discretization on non orthogonal quadrilateral grids for inhomogeneous anisotropic media two classes of discretization methods are proposed for control volume formulations on quadrilateral grids in two space dimensions curvilinear grids are considered as a special case the methods are applicable for any system of conservation laws where the flux is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy and or non orthogonality of the grid further properties of the methods will be discussed and examples of their use will be presented

block partitioned solvers for coupled poromechanics a unified framework abstract coupled poromechanical problems appear in a variety of disciplines from reservoir engineering to biomedical applications this work focuses on efficient strategies for solving the matrix systems that result from discretization and linearization of the governing equations these systems have an inherent block structure due to the coupled nature of the mass and momentum balance equations recently several iterative solution schemes have been proposed that exhibit stable and rapid convergence to the coupled solution these schemes appear distinct but a unifying feature is that they exploit the block partitioned nature of the problem to accelerate convergence this paper analyzes several of these schemes and highlights the fundamental connections that underlie their effectiveness we begin by focusing on two specific methods a fully implicit and a sequential implicit scheme in the first approach the system matrix is treated monolithically and a krylov iteration is used to update pressure and displacement unknowns simultaneously to accelerate convergence a preconditioning operator is introduced based on an approximate block factorization of the linear system next we analyze a sequential implicit scheme based on the fixed stress split in this method one iterates back and forth between updating displacement and pressure unknowns separately until convergence to the coupled solution is reached we re interpret this scheme as a block preconditioned richardson iteration and we show that the preconditioning operator is identical to that used within the fully implicit approach rapid convergence in both the richardson and krylov based methods results from a particular choice for a sparse schur complement approximation this analysis leads to a unified framework for developing solution schemes based on approximate block factorizations many classic fully implicit and sequential implicit schemes are simple sub cases the analysis also highlights several new approaches that have not been previously explored for illustration we directly compare the performance and robustness of several variants on a benchmark problem

small solutions of nonlinear schrodinger equations near first excited states abstract consider a nonlinear schrodinger equation in r 3 whose linear part has three or more eigenvalues satisfying some resonance conditions solutions which are initially small in h 1 l 1 r 3 and inside a neighborhood of the first excited state family are shown to converge to either a first excited state or a ground state at time infinity an essential part of our analysis is on the linear and nonlinear estimates near nonlinear excited states around which the linearized operators have eigenvalues with nonzero real parts and their corresponding eigenfunctions are not uniformly localized in space

eigenstate structure in graphs and disordered lattices we study wave function structure for quantum graphs in the chaotic and disordered regime using measures such as the wave function intensity distribution and the inverse participation ratio the result is much less ergodicity than expected from random matrix theory even though the spectral statistics are in agreement with random matrix predictions instead analytical calculations based on short time semiclassical behavior correctly describe the eigenstate structure

analysis of the linearization around a critical point of an infinite dimensional hamiltonian system partial differential equations that conserve energy can often be written as infinite dimensional hamiltonian systems of the following general form du dt je u where j x x is a symplectic matrix i e jj 1 and e x r is a c 2 functional defined on some hilbert space x

rate of quantum ergodicity in euclidean billiards for a large class of quantized ergodic flows the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in the semiclassical limit in this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the wigner function in the case of ergodic systems of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value this is studied numerically for three euclidean billiards stadium cosine and cardioid billiard using up to 6000 eigenfunctions we find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions such as bouncing ball modes or scarred eigenfunctions we give a detailed discussion and explanation of these effects using a simple but powerful model for the rate of quantum ergodicity in momentum space we observe a slower decay we also study the suitably normalized fluctuations of the expectation values around their mean and find good agreement with a gaussian distribution

