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

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

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

vibroacoustic analysis of rectangular plates with elastic rotational edge restraints this investigation deals with the vibration of and the acoustic radiation from a simply supported rectangular plate with elastic restraints against edge rotations the displacement of the plate is first sought as a series expansion in terms of the beam functions each of the beam functions is then expressed as the linear combination of a fourier sine series and a complementary sufficiently smooth function that is introduced to ensure and improve the convergence of the fourier series expansion to facilitate the acoustic analysis the plate displacement is eventually simplified to a standard fourier sine series the modal and acoustic characteristics of square plates are studied for different restraining stiffnesses and configurations it is shown that the modes of the restrained plates can be considerably different from those in the simply supported case so are the corresponding modal radiation efficiencies the proposed method is generally applicable to rectangular plates elastically restrained along any edge s and the acoustic calculations are valid for an arbitrary acoustic or structural modal wave number

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

a transversal substructuring mode matching method applied to the acoustic analysis of dissipative mufflers to carry out the acoustic analysis of dissipative silencers with uniform cross section the application of the mode matching method at the geometrical discontinuities is an attractive option from a computational point of view the consideration of this methodology assumes in general that the modes associated with the transversal geometry of each element with uniform cross section are known for the excitation frequencies considered in the analysis the calculation of the transversal modes is not however a simple task when the acoustic system involves perforated elements and absorbent materials the current work presents a modal approach to calculate the transversal modes and the corresponding axial wavenumbers for dissipative mufflers of uniform but arbitrary cross section the proposed technique is based on the division of the transversal section into subdomains and the subsequent use of a substructuring procedure with two sets of modes to improve the convergence the former set of modes fulfils the condition of zero pressure at the common boundary between transversal subdomains while the latter satisfies the condition of zero derivative in the direction normal to the boundary the approach leads to a versatile methodology with a moderate computational effort that can be applied to mufflers commonly found in real applications to validate the procedure presented in this work comparisons are provided with finite element predictions and results available in the literature showing a good agreement in addition the procedure is applied to an example of practical interest

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

on the well balanced numerical discretization of shallow water equations on unstructured meshes we consider in this work a finite volume numerical approximation of weak solutions of the shallow water equations with varying topography on unstructured meshes relying on an alternative formulation of the shallow water equations that involves the free surface as a conservative variable instead of the water height we introduce a simple discretization of the bed slope source term together with some suitable conservative variables reconstructions the resulting scheme is automatically consistent and well balanced for any given consistent numerical flux for the homogeneous system we obtain a very simple formulation which do not need to be modified when second order accuracy muscl reconstructions are adopted additionally the positivity of the water height is preserved under a relevant stability condition as soon as the numerical flux for the associated homogeneous system does numerical assessments involving dry areas and complex geometry are performed

a fast and stable well balanced scheme with hydrostatic reconstruction for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom it is a hyperbolic system of conservation laws that approximately describes various geophysical flows such as rivers coastal areas and oceans when completed with a coriolis term or granular flows when completed with friction numerical approximate solutions to this system may be generated using conservative finite volume methods which are known to properly handle shocks and contact discontinuities however in general these schemes are known to be quite inaccurate for near steady states as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions this difficulty can be overcome by using the so called well balanced schemes we describe a general strategy based on a local hydrostatic reconstruction that allows us to derive a well balanced scheme from any given numerical flux for the homogeneous problem whenever the initial solver satisfies some classical stability properties it yields a simple and fast well balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality

fully adaptive multiresolution finite volume schemes for conservation laws the use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by a harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation in this approach the solution is still represented at each time step on the finest grid resulting in an inherent limitation of the potential gain in memory space and computational time the present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes in which the solution is represented and computed in a dynamically evolved adaptive grid a crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages several solutions to this problem are proposed analyzed and compared in terms of accuracy and complexity

a note on vibrations of generally restrained beams exact frequency and normal mode shape expressions are derived in this note for generally restrained bernoulli euler beams with unsymmetrical translations and rotations at either end the eigenfrequencies and mode shape parameters are presented for a wide range of restraint parameters several degenerate cases have been studied and some of these have been compared with those available in the published literature it is believed that the results presented in this paper will be of use in design of beams shafts and piping under dynamic considerations

