discrete breathers in thermal equilibrium distributions and energy gaps we study a discrete two dimensional nonlinear system that allows for discrete breather solutions we perform a spectral analysis of the lattice dynamics at thermal equilibrium and use a cooling technique to measure the amount of breathers at thermal equilibrium our results confirm the existence of an energy threshold for discrete breathers the cooling method provides a novel way of measuring and analyzing discrete breather distribution properties in thermal equilibrium

localization and coherence in nonintegrable systems we study the irreversible dynamics of nonlinear nonintegrable hamiltonian oscillator chains approaching their statistical asymptotic states in systems constrained by more than one conserved quantity the partitioning of the conserved quantities leads naturally to localized and coherent structures if the phase space is compact the final equilibrium state is governed by entropy maximization and the coherent structures are stable lumps in systems where the phase space is not compact the coherent structures can be collapsed represented in phase space by a heteroclinic connection of some unstable saddle to infinity

the notion of persistence applied to breathers in thermal equilibrium abstract we study the thermal equilibrium of nonlinear klein gordon chains at the limit of small coupling anticontinuum limit we show that the persistence distribution associated to the local energy density is a useful tool to study the statistical distribution of the so called thermal breathers mainly when the equilibrium is characterized by long lived pinned excitations in that case the distribution of persistence intervals turns out to be a power law we demonstrate also that this generic behaviour has a counterpart in the power spectra where the high frequencies domains nicely collapse if properly rescaled these results are also compared to nonlinear klein gordon chains with a soft nonlinearity for which the thermal breathers are rather mobile entities finally we discuss the possibility of a breather induced anomalous diffusion law and show that despite a strong slowing down of the energy diffusion there are numerical evidences for a normal asymptotic diffusion mechanism but with exceptionally small diffusion coefficients

simple statistical explanation for the localization of energy in nonlinear lattices with two conserved quantities the localization of energy in the discrete nonlinear schrodinger equation is explained with statistical methods the partition function and the entropy of the system are computed for low amplitude initial conditions detailed predictions for the long time solution are derived localized high amplitude excitations absorb a surplus of energy when they emerge as a by product of the production of entropy in the small fluctuations the thermodynamic interpretation of this process applies to many dynamical systems with two conserved quantities

linear stability of perturbed hamiltonian systems theory and a case example in this paper we present a general theory based on a lyapunov schmidt reduction for the linearized stability of a perturbed hamiltonian system with a number of symmetries the reduction leads to an eigenvalue problem in a small subspace of the original system whose eigenvalues and eigenvectors can be used to derive the ones of the perturbed original system the krein signature of the eigenvalues can also be obtained in this framework the method is applied to the case example of the coupled nonlinear schr dinger equations giving excellent agreement with the full numerical linear stability results

chaotic and homoclinic behavior for numerical discretizations of the nonlinear schro dinger equation abstract certain conservative discretizations of the nls can produce irregular behavior we consider the diagonal discretization as a conservative perturbation of the integrable discretization and study the homoclinic crossings in its nonlinear spectrum we find that irregularity sets in when two homoclinic structures are present and in this case many and continual homoclinic crossings occur throughout the irregular time series we indicate a melnikov analysis to study the consequences of this homoclinic behavior

lp functional and j holomorphic curves in kahler manifolds abstract in this paper we give another variational characterization of j m holomorphic curves in kahler manifolds generalizing our results in 1 we prove that if the functional l p has a critical point or a stable point in the fixed kahler class then the immersion is j m holomorphic or lagrangian

localized breather like solution in a discrete klein gordon model and application to dna abstract non propagating localized oscillating solutions are studied in the case of a discrete one dimensional system with the 2 3 power onsite potential a multiple scale asymptotic perturbation expansion is not adequate for describing the large amplitude oscillations we consider starting from an exact pulse like solution in the continuum we derive an approximate oscillating solution our numerical simulations reveal that whereas the excitations are strongly damped in the strong intersite coupling case they are long lived in the discrete cases an additional modulation phenomenon of the oscillation seen in the simulation is explained by combination of the breather oscillations with the zero group velocity phonons modes then applying the green s functions formalism in analogy with a previous work s takeno et al prog theor phys suppl 94 1988 we derive a very accurate expression for the discrete breather with a quasi infinite lifetime lastly we demonstrate how such excitations can explain why the breathers make the dominant contributions in the dynamics of a simple model of dna