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

equipartition of the eigenfunctions of quantized ergodic maps on the torus we give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area preserving maps on the torus examples are the irrational translations the skew translations the hyperbolic automorphisms and some of their perturbations

on instability of excited states of the nonlinear schrodinger equation we introduce a new notion of linear stability for standing waves of the nonlinear schrodinger equation nls which requires not only that the spectrum of the linearization be real but also that the generalized kernel be not degenerate and that the signature of all the positive eigenvalues be positive we prove that excited states of the nls are not linearly stable in this more restrictive sense we then give a partial proof that this more restrictive notion of linear stability is a necessary condition to have orbital stability

stability of standing waves for nls with perturbed lame potential abstract we perturb a linear schrodinger equation with lame potential with a small positive or negative potential the new perturbed operator has one or more eigenvalues at most one in each spectral gap we then add a nonlinear term and study the stability of the corresponding nonlinear stationary waves

two point spectral correlations for star graphs the eigenvalues of the schrodinger operator on a graph g are related via an exact trace formula to periodic orbits on g this connection is used to calculate two point spectral statistics for a particular family of graphs called star graphs in the limit as the number of edges tends to infinity combinatorial techniques are used to evaluate both the diagonal same orbit and off diagonal different orbit contributions to the sum over pairs of orbits involved in this way a general formula is derived for terms in the short time expansion of the form factor k in powers of and the first few are computed explicitly the result demonstrates that k is neither poissonian nor random matrix but an intermediate between the two off diagonal pairs of orbits are shown to make a significant contribution to all but the first few coefficients

spectral statistics for the dirac operator on graphs we determine conditions for the quantization of graphs using the dirac operator for both two and four component spinors according to the bohigas giannoni schmit conjecture for such systems with time reversal symmetry the energy level statistics are expected in the semiclassical limit to correspond to those of random matrices from the gaussian symplectic ensemble this is confirmed by numerical investigation the scattering matrix used to formulate the quantization condition is found to be independent of the type of spinor we derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation

scarred eigenstates for quantum cat maps of minimal periods in this paper we construct a sequence of eigenfunctions of the quantum arnold s cat map that in the semiclassical limit shows a strong scarring phenomenon on the periodic orbits of the dynamics more precisely those states have a semiclassical limit measure that is the sum of 1 2 the normalized lebesgue measure on the torus plus 1 2 the normalized dirac measure concentrated on any a priori given periodic orbit of the dynamics it is known the schnirelman theorem that most sequences of eigenfunctions equidistribute on the torus the sequences we construct therefore provide an example of an exception to this general rule our method of construction and proof exploits the existence of special values of h for which the quantum period of the map is relatively short and a sharp control on the evolution of coherent states up to this time scale we also provide a pointwise description of these states in phase space which uncovers their hyperbolic structure in the vicinity of the fixed points and yields more precise localization estimates

quantum scars of classical closed orbits in phase space the way in which quantum eigenstates are influenced by the closed orbits of a chaotic classical system is analysed in phase space x q p through the spectral wigner function w x e this is a sum over wigner functions of eigenstates within a range of energy e in the classical limit w is concentrated on the energy surface and smoothly distributed over it closed orbits provide oscillatory corrections scars for which explicit semiclassical formulae are calculated each scar is a fringe pattern decorating the orbit as x moves off the energy surface the fringes form an airy pattern with spacing of order h as x moves off the closed orbit the fringes form a complex gaussian with spacing h

star graphs and seba billiards we derive an exact expression for the two point correlation function for quantum star graphs in the limit as the number of bonds tends to infinity this turns out to be identical to the corresponding result for certain eba billiards in the semiclassical limit the reasons for this are discussed the formula we derive is also shown to be equivalent to a series expansion for the form factor the fourier transform of the two point correlation function previously calculated using periodic orbit theory

finite volume hydromechanical simulation in porous media cell centered finite volume methods are prevailing in numerical simulation of flow in porous media however due to the lack of cell centered finite volume methods for mechanics coupled flow and deformation is usually treated either by coupled finite volume finite element discretizations or within a finite element setting the former approach is unfavorable as it introduces two separate grid structures while the latter approach loses the advantages of finite volume methods for the flow equation recently we proposed a cell centered finite volume method for elasticity herein we explore the applicability of this novel method to provide a compatible finite volume discretization for coupled hydromechanic flows in porous media we detail in particular the issue of coupling terms and show how this is naturally handled furthermore we observe how the cell centered finite volume framework naturally allows for modeling fractured and fracturing porous media through internal boundary conditions we support the discussion with a set of numerical examples the convergence properties of the coupled scheme are first investigated second we illustrate the practical applicability of the method both for fractured and heterogeneous media

scattering theory on graphs we consider the scattering theory for the schrodinger operator dx2 v x on the graphs made of one dimensional wires connected to external leads we derive two expressions for the scattering matrix on arbitrary graphs one involves matrices that couple arcs oriented bonds the other involves matrices that couple vertices we discuss a simple way to tune the coupling between the graph and the leads the efficiency of the formalism is demonstrated with a few known examples

visibility of quantum graph spectrum from the vertices we investigate the relation between the eigenvalues of the laplacian with kirchhoff vertex conditions on a finite metric graph and a corresponding titchmarsh weyl function a parameter dependent ne

