rational singularities and almost split sequences the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the finite dimen sional representations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules of the quotient singularity associated with g over the past decade almost split sequences have been playing an increasingly important role in the representation theories of finite dimensional algebras and classical orders see 5 and 3 8 for basic existence theorems in these contexts while they have been known for some time to exist in higher dimensional situations 3 it has not been at all clear how they related to singularity theory if at all the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the fimte dimensional complex represen tations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules over the quotient singularity r associated with g as in the case of finite dimensional algebras the ar quiver of reflexive r modules is defined in terms of the almost split sequences of reflexive r modules in the case g c sl 2 c mckay observed that the underlying graph of the mckay quiver with the trivial module removed is isomorphic to the desingulariza tion graph of the associated singularity various explanations of this phenomenon have been given by knorrer gonzalez sprinberg verdier 6 and artin verdier 1 which along the way have established most explicitly in 1 a natural one to one correspondence between the indecomposable reflexive r modules and the nodes of the desingularization graph but why the almost split sequences describe the edge of the desingularization graph still remains to be explained an effort has been made to make this paper as self contained as possible in particular no prior knowledge of almost split sequences is required before describing the contents of the six sections of this paper we fix some notation throughout this paper g is a finite group k an algebraically closed field of characteristic not dividing the order of g and v a two dimensional k representation of g setting s k x y the k algebra of formal power series the two dimen sional representation v gives a linear action of g on s as a group of k algebra automorphisms we denote by s g the skew group ring given by this action

on the birational nature of lifting abstract let x and y be proper birational varieties say with only rational double points over a perfect field k of positive characteristic if x lifts to w n k is it true that y has the same lifting property this is true for smooth surfaces but we show by example that this is false for smooth varieties in higher dimension and for surfaces with canonical singularities we also answer a stacky analogue of this question given a canonical surface x with minimal resolution y and stacky resolution x we characterize when liftability of y is equivalent to that of x the main input for our results is a study of how the deformation functor of a canonical surface singularity compares with the deformation functor of its minimal resolution this extends work of burns and wahl to positive characteristic as a byproduct we show that tjurina s vanishing result fails for every canonical surface singularity in every positive characteristic

the versal deformation space of a reflexive module on a rational cone abstract by an approach based on results of a ishii we describe the versal deformation space of any reflexive module on the cone over the rational normal curve of degree m for each component a resolution is given as the total space of a vector bundle on a grassmannian the vector bundle is a sum of copies of the cotangent bundle the canonical sub bundle the dual of the canonical quotient bundle and the trivial line bundle via an embedding in a trivial bundle we obtain the components by projection in particular we give equations for the minimal stratum in the chern class filtration of the versal deformation space we obtain a combinatorial description of the local deformation relation and a classification of the components in particular we give a formula for the number of components

vibration analysis of a continuous system subject to generic forms of actuation forces and sensing devices this work provides a general formulation to solve vibration problems for continuous systems with damping effects including modal transient harmonic and spectrum response analyses in modal analysis the system eigenvalues and corresponding eigenfunctions can be determined the orthogonal relations of eigenfunctions are shown for transient harmonic and spectrum analyses the generic force actuator functions and response sensing operators are introduced respectively and used to derive the system response the time domain response is obtained for transient analysis the frequency response function is derived for harmonic analysis and statistical quantities of response variables due to random excitation are determined in spectrum analysis the solution for each type of analysis can be formulated and expressed in a concise format in terms of generic force actuator and response sensor mode shape functions in particular one dimensional beam and two dimensional plate vibration analyses are illustrated by following the developed generic formulation this work provides the complete analytical solutions of four types of vibration analyses for continuous systems and can be applied to other engineering structures as well

the effect of modal coupling in random vibration analysis when modal analysis techniques are used to determine spectral densities of the responses of distributed parameter linear mechanical structures subjected to stationary random excitation a summation of all modal spectral densities both direct and cross should be performed a very common textbook recommendation is that the modal cross spectral densities may be neglected if certain conditions are satisfied in this paper that recommendation will be discussed using a simple example the influence of modal cross spectral densities on the spectral densities of some responses of a simply supported beam will be investigated response standard deviations will be determined and the importance of the modal cross spectral densities in some frequency ranges covering also resonance frequencies of the beam will be demonstrated special interest is devoted to extreme values of some response processes