q breathers in fermi pasta ulam chains existence localization and stability the fermi pasta ulam fpu problem consists of the nonequipartition of energy among normal modes of a weakly anharmonic atomic chain model in the harmonic limit each normal mode corresponds to a periodic orbit in phase space and is characterized by its wave number q we continue normal modes from the harmonic limit into the fpu parameter regime and obtain persistence of these periodic orbits termed here q breathers qb they are characterized by time periodicity exponential localization in the q space of normal modes and linear stability up to a size dependent threshold amplitude trajectories computed in the original fpu setting are perturbations around these exact qb solutions the qb concept is applicable to other nonlinear lattices as well

existence and stability of breathers in lattices of weakly coupled two dimensional near integrable hamiltonian oscillators a method of proving the existence of breathers in lattices of two dimensional weakly coupled hamiltonian oscillators is presented this method has the advantage that provides a first approximation of the initial conditions of the breather solution together with its linear stability since the proof relies on the use of action angle variables we also present a method to perform the calculations without knowledge of the specific transformation we illustrate the results in a lattice of quartic oscillators and in another one that consists of coupled 2d morse oscillators finally we examine the possibility of extending these results in lattices of more than two degrees of freedom per lattice site

nonlinear chains inside walls abstract the conservative dynamics of a 1d chain of units coupled with fpu type nonlinear interactions is considered stationary patterns in such chains emerge due to a balance of coupling energy between neighbouring units particularly interesting are the nontrivial stationary states which contain segments of positive and negative slope this results in a zig zag pattern in the case of periodic boundary conditions and in kink or anti kink solutions in the case of the free boundary conditions imposing constraints on the chain by way of two confining infinitely high walls has repercussions for the stability of these stationary states here such stationary states commensurable with the available space between the two walls are examined in detail and their respective stability properties are determined analytically by invoking the transfer matrix method strikingly stationary anti kink solutions and periodic zig zag states being unstable in the absence of confining walls become stable when confining walls are introduced furthermore simulations reveal that chains with randomly generated initial conditions can seek out these patterns thus localising energy and persist for considerable time

nonpersistence of breather families for the perturbed sine gordon equation we show that up to one exception and as a consequence of first order perturbation theory only it is impossible that a large portion of the well known family of breather solutions to the sine gordon equation could persist under any nontrivial perturbation of the form u tt u xx sin u varepsilon delta left u right o left varepsilon 2 right where is an analytic function in anarbitrarily small neighbourhood ofu 0 improving known results we analyze and overcome the particular difficulties that arise when one allows the domain of analyticity of to be small the single exception is a one dimensional linear space of perturbation functions under which the full family of breathers does persist up to first order in e

focusing a mechanism for instability of nonlinear finite difference equations abstract a mechanism for the destabilization of numerical algorithms for partial differential equations is suggested the novelty of the work is that it attempts to explain the dynamical process by which noise can localize on a spatial grid and cause finite amplitude instability thresholds to be exceeded at distinct locations

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

numerical study of two dimensional disordered klein gordon lattices with cubic soft anharmonicity localized oscillations appear both in ordered nonlinear lattices breathers and in disordered linear lattices anderson modes numerical studies on a class of two dimensional systems of the klein gordon type show that there exist two different types of bifurcation in the path from nonlinearity order to linearity disorder inverse pitchforks with or without period doubling and saddle nodes this was discovered for a one dimensional system in a previous work of archilla mackay and marin the appearance of a saddle node bifurcation indicates that nonlinearity and disorder begin to interfere destructively and localization is not possible in contrast the appearance of a pitchfork bifurcation indicates that localization persists