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

no quantum ergodicity for star graphs we investigate statistical properties of the eigenfunctions of the schrodinger operator on families of star graphs with incommensurate bond lengths we show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average we further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds we describe how to construct such subsequences explicitly these structures are analogous to scars on short unstable periodic orbits

unitary stochastic matrix ensembles and spectral statistics we propose to study unitary matrix ensembles defined in terms of unitary stochastic transition matrices associated with markov processes on graphs we argue that the spectral statistics of such an ensemble after ensemble averaging depends crucially on the spectral gap between the leading and subleading eigenvalue of the underlying transition matrix it is conjectured that unitary stochastic ensembles follow one of the three standard ensembles of random matrix theory in the limit of infinite matrix size n if the spectral gap of the corresponding transition matrices closes slower than 1 n the hypothesis is tested by considering several model systems ranging from binary graphs to uniformly and non uniformly connected star graphs and diffusive networks in arbitrary dimensions

occupation times distribution for brownian motion on graphs considering a brownian motion on a general graph we study the joint law for the occupation times on all the bonds in particular we show that the laplace transform of this distribution can be expressed as the ratio of two determinants we give two formulations with arc or vertex matrices for this result and discuss a simple example

multichannel nonlinear scattering for nonintegrable equations ii the case of anisotropic potentials and data abstract the nonlinear scattering and stability results of soffer and weinstein comm math phys 1990 are extended to the case of anisotropic potentials and data the range of nonlinearities for which the theory is shown to be valid is also extended considerably

on asymptotic stability in energy space of ground states of nls in 1d abstract we transpose work by t mizumachi to prove smoothing estimates for dispersive solutions of the linearization at a ground state of a nonlinear schrodinger equation nls in 1d as an application we extend to dimension 1d a result on asymptotic stability of ground states of nls proved by cuccagna and mizumachi for dimensions 3

on asymptotic stability of moving kink for relativistic ginzburg landau equation we prove the asymptotic stability of kink for the nonlinear relativistic wave equations of the ginzburg landau type in one space dimension for any odd initial condition in a small neighborhood of the kink the solution asymptotically in time is the sum of the kink and dispersive part described by the free klein gordon equation the remainder converges to zero in a global norm

the continuous galerkin method is locally conservative we examine the conservation law structure of the continuous galerkin method we employ the scalar advection diffusion equation as a model problem for this purpose but our results are quite general and apply to time dependent nonlinear systems as well in addition to global conservation laws we establish local conservation laws which pertain to subdomains consisting of a union of elements as well as individual elements these results are somewhat surprising and contradict the widely held opinion that the continuous galerkin method is not locally conservative

instability of nonlinear bound states we establish a sharp instability theorem for the bound states of lowest energy of the nonlinear klein gordon equation utt u f u 0 and the nonlinear schrodinger equation iut u f u 0

invariant manifolds for a class of dispersive hamiltonian partial differential equations we construct an invariant manifold of periodic orbits for a class of non linear schrodinger equations using standard ideas of the theory of center manifolds we rederive the results of soffer and weinstein comm math phys 133 119 146 1997 j differential equations98 376 390 1992 on the large time asymptotics of small solutions scattering theory

on scattering of small energy solutions of non autonomous hamiltonian nonlinear schrodinger equations abstract we revisit a result by cuccagna kirr and pelinovsky about the cubic nonlinear schrodinger equation nls with an attractive localized potential and a time dependent factor in the nonlinearity we show that under generic hypotheses on the linearization at 0 of the equation small energy solutions are asymptotically free this is yet a new application of the hamiltonian structure continuing a program initiated in a paper by bambusi and cuccagna