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

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

on the time decay of a wave packet in a one dimensional finite band periodic lattice nonstationary schrodinger equation with a periodic finite band potential p x is considered the green s function g x x t of this equation is investigated when t asymptotics for g x x t are specified it is shown that for large velocities v x x t the principal term in asymptotics of g x x t t coincides with the green s function for p 0 the principal term in the asymptotics of g x x t in the case v is equal to a sum of green s functions of unperturbed problems for particles whose masses are equal to effective masses of the hill operator under investigation

lp lp estimates for the schrodinger equation on the line and inverse scattering for the nonlinear schrodinger equation with a potential abstract in this paper i prove a l p l p estimate for the solutions to the one dimensional schrodinger equation with a potential in l 1 where in the generic case 3 2 and in the exceptional case i e when there is a half bound state of zero energy 5 2 i use this estimate to construct the scattering operator for the nonlinear schrodinger equation with a potential i prove moreover that the low energy limit of the scattering operator uniquely determines the potential and the coupling constant of the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonlinearity

surface singularities of finite buchsbaum representation type the purpose of this paper is to determine the structure of surface singularities of finite buchsbaum representation type and the main results are summarized into the following let r be a noetherian complete local ring of dim r 2 and assume that the residue class field of r is algebraically closed let e r and v r denote respectively the multiplicity and the embedding dimension of r then the following three conditions are equivalent 1 r has a finite buchsbaum representation type that is r possesses only finitely many isomorphism classes of indecomposable maximal buchsbaum r modules 2 e r 1 and v r 3 3 r p xi where p is a three dimensional complete regular local ring with maximal ideal n x n n 2 and i is an ideal of p such that ht p i 2

approximate solutions of non classically damped linear systems in normalized and physical co ordinates the goal of this note was to show that the accuracy of the approximate solution in the physical coordinates cannot be determined easily and hence to show that the approximate technique can possibly lead to inaccurate solutions in the physical coordinates

the classification of special cohen macaulay modules in this paper we completely classify all the special cohen macaulay cm modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities in every case we exhibit the specials explicitly in a combinatorial way our result relies on realizing the specials as those cm modules whose first ext group vanishes against the ring r thus reducing the problem to combinatorics on the ar quiver such possible ar quivers were classified by auslander and reiten we also give some general homological properties of the special cm modules and their corresponding reconstruction algebras

chern classes of reflexive sheaves on normal surfaces we define chern classes of reflexive sheaves using wahl s relative local chern classes of vector bundles the main result of the paper bounds contributions of singularities of a sheaf to the riemann roch formula using it we are able to prove inequality in wahl s conjecture on relative asymptotic rr formula for rank 2 vector bundles moreover we prove that if wahl s conjecture is true for a singularity then it is true for any its quotient this implies wahl s conjecture for quotient singularities and for quotients of cones over elliptic curves

mckay correspondence for canonical orders canonical orders introduced in the minimal model program for orders are simultaneous generalisations of kleinian singularities k s t g g sl 2 and their associated skew group rings k s t g in this paper we construct minimal resolutions of canonical orders via non commutative cyclic covers and skew group rings this allows us to exhibit a derived equivalence between minimal resolutions of canonical orders and the skew group ring form of the canonical order in all but one case the fourier mukai transform used to construct this equivalence allows us to make explicit the numerical version of the mckay correspondence for canonical orders noted in chan hacking and ingalls which relates the exceptional curves of the minimal resolution to the indecomposable reflexive modules of the canonical order

surface singularities dominated by smooth varieties we give a version in characteristic p 0 of mumford s theorem charac terizing a smooth complex germ of surface x x by the triviality of the topological funda mental group of u xnfxg