discrete breathers classical and quantum abstract discrete breathers are time periodic spatially localised motions in networks of oscillators their occurrence in three principal contexts is reviewed i autonomous hamiltonian or reversible systems ii autonomous forced damped systems and iii time periodically forced systems the main proposed area of application however is to molecular crystals where quantum effects must be taken into account thus an idea for a theory of quantum discrete breathers is also presented though some details remain to be worked out

discrete compactons some exact results in this letter we use the method of constructing exact solutions on lattices proposed by kinnersley and described in schmidt 1979 phys rev b 20 4397 to obtain exact compacton solutions in discrete models we examine the linear stability of such solutions both for the bright compacton and for the dark compacton cases we focus on a quantization condition that the width of the profile should satisfy we also use this quantization condition to examine the possibility of compact coherent structures travelling in discrete settings our results are obtained for sinusoidal profiles and then generalized to elliptic functions of arbitrary modulus the possibility of multi compacton solutions is considered

brownian asymptotics in a poissonian environment we consider a brownian motion moving in a random potential obtained by translating a given fixed non negative shape function at the points of a poisson cloud we derive the almost sure principal long time behavior of the expectation of the natural feynman kac functional which is insensitive to the detail of the shape function we also study the situation of hard obstacles where brownian motion is killed once it comes within distancea of a point of the cloud the nature of the results then changes between the case whena is small or large in connection with the presence or absence of an infinite component in the complement of the obstacles

grid resonances focusing and benjamin feir instabilities in leapfrog time discretizations abstract w l briggs a c newell and t sarie j comput phys 50 83 1983 studied the solutions of a class of discretizations of u t au x uu x 0 where a is a constant they discovered that the numerical solution of the discretization becomes unbounded despite the fact that theoretically the solution should remain bounded they attributed this anomalous behavior to a focusing mechanism in this paper we make use of a multiple scales analysis and show that the instabilities are caused by a resonance effect introduced by the discretizations the contributions from the space and time discretizations are analysed separately and in detail thus the structure of the focusing mechanism becomes transparent

nonlinear modes in coupled rotator models abstract we compare the properties of generalized nonlinear klein gordon equations with a sinusoidal coupling sine lattice equations to those of the usual nonlinear klein gordon equation with harmonic coupling for three equations leading to the same sine gordon equation in the continuum limit using hirota s bilinear method we find a strong analogy between the properties of nonlinear oscillatory modes in the three equations numerical simulations confirm this analogy even for very discrete cases however we point out a fundamental difference between the harmonic and sinusoidal coupling by exhibiting a new class of localized nonlinear excitations which are specific to the sinusoidal coupling the rotating modes they can be created thermally an approximate analytical solution is obtained for these modes which are intrinsically discrete and appear to be extremely stable even in collisions with breathers possible implications for physics are discussed

chaotic trajectories in the standard map the concept of anti integrability a rigorous proof is given in the standard map associated with a frenkel kontorowa model for the existence of chaotic trajectories with unbounded momenta for large enough coupling constant k k0 these chaotic trajectories with finite entropy per site are coded by integer sequences mi such that the sequence bi mi 1 mi 1 2mi be bounded by some integer b the bound k0 in k depends on b and can be lowered for coding sequences mi fulfilling more restrictive conditions the obtained chaotic trajectories correspond to stationary configurations of the frenkel kontorowa model with a finite non zero photon gap called gap parameter in dimensionless units this property implies that the trajectory or the configuration ui can be uniquely continued as a uniformly continuous function of the model parameter k in some neighborhood of the initial configuration a non zero gap parameter implies that the lyapunov coefficient is strictly positive when it is defined in addition the existence of dilating and contracting manifolds is proven for these chaotic trajectories exotic trajectories such as ballistic trajectories are also proven to exist as a consequence of these theorems the concept of anti integrability emerges from these theorems in the anti integrable limit which can be only defined for a discrete time dynamical system the coordinates of the trajectory at time i do not depend on the coordinates at time i 1 thus at this singular limit the existence of chaotic trajectories is trivial and the dynamical system reduces to a bernoulli shift it is well known that the kam tori of symplectic dynamical originates by continuity from the invariant tori which exists in the integrible limit under certain conditions in a similar way it appears that the chaotic trajectories of dynamical systems originate by continuity from those which exists at the anti integrable limits also under certain conditions