a kam algorithm for the resonant non linear schrodinger equation abstract we prove through a kam algorithm the existence of large families of stable and unstable quasi periodic solutions for the nls in any number of independent frequencies the main tools are the existence of a non degenerate integrable normal form proved in 26 27 and a generalization of the quasi toplitz functions defined in 31

free vibration of two elastically coupled rectangular plates with uniform elastic boundary restraints abstract an analytical method is derived for determining the vibrations of two plates which are generally supported along the boundary edges and elastically coupled together at an arbitrary angle the interactions of all four wave groups bending waves out of plane shearing waves in plane longitudinal waves and in plane shearing waves have been taken into account at the junction via four types of coupling springs of arbitrary stiffnesses each of the transverse and in plane displacement functions is expressed as the superposition of a two dimensional 2 d fourier cosine series and several supplementary functions which are introduced to ensure and improve the convergence of the series representation by removing the discontinuities that the original displacement and its derivatives will potentially exhibit at the edges when they are periodically expanded onto the entire x y plane as mathematically implied by a 2 d fourier series the unknown expansions coefficients are calculated using the rayleigh ritz procedure which is actually equivalent to solving the governing equation and the boundary and coupling conditions directly when the assumed solutions are sufficiently smooth over the solution domains numerical examples are presented for several different coupling configurations a good comparison is observed between the current results and the fea models although this study is specifically focused on the coupling of two plates the proposed method can be directly extended to structures consisting of any number of plates

kam for reversible derivative wave equations we prove the existence of cantor families of small amplitude analytic linearly stable quasi periodic solutions of reversible derivative wave equations

well balanced adaptive mesh refinement for shallow water flows well balanced shock capturing wbsc schemes constitute nowadays the state of the art in the numerical simulation of shallow water flows they allow to accurately represent discontinuous behavior known to occur due to the non linear hyperbolic nature of the shallow water system and at the same time numerically maintain stationary solutions in situations of practical interest these schemes often need to be combined with some kind of adaptivity in order to speed up computing times in this paper we discuss what ingredients need to be modified in a block structured amr technique in order to ensure that when combined with a wbsc scheme the so called water at rest stationary solutions are exactly preserved

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

a subsonic well balanced reconstruction scheme for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom in past years efficient well balanced methods have been proposed in order to well resolve solutions close to steady states at rest here we describe a strategy based on a local subsonic steady state reconstruction that allows one to derive a subsonic well balanced scheme preserving exactly all the subsonic steady states it generalizes the now well known hydrostatic solver and like the latter it preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality an application to the euler poisson system is proposed

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

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

acoustic analysis of a rectangular cavity with general impedance boundary conditions a fourier series method is proposed for the acoustic analysis of a rectangular cavity with impedance boundary conditions arbitrarily specified on any of the walls the sound pressure is expressed as the combination of a three dimensional fourier cosine series and six supplementary two dimensional expansions introduced to ensure accelerate the uniform and absolute convergence rate of the series representation in the cavity including the boundary surfaces the expansion coefficients are determined using the rayleigh ritz method since the pressure field is constructed adequately smooth throughout the entire solution domain the rayleigh ritz solution is mathematically equivalent to what is obtained from a strong formulation based on directly solving the governing equations and the boundary conditions to unify the treatments of arbitrary nonuniform impedance boundary conditions the impedance distribution function on each specified surface is invariantly expressed as a double fourier series expansion so that all the relevant integrals can be calculated analytically the modal parameters for the acoustic cavity can be simultaneously obtained from solving a standard matrix eigenvalue problem instead of iteratively solving a nonlinear transcendental equation as in the existing methods several numerical examples are presented to demonstrate the effectiveness and reliability of the current method for various impedance boundary conditions including nonuniform impedance distributions

application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes we consider the numerical solution of the shallow water equations on unstructured grids we focus on flows over wet areas the extension to the case of dry bed will be reported elsewhere the shallow water equations fall into the category of systems of conservation laws which can be symmetrized thanks to the existence of a mathematical entropy coinciding in this case with the total energy our aim is to show the application of a particular class of conservative residual distribution rd schemes to the discretization of the shallow water equations and to analyze their discrete accuracy and stability properties we give a review of conservative rd schemes showing relations between different approaches previously published and recall l stability and accuracy criteria characterizing the schemes in particular the accuracy of the rd method in presence of source terms is analyzed and conditions to construct rth order discretizations on irregular triangular grids are proved it is shown that the rd approach gives a natural way of obtaining high order discretizations which moreover preserves exactly the steady lake at rest solution independently on mesh topology nature of the variation of the bottom and polynomial order of interpolation used for the unknowns we also consider more general analytical solutions which are less investigated from the numerical view point on irregular grids linearity preserving rd schemes yield a truly second order approximation we also sketch a strategy to achieve discretizations which preserve exactly some of these solutions numerical results on steady and time dependent problems involving smooth and non smooth variations of the bottom topology show very promising features of the approach