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

relaxation of solitons in nonlinear schrodinger equations with potential abstract in this paper we study dynamics of solitons in the generalized nonlinear schrodinger equation nls with an external potential in all dimensions except for 2 for a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space centered near different critical points of the potential we call those solutions which are centered near the minima of the potential and which minimize energy restricted to l 2 unit sphere trapped solitons or just solitons in this paper we prove under certain conditions on the potentials and initial conditions that trapped solitons are asymptotically stable moreover if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position the dynamical law of motion of the soliton i e effective equations of motion for the soliton s center and momentum is close to newton s equation but with a dissipative term due to radiation of the energy to infinity

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

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

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

three dimensional flops and noncommutative rings limburgs univ ctr dept math b 3590 diepenbeek belgium van den bergh m limburgs univ ctr dept math univ campus b 3590 diepenbeek belgium vdbergh luc ac be

the fundamental module of a normal local domain of dimension 2 the fundamental module e of a normal local domain r m of dimension 2 is defined by the nonsplit exact sequence 0 a k a e a m a 0 where k is the canonical module of r we prove that if r is complete with r m c then e is decomposable if and only if r is a cyclic quotient singularity various other properties of fundamental modules will be discussed 0 introduction let r m be a normal local domain of dimension 2 which possesses the canonical module k let c r denote the category of finitely generated reflexive r modules note that a finitely generated r module is an object in c r if and only if it is a maximal cohen macaulay module over r by definition k is a reflexive module of rank 1 and it satisfies extr r m k r m see herzog and kunz 8 for the details we denote the duality with respect to r resp k by resp that is homr r and homr k remark that a finitely generated r module m lies in c r if and only if m m or equivalently m m we make the following definition 0 1 since ext m k ext2r r m k r m there uniquely exists the nonsplit exact sequence 0 k e m 0 which is called the fundamental sequence in 3 in particular the module e appearing in the middle term of this sequence is also unique up to isomorphism we call e the fundamental module of r e is said to be decomposable if there is an isomorphism e a e b where a and b are nontrivial ideals of r otherwise e is indecomposable in this paper we are interested in the properties of the fundamental modules in particular we are mostly concerned with its decomposability in fact we can prove that if r is complete with the residue field c then r has the decomposable fundamental module if and only if r is a cyclic quotient singularity see theorem 2 1 and if this is the case then the ar quiver of r will be easily described in 1 we summarize some elementary properties of fundamental modules in particular we can prove that if r is a hypersurface then the fundamental module of r is isomorphic to the third syzygy of k in 2 our main theorem 2 1 stated above will be proved the reader will notice that the keys of the proof are the first author s theorem 11 theorem 1 1 and the theorem of herzog auslander auslander 3 theorem 4 9 herzog 71 received by the editors june 26 1986 and in revised form july 29 1987 1980 mathematics subject classification 1985 revision primary 13h10 13c13 secondary 14b05 14j17

deformations of rational surface singularities and reflexive modules with an application to flops abstract blowing up a rational surface singularity in a reflexive module gives a any partial resolution dominated by the minimal resolution the main theorem shows how deformations of the pair singularity module relates to deformations of the corresponding pair of partial resolution and locally free strict transform and to deformations of the underlying spaces the results imply some recent conjectures on small resolutions and flops

response errors of non proportionally lightly damped structures in this letter we show that for flexible structures with small non proportional damping neglecting the off diagonal terms of the modal damping matrix in most practical cases imposes negligible errors in the system dynamics the shahruz criterion which allows neglecting the off diagonal terms of the modal matrix was relaxed

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

maximal functions associated to filtrations abstract let t be a bounded linear or sublinear operator from lp y to lq x a maximal operator t f x supj t f yj x is associated to any sequence of subsets yj of y under the hypotheses that q p and the sets yj are nested we prove that t is also bounded classical theorems of menshov and zygmund are obtained as corollaries multilinear generalizations of this theorem are also established these results are motivated by applications to the spectral analysis of schrodinger operators

blt azumaya algebras and moduli of maximal orders we study moduli spaces of maximal orders in a ramified division algebra over the function field of a smooth projective surface as in the case of moduli of stable commutative surfaces we show that there is a kollar type condition giving a better moduli problem with the same geometric points the stack of blt azumaya algebras one virtue of this refined moduli problem is that it admits a compactification with a virtual fundamental class