on the variational structure of breather solutions i sine gordon equation abstract in this paper we describe stability properties of the sine gordon breather solution these properties are first described by suitable variational elliptic equations which also implies that the stability problem reduces in some sense to i the study of the spectrum of explicit linear systems and ii the understanding of how bad directions if any can be controlled using low regularity conservation laws then we discuss spectral properties of a fourth order linear matrix system using numerical methods we confirm that all spectral assumptions leading to the h 2 h 1 stability of sg breathers are numerically satisfied even in the ultra relativistic singular regime

targeted energy transfer between a rotor and a morse oscillator a model for selective chemical dissociation abstract standard kramers theory of chemical reactions involves a coupling with a langevin thermal bath which intrinsically forbids the possible existence of discrete breathers dbs i e local modes however it is now known that in complex systems that energy may focus for long time as discrete breathers local mode in very special systems targeted energy transfer may occur subsequently to another selected site and induces an ultraselective chemical reaction operating at low temperature the dynamics of the reaction is non brownian but highly coherent along a specific path in the phase space where the system is nearly integrable chemical expressway a simple toy model illustrating this idea is reduced to a rotor weakly coupled to a morse oscillator supposed to represent two specific local modes in a complex system which are appropriately tuned for targeted energy transfer when the rotor is initially rotating with a frequency resonant with those of the morse oscillator at rest the energy of the rotor is almost completely transferred to the morse oscillator and induces chemical dissociation the periodic oscillations of the rotor and morse oscillator remain coherent and their frequencies simultaneously vary but always remain resonant this process is analytically described within an integrable approximation numerical investigations of this model confirm that in the appropriate conditions the particle in the morse oscillator is indeed promptly ejected at infinity with a finite velocity chemical dissociation despite some chaotic transient manifesting imperfect integrability

parabolic problems for the anderson model i intermittency and related topics the objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics in particular in hydrodynamics on a qualitative level intermittent random fields are distinguished by the appearance of sparsely distributed sharp peaks which give the main contribution to the formation of the statistical moments the paper deals with the cauchy problem t u t x hu t x u 0 x t 0 x 0 t x d for the anderson hamiltonianh x x d where is a generally unbounded spatially homogeneous random potential this first part is devoted to some basic problems using percolation arguments a complete answer to the question of existence and uniqueness for the cauchy problem in the class of all nonnegative solutions is given in the case of i i d random variables necessary and sufficient conditions for intermittency of the fieldsu t ast are found in spectral terms ofh rough asymptotic formulas for the statistical moments and the almost sure behavior ofu t x ast are also derived

a mean field statistical theory for the nonlinear schrodinger equation abstract a statistical model of self organization in a generic class of one dimensional nonlinear schrodinger nls equations on a bounded interval is developed the main prediction of this model is that the statistically preferred state for such equations consists of a deterministic coherent structure coupled with fine scale random fluctuations or radiation the model is derived from equilibrium statistical mechanics by using a mean field approximation of the conserved hamiltonian and particle number l2 norm squared for finite dimensional spectral truncations of the nls dynamics the continuum limits of these approximated statistical equilibrium ensembles on finite dimensional phase spaces are analyzed holding the energy and particle number at fixed finite values the analysis shows that the coherent structure minimizes total energy for a given value of particle number and hence is a solution to the nls ground state equation and that the remaining energy resides in gaussian fluctuations equipartitioned over wave numbers some results of direct numerical integration of the nls equation are included to validate empirically these properties of the most probable states for the statistical model moreover a theoretical justification of the mean field approximation is given in which the approximate ensembles are shown to concentrate on the associated microcanonical ensemble in the continuum limit