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

free vibration analysis of circular cylindrical shells abstract a general analytical method is presented for evaluating the free vibration characteristics of a circular cylindrical shell with classical boundary conditions of any type the solution is obtained through a direct solution procedure in which sanders shell equations are used with the axial modal displacements represented as simple fourier series expressions stokes transformation is exploited to obtain correct series expressions for the derivatives of the fourier series an explicit expression of the exact frequency equation can be obtained for any kind of boundary conditions the accuracy of the method is checked against available data the method is used to find the modal characteristics of the thermal liner model of the u s fast test reactor ftp the numerical results obtained are compared with finite element method solutions

growth of sobolev norms for the analytic nls on t2 we consider the completely resonant non linear schrodinger equation on the two dimensional torus with any analytic gauge invariant nonlinearity fix s 1 we show the existence of solutions of this equation which achieve arbitrarily large growth of hs sobolev norms we also give estimates for the time required to attain this growth

a well balanced scheme for the numerical processing of source terms in hyperbolic equations in a variety of physical problems one encounters source terms that are balanced by internal forces and this balance supports multiple steady state solutions that are stable typical of these are gravity driven flows such as those described by the shallow water equations over a nonuniform ocean bottom 1 10 qquad h t hu x 0 text and hu t left hu 2 frac gh 2 2 right x ga x x h 0 many classic numerical schemes cannot maintain these steady solutions or achieve them in the long time limit with an acceptable level of accuracy because they do not preserve the proper balance between the source terms and internal forces we propose here a numerical scheme adapted to a scalar conservation law that preserves this balance and that can it is hoped be extended to more general hyperbolic systems the proof of convergence of this scheme toward the entropy solution is given and some numerical tests are reported

kam theory for the reversible perturbations of 2d linear beam equations in the present paper we prove an infinite dimensional reversible kolmogorov arnold moser kam theorem as an application we study the existence of kam tori for a class of two dimensional 2d non hamiltonian completely resonant beam equations with derivative nonlinearities the birkhoff normal form theory is also used since there are no external parameters in the equations

vibration analysis of rectangular plates with general elastic boundary supports abstract in this investigation the rayleigh ritz method is used to determine the modal characteristics of a rectangular plate with general elastic supports alone its edges each of the admissible functions here is composed of a trigonometric function and an arbitrary continuous function that is introduced to ensure the sufficient smoothness of the so called residual displacement function at the edges as a result a drastic improvement of the convergence can be expected of the solution expressed as a series expansion in terms of the admissible functions perhaps more importantly this study has developed a general approach for deriving a complete set of admissible functions that can be universally applied to various boundary conditions several numerical examples are given to demonstrate the accuracy and convergence of the current solution

arbitrary precision frequencies of a free rectangular thin plate abstract a variational method developed by the authors and named whole element method wem is used to find the arbitrary precision frequencies of a rectangular thin plate within the germain lagrange theory having its four borders free of constraint wem consists is proposing an adequate functional and a sequence representing the plate transversal displacementw x y such a sequence is made of a linear combination of functions belonging to a complete set inl2 the sequence and not each co ordinate function is required to satisfy the essential or geometric conditions the sequence generation is systematic and no analysis of the classical natural modes of the plate is needed in particular trigonometric functions which a priori belong to a complete set in the domain are used in the present analysis the solving equations involving very simple sums arise from the minimization of the functional wem is based on theorems which show the ultimate exactness of the eigenvalues and the uniform convergence of the essential functions of the problem to the authors knowledge this problem has no classical solution