recovering a quantum graph spectrum from vertex data we study the question to what extent spectral information on a schrodinger operator on a finite compact metric graph subject to standard or type matching conditions can be recovered from a corresponding titchmarsh weyl function on the boundary of the graph in contrast to the case of ordinary or partial differential operators the knowledge of the titchmarsh weyl function is in general not sufficient for recovering the complete spectrum of the operator or the potentials on the edges however it is shown that those eigenvalues with sufficiently high depending on the cyclomatic number of the graph multiplicities can be recovered moreover we prove that under certain additional conditions the titchmarsh weyl function even contains the full spectral information

monotonicity of control volume methods robustness of numerical methods for multiphase flow problems in porous media is important for development of methods to be used in a wide range of applications here we discuss monotonicity for a simplified problem of single phase flow but where the simulation grids and media are allowed to be general posing challenges to control volume methods we discuss discrete formulations of the maximum principle and derive sufficient criteria for discrete monotonicity for arbitrary nine point control volume discretizations for conforming quadrilateral grids in 2d these criteria are less restrictive than the m matrix property it is shown that it is impossible to construct nine point methods which unconditionally satisfy the monotonicity criteria when the discretization satisfies local conservation and exact reproduction of linear potential fields numerical examples are presented which show the validity of the criteria for monotonicity further the impact of nonmonotonicity is studied different behavior for different discretization methods is illuminated and simple ideas are presented for improvement in terms of monotonicity

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

extended finite element method for three dimensional crack modelling an extended finite element method x fem for three dimensional crack modelling is described a discontinuous function and the two dimensional asymptotic crack tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity this enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed stress intensity factors sifs for planar three dimensional cracks are presented which are found to be in good agreement with benchmark solutions copyright 2000 john wiley sons ltd

on eigenfunction expansion of solutions to the hamilton equations we establish a spectral representation for solutions to linear hamilton equations with positive definite energy in a hilbert space our approach is a special version of m krein s spectral theory of j selfadjoint operators in the hilbert spaces with indefinite metric our main result is an application to the eigenfunction expansion for the linearized relativistic ginzburg landau equation

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

value distribution of the eigenfunctions and spectral determinants of quantum star graphs we compute the value distributions of the eigenfunctions and spectral determinant of the schrodinger operator on families of star graphs the values of the spectral determinant are shown to have a cauchy distribution with respect both to averages over bond lengths in the limit as the wavenumber tends to infinity and to averages over wavenumber when the bond lengths are fixed and not rationally related this is in contrast to the spectral determinants of random matrices for which the logarithm is known to satisfy a gaussian limit distribution the value distribution of the eigenfunctions also differs from the corresponding random matrix result we argue that the value distributions of the spectral determinant and of the eigenfunctions should coincide with those of seba type billiards

the spin contribution to the form factor of quantum graphs following the quantization of a graph with the dirac operator spin 1 2 we explain how additional weights in the spectral form factor k due to spin propagation around orbits produce higher order terms in the small asymptotics in agreement with symplectic random matrix ensembles we determine conditions on the group of spin rotations sufficient to generate cse statistics

on asymptotic stability of moving ground states of the nonlinear schrodinger equation we extend to the case of moving solitons the result on asymptotic stability of ground states of the nls obtained by the author in cu1 for technical reasons we consider only smooth solutions the proof is similar to the earlier paper however now the flows required for the darboux theorem and the birkhoff normal forms instead of falling within the framework of standard theory of ode s are related to quasilinear hyperbolic symmetric systems it is also not obvious that darboux theorem can be applied since we need to compare two symplectic forms in a neighborhood of the ground states not in h r but rather in the space where also the variance is bounded but the nls does not preserve small neighborhoods of the ground states in

a borg levinson theorem for trees we prove that the dirichelet to neumann map for a schrodinger operator on a finite simply connected tree determines uniquely the potential on that tree

simplicity of eigenvalues and non vanishing of eigenfunctions of a quantum graph we prove that after an arbitrarily small adjustment of edge lengths the spectrum of a compact quantum graph with type vertex conditions can be simple we also show that the eigenfunctions with the exception of those living entirely on a looping edge can be made to be non vanishing on all vertices of the graph as an application of the above result we establish that the secular manifold also called determinant manifold of a large family of graphs has exactly two smooth connected components