turbulent transport and the random occurrence of coherent events abstract we suggest that the transport properties and dissipation rates of a wide class of turbulent flows are determined by the random occurrence of coherent events which correspond to certain orbits which we call homoclinic excursions in the high dimensional strange attractor homoclinic excursions are trajectories in the noncompact phase space that are attracted to special orbits which connect saddle points in the finite region of phase space to infinity and represent organized structures in the flow field this picture also suggests that one can compute fluxes using a relatively low dimensional description of the flow a method for extracting the organized structures from a time series is given and provides a local analogue of the notion of lyapunov exponents

mel nikov analysis of numerically induced chaos in the nonlinear schro dinger equation abstract certain hamiltonian discretizations of the periodic focusing nonlinear schrodinger equation nls have been shown to be responsible for the generation of numerical instabilities and chaos in this paper we undertake a dynamical systems type of approach to modeling the observed irregular behavior of a conservative discretization of the nls using heuristic mel nikov methods the existence of a pair of isolated homoclinic orbits is indicated for the perturbed system the structure of the persistent homoclinic orbits that are predicted by the mel nikov theory possesses the same features as the wave form observed numerically in the perturbed system after the onset of chaotic behavior and appears to be the main structurally stable feature of this type of chaos the mel nikov analysis implemented in the pde context appears to provide relevant qualitative information about the behavior of the pde in agreement with the numerical experiments in a neighborhood of the persistent homoclinic orbits the existence of a horseshoe is investigated and related with the onset of chaos in the numerical study

quenched asymptotics for brownian motion of renormalized poisson potential and for the related parabolic anderson models let bs be a d dimensional brownian motion and dx be an independent poisson field on rd the almost sure asymptotics for the logarithmic moment generating function loge0exp t0v bs ds t are investigated in connection with the renormalized poisson potential of the form v x rd1 y x p dy dy x rd the investigation is motivated by some practical problems arising from the models of brownian motion in random media and from the parabolic anderson models

localized excitations in two dimensional hamiltonian lattices we analyze the origin and features of localized excitations in a discrete two dimensional hamiltonian lattice the lattice obeys discrete translational symmetry and the localized excitations exist because of the presence of nonlinearities we connect the presence of these excitations with the existence of local integrability of the original n degree of freedom system on the basis of this explanation we make several predictions about the existence and stability of these excitations this work is an extension of previously published results on vibrational localization in one dimensional nonlinear hamiltonian lattices phys rev e 49 1994 836 thus we confirm earlier suggestions about the generic property of hamiltonian lattices to exhibit localized excitations independent on the dimensionality of the lattice

the mild and weak solutions of a stochastic parabolic anderson equation abstract in this paper we investigate the existence of the solutions to the stochastic parabolic anderson equation with gaussian potential and perturbed by external force that driven by classical brownian motion we show the sufficient conditions that the shape function must satisfy for the existence of the mild solution and weak solution to this type of equation

discrete nonlinear schrodinger equations with superlinear nonlinearities abstract in this paper we study the discrete nonlinear equation lu n u n n g n u n n z where belongs to a finite spectral gap of the operator l or it is a lower edge of a finite spectral gap by using a variant generalized weak linking theorem for strongly indefinite problem developed by schechter and zou we will study the existence of nontrivial solitons for this equation with gn is super linear

conditions on the existence of localized excitations in nonlinear discrete systems we use recent results that localized excitations in nonlinear hamiltonian lattices can be viewed and described as multiple frequency excitations their dynamics in phase space takes place on tori of corresponding dimension for a one dimensional hamiltonian lattice with nearest neighbour interaction we transform the problem of solving the coupled differential equations of motion into a certain mapping m l 1 f m l m l 1 where m l for every l lattice site is a function defined on an infinite discrete space of the same dimension as the torus we consider this mapping in the tails of the localized excitation i e for l rightarrow pm infty for a generic hamiltonian lattice the thus linearized mapping is analyzed we find conditions of existence of periodic one frequency localized excitations as well as of multiple frequency excitations the symmetries of the solutions are obtained as a result we find that the existence of localized excitations can be a generic property of nonlinear hamiltonian lattices in contrast to nonlinear hamiltonian fields