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

multiwavelet based grid adaptation with discontinuous galerkin schemes for shallow water equations we provide an adaptive strategy for solving shallow water equations with dynamic grid adaptation including a sparse representation of the bottom topography a challenge in computing approximate solutions to the shallow water equations including wetting and drying is to achieve the positivity of the water height and the well balancing of the approximate solution a key property of our adaptive strategy is that it guarantees that these properties are preserved during the refinement and coarsening steps in the adaptation process the underlying idea of our adaptive strategy is to perform a multiresolution analysis using multiwavelets on a hierarchy of nested grids this provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients furthermore we use the multiresolution analysis of the underlying data as an additional indicator of whether the limiter has to be applied on a cell or not by this the number of cells where the limiter is applied is reduced without spoiling the accuracy of the solution by means of well known 1d and 2d benchmark problems we verify that multiwavelet based grid adaptation can significantly reduce the computational cost by sparsening the computational grids while retaining accuracy and keeping well balancing and positivity

an infinite dimensional kam theorem and its application to the two dimensional cubic schrodinger equation abstract we prove an infinite dimensional kam theorem as an application we use the theorem to study the two dimensional nonlinear schrodinger equation i u t u u 2 u 0 t r x t 2 with periodic boundary conditions we obtain for the equation a whitney smooth family of small amplitude quasi periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system

free vibrations of beams with general boundary conditions a simple and unified approach is presented for the vibration analysis of a generally supported beam the flexural displacement of the beam is sought as the linear combination of a fourier series and an auxiliary polynomial function the polynomial function is introduced to take all the relevant discontinuities with the original displacement and its derivatives at the boundaries and the fourier series now simply represents a residual or conditioned displacement that has at least three continuous derivatives as a result not only is it always possible to expand the displacement in a fourier series for beams with any boundary conditions but also the solution converges at a much faster speed the reliability and robustness of the proposed technique are demonstrated through numerical examples

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

holder continuity of the integrated density of states for quasi periodic schrodinger equations and averages of shifts of subharmonic functions in this paper we consider various regularity results for discrete quasiperiodic schr6dinger equations n l pn 1 v 9 nw on eon with analytic potential v we prove that on intervals of positivity for the lyapunov exponent the integrated density of states is holder continuous in the energy provided w has a typical continued fraction expansion the proof is based on certain sharp large deviation theorems for the norms of the monodromy matrices and the avalanche principle the latter refers to a mechanism that allows us to write the norm of a monodromy matrix as the product of the norms of many short blocks in the multi frequency case the integrated density of states is shown to have a modulus of continuity of the form exp log tl for some 0 a 1 but currently we do not obtain holder continuity in the case of more than one frequency we also present a mechanism for proving the positivity of the lyapunov exponent for large disorders for a general class of equations the only requirement for this approach is some weak form of a large deviation theorem for the lyapunov exponents in particular we obtain an independent proof of the herman sorets spencer theorem in the multi frequency case the approach in this paper is related to the recent nonperturbative proof of anderson localization in the quasi periodic case by j bourgain and m goldstein

analysis and approximation of conservation laws with source terms we consider a conservation law of the form cl i ut f u x ax where a cdot is a bounded piecewise smooth source term and f an even convex function we first characterize the solution to the riemann problem through a new lax type formula then we prove that for a cdot fixed the semigroup associated with cl is an l1 contraction and we obtain an existence theorem for weak solutions to cl we conclude by constructing godunov type difference schemes and proving that these schemes are l infty stable and have stable steady solutions similar in structure to those of cl some numerical tests are reported

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

quasi periodic solutions for quasi linear generalized kdv equations abstract we prove the existence of cantor families of small amplitude linearly stable quasi periodic solutions of quasi linear autonomous hamiltonian generalized kdv equations we consider the most general quasi linear quadratic nonlinearity the proof is based on an iterative nash moser algorithm to initialize this scheme we need to perform a bifurcation analysis taking into account the strongly perturbative effects of the nonlinearity near the origin in particular we implement a weak version of the birkhoff normal form method the inversion of the linearized operators at each step of the iteration is achieved by pseudo differential techniques linear birkhoff normal form algorithms and a linear kam reducibility scheme