universal spectral statistics in quantum graphs we prove that the spectrum of an individual chaotic quantum graph shows universal spectral correlations as predicted by random matrix theory the stability of these correlations with regard to nonuniversal corrections is analyzed in terms of the linear operator governing the classical dynamics on the graph

form factor for a family of quantum graphs an expansion to third order for certain types of quantum graphs we show that the random matrix form factor can be recovered to at least third order in the scaled time from periodic orbit theory we consider the contributions from pairs of periodic orbits represented by diagrams with up to two self intersections connected by up to four arcs and explain why all other diagrams are expected to give higher order corrections only for a large family of graphs with ergodic classical dynamics the diagrams that exist in the absence of time reversal symmetry sum to zero the mechanism for this cancellation is rather general which suggests that it also applies at higher orders in the expansion this expectation is in full agreement with the fact that in this case the linear contribution the diagonal approximation already reproduces the random matrix form factor for 1 for systems with time reversal symmetry there are more diagrams which contribute at third order we sum these contributions for quantum graphs with uniformly hyperbolic dynamics obtaining 2 3 in agreement with random matrix theory as in the previous calculation of the leading order correction to the diagonal approximation we find that the third order contribution can be attributed to exceptional orbits representing the intersection of diagram classes

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

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

smoothed wave functions of chaotic quantum systems abstract it is shown that wave functions of quantum systems as 0 have an extra density near unstable periodic trajectories of the classical problem the averaged wave function square is represented as the sum over a finite number of periodic trajectories the contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory the results are compared with the numerical calculations of the wave functions for the stadium billiard

the hamiltonian structure of the nonlinear schrodinger equation and the asymptotic stability of its ground states in this paper we prove that ground states of the nls which satisfy the sufficient conditions for orbital stability of m weinstein are also asymptotically stable for seemingly generic equations the key issue is to prove that a certain coefficient is non negative because is a square power we assume that the nls has a smooth short range nonlinearity we assume also the presence of a very short range and smooth linear potential to avoid translation invariance the basic idea is to perform a birkhoff normal form argument on the hamiltonian as in a paper by bambusi and cuccagna on the stability of the 0 solution for nlkg but in our case the natural coordinates arising from the linearization are not canonical so we need also to apply the darboux theorem with some care though in order not to destroy some nice features of the initial hamiltonian

stable directions for small nonlinear dirac standing waves we prove that for a dirac operator with no resonance at thresholds nor eigenvalue at thresholds the propagator satisfies propagation and dispersive estimates when this linear operator has only two simple eigenvalues sufficiently close to each other we study an associated class of nonlinear dirac equations which have stationary solutions as an application of our decay estimates we show that these solutions have stable directions which are tangent to the subspaces associated with the continuous spectrum of the dirac operator this result is the analogue in the dirac case of a theorem by tsai and yau about the schrodinger equation to our knowledge the present work is the first mathematical study of the stability problem for a nonlinear dirac equation

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

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

on asymptotic stability in energy space of ground states for nonlinear schrodinger equations we consider nonlinear schrodinger equations iu t delta u beta u 2 u 0 text for t x in mathbb r times mathbb r d where d 3 and is smooth we prove that symmetric finite energy solutions close to orbitally stable ground states converge to a sum of a ground state and a dispersive wave as t assuming the so called the fermi golden rule fgr hypothesis we improve the sign condition required in a recent paper by gang zhou and i m sigal

spectra of positive and negative energies in the linearized nls problem we study the spectrum of the linearized nls equation in three dimensions in association with the energy spectrum we prove that unstable eigenvalues of the linearized nls problem are related to negative eigenvalues of the energy spectrum while neutrally stable eigenvalues may have both positive and negative energies the nonsingular part of the neutrally stable essential spectrum is always related to the positive energy spectrum we derive bounds on the number of unstable eigenvalues of the linearized nls problem and study bifurcations of embedded eigenvalues of positive and negative energies we develop the l 2 scattering theory for the linearized nls operators and recover results of grillakis 5 with a fermi golden rule c