stable and metastable states and the formation and destruction of breathers in the discrete nonlinear schrodinger equation statistical mechanics explains many localization phenomena of lattices such as the discrete nonlinear schrodinger equation however numerical simulations show that the complete thermalization is rarely achieved instead one observes metastable statistical states that are robust when excited locally this paper investigates thermodynamically metastable states where the trajectory is confined to some part of the energy shell the partition function and the entropy are computed with a perturbation method this method is applicable to stable and metastable states and it allows us to give approximative analytic expressions for the entropy in the complete thermodynamic state space

on homoclinic structure and numerically induced chaos for the nonlinear schro dinger equation it has recently been demonstrated that standard discretizations of the cubic nonlinear schrodinger nls equation may lead to spurious numerical behavior in particular the origins of numerically induced chaos and the loss of spatial symmetry are related to the homoclinic structure associated with the nls equation in this paper an analytic description of the homoclinic structure via soliton type solutions is provided and some consequences for numerical computations are demonstrated differences between an integrable discretization and standard discretizations are highlighted

equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for frenkel kontorova models there is a well known correspondence between the dynamics of symplectic twist maps which represent an important class of hamiltonian systems and the equilibrium states of a class of variational problems in solid state physics known as frenkel kontorova models in this paper it is shown that the key concepts of uniform hyperbolicity in the first context and phonon gap in the second context are equivalent this allows one to transfer many ideas between the two and hence to deduce new results for example we prove the uniform hyperbolicity of certain invariant sets for symplectic twist maps constructed from so called non degenerate anti integrable limits by aubry and abramovici using the concept of phonon gap and deduce that they have measure zero we prove existence of many multi defect equilibrium states for a frenkel kontorova system if it has a single non degenerate defect equilibrium state via standard results in the theory of uniform hyperbolicity

discrete breathers in vibroimpact chains analytic solutions we present exact analytic solutions for discrete breathers in essentially nonlinear oscillatory chains belonging to both of the most common universality classes klein gordon and fermi pasta ulam the exact solutions can be obtained due to use of vibroimpact potentials combining extreme nonlinearity with the possibility of description in terms of a forced linear model under conditions of self consistency a crossover between the cases of high and low energies can be studied directly the solutions obtained may be used as a high energy limit for models with other realistic potentials as well as benchmarks for the testing of approximate approaches in the theory of discrete breathers

large memory capacity in chaotic artificial neural networks a view of the anti integrable limit in the literature it was reported that the chaotic artificial neural network model with sinusoidal activation functions possesses a large memory capacity as well as a remarkable ability of retrieving the stored patterns better than the conventional chaotic model with only monotonic activation functions such as sigmoidal functions this paper from the viewpoint of the anti integrable limit elucidates the mechanism inducing the superiority of the model with periodic activation functions that includes sinusoidal functions particularly by virtue of the anti integrable limit technique this paper shows that any finite dimensional neural network model with periodic activation functions and properly selected parameters has much more abundant chaotic dynamics that truly determine the model s memory capacity and pattern retrieval ability to some extent this paper mathematically and numerically demonstrates that an appropriate choice of the activation functions and control scheme can lead to a large memory capacity and better pattern retrieval ability of the artificial neural network models

confinement of brownian motion among poissonian obstacles in ℝ d d 3 consider brownian motion among random obstacles obtained by translating a fixed compact nonpolar subset of d d 1 at the points of a poisson cloud of constant intensity v 0 assume that brownian motion is absorbed instantaneously upon entering the obstacle set in szn conf sznitman has shown that in d 2 conditionally on the event that the process does not enter the obstacle set up to time t the probability that brownian motion remains within distance t1 4 from its starting point is going to 1 as t goes to infinity we show that the same result holds true for d 3 with t1 4 replaced by t1 d 2 the proof is based on sznitmans refined method of enlargement of obstacles 10 as well as on a quantitative isoperimetric inequality due to hall 4