random vibration analysis of axially compressed cylindrical shells under turbulent boundary layer in a symplectic system a random vibration analysis of an axially compressed cylindrical shell under a turbulent boundary layer tbl is presented in the symplectic duality system by expressing the cross power spectral density psd of the tbl as a fourier series in the axial and circumferential directions the problem of structures excited by a random distributed pressure due to the tbl is reduced to solving the harmonic response function which is the response of structures to a spatial and temporal harmonic pressure of unit magnitude the governing differential equations of the axially compressed cylindrical shell are derived in the symplectic duality system and then a symplectic eigenproblem is formed by using the method of separation of variables expanding the excitation vector and unknown state vector in symplectic space decoupled governing equations are derived and then the analytical solution can be obtained in contrast to the modal decomposition method mdm the present method is formulated in the symplectic duality system and does not need modal truncation and hence the computations are of high precision and efficiency in numerical examples harmonic response functions for the axially compressed cylindrical shell are studied and a comparison is made with the mdm to verify the present method then the random responses of the shell to the tbl are obtained by the present method and the convergence problems induced by fourier series expansion are discussed finally influences of the axial compression on random responses are investigated

the length classification of threefold flops via noncommutative algebras abstract smooth threefold flops with irreducible centres are classified by the length invariant which takes values 1 2 3 4 5 or 6 this classification by katz and morrison identifies 6 possible partial resolutions of kleinian singularities that can occur as generic hyperplane sections and the simultaneous resolutions associated to such a partial resolution produce the universal flop of length l in this paper we translate these ideas into noncommutative algebra we introduce the universal flopping algebra of length l from which the universal flop of length l can be recovered by a moduli construction and we present each of these algebras as the path algebra of a quiver with relations this explicit realisation can then be used to construct examples of nccrs associated threefold flops of any length as quiver with relations defined by superpotentials to recover the matrix factorisation description of the universal flop conjectured by curto and morrison and to realise examples of contraction algebras

coupling constant thresholds of perturbed periodic hamiltonians we consider schrodinger operators of the form h v w on l2 r 1 2 or 3 with v periodic w short range and a real non negative parameter then the continuous spectrum of h has the typical band structure consisting of intervals separated by gaps in the gaps there may be discrete eigenvalues of h that are functions of the parameter let a b be a gap and e a b an eigenvalue of h we study the asymptotic behavior of e as approaches a critical value 0 called a coupling constant threshold at which the eigenvalue either emerges from or is absorbed into the continuous spectrum a typical question is the following assuming e a as 0 is e a c 0 for some 0 and c 0 or is there an expansion in some other quantity as one expects from previous work in the case v 0 the answer strongly depends on

asymptotic stability of ground states in 3d nonlinear schrodinger equation including subcritical cases we consider a class of nonlinear schrodinger equation in three space dimensions with an attractive potential the nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical in l2 nonlinearities we study the asymptotic stability of the nonlinear bound states i e periodic in time localized in space solutions our result shows that all solutions with small initial data converge to a nonlinear bound state therefore the nonlinear bound states are asymptotically stable the proof hinges on dispersive estimates that we obtain for the time dependent hamiltonian linearized dynamics around a careful chosen one parameter family of bound states that shadows the nonlinear evolution of the system due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations

stable manifolds for all monic supercritical focusing nonlinear schrodinger equations in one dimension keywords critical schrodinger equation stable manifolds modulation theory spectral theory klein gordon equations non linear schrodinger blow up rate standing waves nonintegrable equations invariant manifolds stability theory solitary waves ground states instability reference epfl article 155183doi 10 1090 s0894 0347 06 00524 8 record created on 2010 11 19 modified on 2017 05 12

the peak factor of a short sample of a stationary gaussian process the peak factor of a short sample of a stationary narrowband gaussian random process is discussed several authors have shown by simulation that formulae valid when the maxima of the process are independent will overestimate the peak factor correction functions have been proposed most of them being empirically based on a bandwidth measure of the random process in this paper it is shown that the largest maximum or extremum of the autocorrelation function of the random process may supply a better characterization of the process than does the bandwidth measure simulated values of peak factors and their standard deviations of processes with different spectral shapes are replotted as functions of the largest extremum of the autocorrelogram the values of the peak factor then fall closer together than when they are plotted as functions of the bandwidth measure an alternative semi empirical formula for the peak factor is suggested

