r 3 sup x in mathbb r 3 int mathbb r 3 int infty infty frac v hat tau x x y d tau dy c 0 here c0 is some small constant and v hat tau x denotes the fourier transform with respect to the first variable we show that under these conditions 0 1 admits solutions l t l2x 3 l2t l6x 3 for any f l2 3 satisfying the dispersive inequality psi t infty le c t s frac32 f 1 text for all times t s 0 2 for the case of time independent potentials v x 0 2 remains true if int mathbb r 6 frac v x v y x y 2 dxdy 4 pi 2 text and v mathcal k sup x in mathbb r 3 int mathbb r 3 frac v y x y dy 4 pi we also establish the dispersive estimate with an e loss for large energies provided v mathcal k v 2 infty

on quasi periodic solutions for generalized boussinesq equation with quadratic nonlinearity in this paper one dimensional generalized boussinesq equation utt uxx u2 uxx xx 0 with boundary conditions ux 0 t ux t uxxx 0 t uxxx t 0 is considered it is proved that the equation admits a whitney smooth family of small amplitude quasi periodic solutions with 2 dimensional diophantine frequencies the proof is based on an infinite dimensional kolmogorov arnold moser theorem and birkhoff normal form

pure point spectrum of the floquet hamiltonian for the quantum harmonic oscillator under time quasi periodic perturbations we prove that the 1 d quantum harmonic oscillator is stable under spatially localized time quasi periodic perturbations on a set of diophantine frequencies of positive measure this proves a conjecture raised by enss veselic in their 1983 paper ev in the general quasi periodic setting the motivation of the present paper also comes from construction of quasi periodic solutions for the corresponding nonlinear equation

periodic solutions for the schrodinger equation with nonlocal smoothing nonlinearities in higher dimension abstract we consider the nonlinear schrodinger equation in higher dimension with dirichlet boundary conditions and with a nonlocal smoothing nonlinearity we prove the existence of small amplitude periodic solutions in the fully resonant case we find solutions which at leading order are wave packets in the sense that they continue linear solutions with an arbitrarily large number of resonant modes the main difficulty in the proof consists in a small divisor problem which we solve by using a renormalisation group approach

diffusion in the mean for an ergodic schrodinger equation perturbed by a fluctuating potential diffusive scaling of position moments and a central limit theorem are obtained for the mean position of a quantum particle hopping on a cubic lattice and subject to a random potential consisting of a large static part and a small part that fluctuates stochastically in time if the static random potential is strong enough to induce complete localization in the absence of time dependent noise then the diffusion constant is shown to go to zero proportional to the square of the strength of the time dependent part

long time dynamics of schrodinger and wave equations on flat tori abstract we consider a class of linear time dependent schrodinger equations and quasi periodically forced nonlinear hamiltonian wave klein gordon and schrodinger equations on arbitrary flat tori for the linear schrodinger equation we prove a t 0 upper bound for the growth of the sobolev norms as the time goes to infinity for the nonlinear hamiltonian pdes we construct families of time quasi periodic solutions both results are based on clusterization properties of the eigenvalues of the laplacian on a flat torus and on suitable separation properties of the singular sites of schrodinger and wave operators which are integers in space time fourier lattice close to a cone or a paraboloid thanks to these properties we are able to apply delort abstract theorem 20 to control the speed of growth of the sobolev norms and berti corsi procesi abstract nash moser theorem 8 to construct quasi periodic solutions