mild solution to parabolic anderson model in gaussian and poisson potential we consider the parabolic anderson model in a special random homogeneous potential that is generated by the gaussian and poisson random medium together in our model the coefficient v x is not holder continuous with positive probability and thus the model is unlikely to have a path wise solution we construct a mild solution for the parabolic anderson model with random potential of the form v x rdk y x dy w dy where and w denote the independent standard poisson point process and centred gaussian field respectively the case where the potential switches in sign and the poisson field is absent is handled as well

obtaining breathers in nonlinear hamiltonian lattices we present a numerical method for obtaining high accuracy numerical solutions of spatially localized time periodic excitations on a nonlinear hamiltonian lattice we compare these results with analytical considerations of the spatial decay we show that nonlinear contributions have to be considered and we obtain very good agreement between the latter and numerical results we discuss further applications of the method and results

lower and upper estimates on the excitation threshold for breathers in discrete nonlinear schrodinger lattices we propose analytical lower and upper estimates on the excitation threshold for breathers in the form of spatially localized and time periodic solutions in discrete nonlinear schrodinger dnls lattices with power nonlinearity the estimation depending explicitly on the lattice parameters is derived by a combination of a comparison argument on appropriate lower bounds depending on the frequency of each solution with a simple and justified heuristic argument the numerical studies verify that the analytical estimates can be of particular usefulness as a simple analytical detection of the activation energy for breathers in dnls lattices

standing waves for discrete schrodinger equations in infinite lattices with saturable nonlinearities abstract in this paper we consider the periodic discrete nonlinear equation l u n u n g n u n n z lim n u n 0 where l is a jacobi operator and the nonlinearities g n s are asymptotically linear as s in the two different cases is a spectral endpoint of l or it belongs to a finite spectral gap of l we obtain the existence of nontrivial solitons of this equation by using variational methods in particular a necessary and sufficient condition is obtained for the existence of gap solitons of the nonlinear equation here solitons appear when we look for standing waves of some discrete nonlinear schrodinger equations

properties of one dimensional anharmonic lattice solitons abstract the existence of bell and kink shaped solitons moving at constant velocity while keeping a permanent profile is studied in infinite periodic monoatomic chains of arbitrary anharmonicity by taking advantage of the equation of motion being integrable with respect to solitons a second order non linear differential equation involving advanced and retarded terms must be solved which is done by implementing a scheme based on the finite element and newton s methods if the potential has a harmonic limit the asymptotic time decay behaves exponentially and there is a dispersion relation between propagation velocity and decay time inversely if the potential has no harmonic limit the asymptotic regime shows up either as a power law or faster than exponential excellent agreement is achieved with toda s model illustrative examples are also given for the fermi pasta ulam and sine gordon potentials owing to integrability an effective one body potential is worked out in each case lattice and continuum solitons differ markedly from one another as regards the amplitude versus propagation velocity relationship and the asymptotic time behavior the relevance of the linear stability analysis when applied to solitons propagating in an infinite crystal is questioned the reasons preventing solitons from arising in a diatomic lattice are discussed

discrete breathers on symmetry determined invariant manifolds for scalar models on the plane square lattice a group theoretical approach for studying localized periodic and quasiperiodic vibrations in two and three dimensional lattice dynamical models is developed this approach is demonstrated for the scalar models on the plane square lattice the symmetry determined invariant manifolds admitting existence of localized vibrations are found and some types of discrete breathers are constructed on these manifolds a general method using the apparatus of matrix representations of symmetry groups to simplify the standard linear stability analysis is discussed this method allows one to decompose the corresponding system of linear differential equations with time dependent coefficients into a number of independent subsystems whose dimensions are less than the full dimension of the considered system