an abstract nash moser theorem and quasi periodic solutions for nlw and nls on compact lie groups and homogeneous manifolds we prove an abstract implicit function theorem with parameters for smooth operators defined on scales of sequence spaces modeled for the search of quasi periodic solutions of pdes the tame estimates required for the inverse linearised operators at each step of the iterative scheme are deduced via a multiscale inductive argument the cantor like set of parameters where the solution exists is defined in a non inductive way this formulation completely decouples the iterative scheme from the measure theoretical analysis of the parameters where the small divisors non resonance conditions are verified as an application we deduce the existence of quasi periodic solutions for forced nlw and nls equations on any compact lie group or manifold which is homogeneous with respect to a compact lie group extending previous results valid only for tori a basic tool of harmonic analysis is the highest weight theory for the irreducible representations of compact lie groups

accurate approximate analytical solutions for nonlinear free vibration of systems with serial linear and nonlinear stiffness this paper deals with free vibration of a nonlinear system having combined linear and nonlinear springs in series the conservative oscillation system is formulated as a nonlinear ordinary differential equation having linear and nonlinear stiffness components the governing equation is linearized and associated with the harmonic balance method to establish new and accurate higher order analytical approximate solutions unlike the perturbation method which is restricted to nonlinear conservative systems with a small perturbed parameter and also unlike the classical harmonic balance method which results in a complicated set of algebraic equations the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation some examples are solved and compared with numerical integration solutions and published results new solutions to the nonlinear systems are also presented and discussed

kam for the nonlinear schrodinger equation we consider the d dimensional nonlinear schrodinger equation under periodic boundary conditions i dot u delta u v x u ep frac p f p bar u x u bar u quad u u t x x in t d where v x sum hat v a e i sc a x is an analytic function with hat v real and f is a real analytic function in re u im u and x this equation is a popular model for the real nls equation where instead of the convolution term v u we have the potential term vu for ep 0 the equation is linear and has time quasi periodic solutions u u t x sum a in a hat u a e i a 2 hat v a t e i sc a x quad hat u a 0 where a is any finite subset of z d we shall treat omega a a 2 hat v a a ina as free parameters in some domain u subset r a this is a hamiltonian system in infinite degrees of freedom degenerate but with external parameters and we shall describe a kam theory which under general conditions will have the following consequence if ep is sufficiently small then there is a large subset u of u such that for all omega in u the solution u persists as a time quasi periodic solution which has all lyapounov exponents equal to zero and whose linearized equation is reducible to constant coefficients

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

dynamic analysis of generally supported beams using fourier series the applicability of fourier series to the dynamic analysis of beams having arbitrary boundary conditions is explored for the simplicity of presenting the analysis procedure beams having zero end displacement are used in the discussion numerical results for free vibration and dynamic response based on the present analysis procedure are compared with existing results for natural frequencies and dynamic behavior obtained by the conventional modal analysis these results indicate that the present analysis is accurate and provides a unified and systematic procedure which is seemingly simpler and more straightforward than the conventional modal analysis

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

multiresolution algorithms for the numerical solution of hyperbolic conservation laws given any scheme in conservation form and an appropriate uniform grid for the numerical solution of the initial value problem for one dimensional hyperbolic conservation laws we describe a multiresolution algorithm that approximates this numerical solution to a prescribed tolerance in an efficient manner to do so we consider the grid averages of the numerical solution for a hierarchy of nested diadic grids in which the given grid is the finest and introduce an equivalent multiresolution representation the multiresolution representation of the numerical solution consists of its grid averages for the coarsest grid and the set of errors in predicting the grid averages of each level of resolution in this hierarchy from those of the next coarser one once the numerical solution is resolved to our satisfaction in a certain locality of some grid then the prediction errors there are small for this particular grid and all finer ones this enables us to compress data by setting to zero small components of the representation which fall below a prescribed tolerance therefore instead of computing the time evolution of the numerical solution on the given grid we compute the time evolution of its compressed multiresolution representation algorithmically this amounts to computing the numerical fluxes of the given scheme at the points of the given grid by a hierarchical algorithm which starts with the computation of these numerical fluxes at the points of the coarsest grid and then proceeds through diadic refinements to the given grid at each step of refinement we add the values of the numerical flux at the center of the coarser cells the information in the multiresolution representation of the numerical solution is used to determine whether the solution is locally well resolved when this is the case we replace the costly exact value of the numerical flux with an accurate enough approximate value which is obtained by an inexpensive interpolation from the coarser grid the computational efficiency of this multiresolution algorithm is proportional to the rate of data compression for a prescribed level of tolerance that can be achieved for the numerical solution of the given scheme