asymptotic dynamics of nonlinear schrodinger equations with many bound states abstract we consider a nonlinear schrodinger equation with a bounded localized potential in r 3 the linear hamiltonian is assumed to have three or more bound states with the eigenvalues satisfying some resonance conditions suppose that the initial data is localized and small of order n in h1 and that its ground state component is larger than n3 e with e 0 small we prove that the solution will converge locally to a nonlinear ground state as the time tends to infinity

orbit bifurcations and the scarring of wave functions we extend the semiclassical theory of scarring of quantum eigenfunctions n q by classical periodic orbits to include situations where these orbits undergo generic bifurcations it is shown that n q 2 averaged locally with respect to position q and the energy spectrum en has structure around bifurcating periodic orbits with an amplitude and length scale whose dependence is determined by the bifurcation in question specifically the amplitude scales as and the length scale as and values of the scar exponents and are computed for a variety of generic bifurcations in each case the scars are semiclassically wider than those associated with isolated and unstable periodic orbits moreover their amplitude is at least as large and in most cases larger in this sense bifurcations may be said to give rise to superscars the competition between the contributions from different bifurcations to determine the moments of the averaged eigenfunction amplitude is analysed we argue that there is a resulting universal scaling in the semiclassical asymptotics of these moments for irregular states in systems with mixed phase space dynamics finally a number of these predictions are illustrated by numerical computations for a family of perturbed cat maps

inverse scattering for lasso graph the inverse problem for the magnetic schrodinger operator on the lasso graph with different matching conditions at the vertex is investigated it is proven that the titchmarsh weyl function known f

robust convergence of multi point flux approximation on rough grids this paper establishes the convergence of a multi point flux approximation control volume method on rough quadrilateral grids by rough grids we refer to a family of refined quadrilateral grids where the cells are not required to approach parallelograms in the asymptotic limit in contrast to previous convergence results for these methods we consider here a version where the flux approximation is derived directly in the physical space and not on a reference cell as a consequence less regular grids are allowed however the extra cost is that the symmetry of the method is lost

asymptotic dynamics of nonlinear schrodinger equations resonance dominated and dispersion dominated solutions we consider a linear schrodinger equation with a nonlinear perturbation in r 3 assume that the linear hamiltonian has exactly two bound states and its eigenvalues satisfy some resonance condition we prove that if the initial data is sufficiently small and is near a nonlinear ground state then the solution approaches to certain nonlinear ground state as the time tends to infinity furthermore the difference between the wave function solving the nonlinear schrodinger equation and its asymptotic profile can have two different types of decay the resonancedominated solutions decay as t 1 2 or the dispersion dominated solutions decay at least like t 3 2 c 2002 john wiley sons inc

non linear wave and schrodinger equations i instability of periodic and quasiperiodic solutions we investigate stability of periodic and quasiperiodic solutions of linear wave and schrodinger equations under non linear perturbations we show in the case of the wave equations that such solutions are unstable for generic perturbations for the schrodinger equations periodic solutions are stable while the quasiperiodic ones are not we extend these results to periodic solutions of non linear equations

mixed finite elements for elasticity there have been many efforts dating back four decades to develop stable mixed finite elements for the stress displacement formulation of the plane elasticity system this requires the development of a compatible pair of finite element spaces one to discretize the space of symmetric tensors in which the stress field is sought and one to discretize the space of vector fields in which the displacement is sought although there are number of well known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields the symmetry of the stress field is a substantial additional difficulty and the elements presented here are the first ones using polynomial shape functions which are known to be stable we present a family of such pairs of finite element spaces one for each polynomial degree beginning with degree two for the stress and degree one for the displacement and show stability and optimal order approximation we also analyze some obstructions to the construction of such finite element spaces which account for the paucity of elements available

linearized instability for nonlinear schrodinger and klein gordon equations in this paper i am introducing a new technique for proving instability of bound states for hamiltonian systems there are already two disparate types of instability results in the literature the approach developed by strauss shatah 20 gave an instability criterion coming from the variational structure of the problem on the other hand jones approach 11 produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques it turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results finally in order to demonstrate how this method works i apply the instability criterion in some specific examples