extreme events in discrete nonlinear lattices we perform statistical analysis on discrete nonlinear waves generated through modulational instability in the context of the salerno model that interpolates between the integrable ablowitz ladik al equation and the nonintegrable discrete nonlinear schrodinger equation we focus on extreme events in the form of discrete rogue or freak waves that may arise as a result of rapid coalescence of discrete breathers or other nonlinear interaction processes we find power law dependence in the wave amplitude distribution accompanied by an enhanced probability for freak events close to the integrable limit of the equation a characteristic peak in the extreme event probability appears that is attributed to the onset of interaction of the discrete solitons of the al equation and the accompanied transition from the local to the global stochasticity monitored through the positive lyapunov exponent of a nonlinear map

finite size effects on instabilities of discrete breathers abstract when the two arcs of the continuous phonon spectrum of the floquet matrix of a discrete breather overlap on the unit circle the breather solution in the infinite lattice might be stable while the corresponding solutions in finite systems appear to be unstable more precisely when the model parameters vary the breather in the finite system exhibits a large number of collisions between the floquet eigenvalues belonging to the phonon spectrum these collisions correspond to complex cascades of instability thresholds followed near after by re entrant stability thresholds we interpret this complex structure on the basis of the band analysis of the matrix of the second variation of the action then we can predict that in the limit of an infinite system the number of instability and stability thresholds in the cascade diverges but simultaneously the maximum amplitude of the instabilities vanishes so that the breather in the infinite system recovers its linear stability as long as all its other localized modes remain stable this is the situation which is required in cretegny et al physica d 119 1998 73 87 for having inelastic phonon scattering with two channels we also analyze the size effects when a floquet eigenvalue associated with a localized mode collides with the floquet continuous phonon spectrum with different krein signature in contrast to the previous case the infinite system is unstable after the collision

chaotic breathers of two types in a two dimensional morse lattice with an on site harmonic potential chaotic breathers of two types are generated in two dimensional morse lattices with on site harmonic potentials in both a triangular and a square configuration initial highest frequency mode disturbances evolve into chaotic breathers depending on the magnitude of the parameters in the morse potential either quasi one dimensional localized chaotic breathers or two dimensional localized bell shaped ones are numerically observed when the on site harmonic potential becomes significant both types of chaotic breather move irregularly within the lattice and finally decay out into a thermalization state after a fairly long time quasi one dimensional localized chaotic breathers are favored as the strength of nonlinearity increases and they decay faster than two dimensional localized ones

exact static solutions of a two dimensional discrete ϕ4 model for a two dimensional scalar discrete 4 model we obtain several exact static solutions in the form of the jacobi elliptic functions jef with arbitrary shift along the lattice the quispel roberts thompson type quadratic maps are identified for the considered two dimensional model by using a jef solution we also show that many of the static solutions can be constructed iteratively from these quadratic maps by starting from an admissible initial value the kink solution having the form of tanh is numerically demonstrated to be generically stable

stationary parabolic anderson model and intermittency this paper is devoted to the analysis of the large time behavior of the solutions of the anderson parabolic problem frac partial u partial t kappa delta u xi x u when the potential x is a homogeneous ergodic random field on d our goal is to prove the asymptotic spatial intermittency of the solution and for this reason we analyze the large time properties of all the moments of the positive solutions this provides an extension to the continuous space d of the work done originally by gartner and molchanov in the case of the lattice d in the process of our moment analysis we show that it is possible to exhibit new asymptotic regimes by considering a special class of generalized gaussian fields interpolating continuously between the exponent 2 which is found in the case of bona fide continuous gaussian fields x and the exponent 3 2 appearing in the case of a one dimensional white noise finally we also determine the precise almost sure large time asymptotics of the positive solutions

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

almost sure asymptotics for the continuous parabolic anderson model we consider the parabolic anderson problem t u u x u on d with initial condition u 0 x 1 here 0 is a diffusion constant and is a random homogeneous potential we concentrate on the two important cases of a gaussian potential and a shot noise poisson potential under some mild regularity assumptions we derive the second order term of the almost sure asymptotics of u t 0 as t