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

application of the spectral finite element method to turbulent boundary layer induced vibration of plates the spectral finite element method and equally the dynamic stiffness method use exponential functions as basis functions thus it is possible to find exact solutions to the homogeneous equations of motion for simple rod beam plate and shell structures normally this restricts the analysis to elements where the excitation is at the element ends this study removes the restriction for distributed excitation that in particular has an exponential spatial dependence by the inclusion of the particular solution in the set of basis functions these elementary solutions in turn build up the solution for an arbitrary homogeneous random excitation a numerical implementation for the vibration of a plate excited by a turbulent boundary layer flow is presented the results compare favourably with results from conventional modal analysis

center manifold for nonintegrable nonlinear schrodinger equations on the line in this paper we study the following nonlinear schrodinger equation on the line where f is real valued and it satisfies suitable conditions on regularity on growth as a function of u and on decay as x the generic potential v is real valued and it is chosen so that the spectrum of consists of one simple negative eigenvalue and absolutely continuous spectrum filling 0 the solutions to this equation have in general a localized and a dispersive component the nonlinear bound states that bifurcate from the zero solution at the energy of the eigenvalue of h define an invariant center manifold that consists of the orbits of time periodic localized solutions we prove that all small solutions approach a particular periodic orbit in the center manifold as t in general the periodic orbits are different for t our result implies also that the nonlinear bound states are asymptotically stable in the sense that each solution with initial data near a nonlinear bound state is asymptotic as t to the periodic orbits of nearby nonlinear bound states that are in general different for t

on the asymptotic stability of bound states in 2d cubic schrodinger equation we consider the cubic nonlinear schrodinger equation in two space dimensions with an attractive potential we study the asymptotic stability of the nonlinear bound states i e periodic in time localized in space solutions our result shows that all solutions with small localized in space initial data converge to the set of bound states therefore the center manifold in this problem is a global attractor the proof hinges on dispersive estimates that we obtain for the non autonomous non hamiltonian linearized dynamics around the bound states

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

modal cross spectral terms may be important and an alternative method of analysis be preferable the influence of modal cross spectral densities on structural responses is discussed for linear structures excited by narrow band stationary random processes for a simple beam structure it is shown that these densities play an important role if the narrow band excitation falls close to an anti resonance of the structure an alternative formulation of the solution method in which also modal analysis is used is given it is shown that in some cases of interest the alternative formulation yields better accuracy and involves less computational work than the traditional solution method results obtained by modal analysis of the beam structure are compared with the known exact response

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

higher dimensional auslander reiten theory on maximal orthogonal subcategories abstract we introduce the concept of maximal n 1 orthogonal subcategories over artin algebras and orders and develop n 1 dimensional auslander reiten theory on them we give the n auslander reiten translation and the n auslander reiten duality then show the existence of n almost split sequences and n fundamental sequences we give some examples

chow group of 0 cycles with modulus and higher dimensional class field theory one of the main results of this article is a proof of the rank one case of an existence conjecture on lisse q over bar l sheaves on a smooth variety u over a finite field due to deligne and drinfeld the problem is translated into the language of higher dimensional class field theory over finite fields which describes the abelian fundamental group of u by chow groups of 0 cycles with moduli a key ingredient is the construction of a cycle theoretic avatar of a refined artin conductor in ramification theory originally studied by kazuya kato

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

stability of small periodic waves for the nonlinear schrodinger equation abstract the nonlinear schrodinger equation possesses three distinct six parameter families of complex valued quasiperiodic traveling waves one in the defocusing case and two in the focusing case all these solutions have the property that their modulus is a periodic function of x c t for some c r in this paper we investigate the stability of the small amplitude traveling waves both in the defocusing and the focusing case our first result shows that these waves are orbitally stable within the class of solutions which have the same period and the same floquet exponent as the original wave next we consider general bounded perturbations and focus on spectral stability we show that the small amplitude traveling waves are stable in the defocusing case but unstable in the focusing case the instability is of side band type and therefore cannot be detected in the periodic set up used for the analysis of orbital stability