connections between kelvin functions and zeta functions with applications a very simple method for resuming infinite series of the type sup approximately l 1 or 1 l 1 l n 1 2 cosh la k n 1 2 lb is presented as a result a series expansion in powers of a and b is obtained relevant to many physical applications

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

on the dynamics of the holstein model from the anticontinuous limit we consider the holstein model describing an electron interacting with a lattice of identical oscillators we remark that the on site system i e the system in which the interaction between the different sites of the lattice vanishes is integrable and anisocronous this allows us to apply some recent nekhoroshev type results to show that corresponding to the majority of initial data in which the electron probability is concentrated on a finite number of sites the electron probability distribution is approximatively constant for times growing exponentially with the inverse of the coupling parameter moreover for the same times the total energy of the oscillator system is approximatively constant

generalized multidimensional epstein zeta functions explicit formulas are given for the expression of multiple sum zeta functions of the type l1 ln a1 l1 c1 2 an ln cn 2 m2 s where ai 0 m2 0 i 1 n in terms of riemann and hurwitz zeta functions and in terms of kelvin functions using different analytical continuation methods several formulas especially useful for different values of the involved parameters are obtained

inhomogeneous multidimensional epstein zeta functions the pole structure of the inhomogeneous multidimensional epstein zeta function em2n s a1 an n1 nn 1 a1n21 ann2n m2 s is determined using heat kernel techniques the poles of em2n s a1 an are found to be s n 2 n 1 2 1 2 2l 1 2 l n0 furthermore their residues and em2n p a1 an p n0 are given explicitly these results are used to find the high temperature expansion of the helmholtz free energy of a massive spin 0 and spin 1 2 gas subject to dirichlet boundary conditions on hypercuboids in a flat n dimensional space time

bifurcation analysis of the localized modes dynamics in lattices with saturable nonlinearity abstract the dynamics of localized modes in discrete media with saturable nonlinearity are investigated the stability of stationary bright solitons is discussed from the view point of the energy minimum principle and mapping analysis due to the cascade saturation mechanism a bifurcation from trapped to transversely moving localized mode is found for particular values of high power the bifurcation coincides with the existence of the almost perfect separatrix in the corresponding area preserving map in addition the definition of the peierls nabarro effective potential is reconsidered

breather solutions of the discrete nonlinear schrodinger equations with unbounded potentials in this paper i investigate the existence of nontrivial breather solutions of the discrete nonlinear schrodinger equation with the unbounded potential at infinity first i derive a discrete version of compact embedding theorem then combining the nehari manifold approach and the compact embedding theorem i show the existence of breather solutions without palais smale condition the results on the exponential decay of breather solutions are also provided in this paper

limit operators compactness and essential spectra on bounded symmetric domains abstract this paper is a follow up to a recent article about the essential spectrum of toeplitz operators acting on the bergman space over the unit ball as mentioned in the said article some of the arguments can be carried over to the case of bounded symmetric domains and some cannot the aim of this paper is to close the gaps to obtain comparable results for general bounded symmetric domains in particular we show that a toeplitz operator on the bergman space a p is fredholm if and only if all of its limit operators are invertible even more generally we show that this is in fact true for all band dominated operators an algebra that contains the toeplitz algebra moreover we characterize compactness and explain how the berezin transform comes into play in particular we show that a bounded linear operator is compact if and only if it is band dominated and its berezin transform vanishes at the boundary for p 2 band dominated can be replaced by contained in the toeplitz algebra

on algebras generated by toeplitz operators and their representations abstract we study banach and c algebras generated by toeplitz operators acting on weighted bergman spaces a 2 b 2 over the complex unit ball b 2 c 2 our key point is an orthogonal decomposition of a 2 b 2 into a countable sum of infinite dimensional spaces each one of which can be identified with a differently weighted bergman space a 2 d over the complex unit disk d moreover all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a toeplitz operator on a 2 d the symbols of the generating toeplitz operators are chosen to be suitable extensions to b 2 of families s of bounded functions on d symbol classes s that generate important classical commutative and non commutative toeplitz algebras in l a 2 d are of particular interest in this paper we discuss various examples in the case of s c d and s c d l 0 1 we characterize all irreducible representations of the resulting toeplitz operator c algebras their calkin algebras are described and index formulas are provided

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

a simple recurrence for the higher derivatives of the hurwitz zeta function a recurrent formula which allows the calculation of the asymptotic series expansion of any derivative m z a m z a zm of the hurwitz zeta function z a is obtained in particular the first terms of the series corresponding to n a in inverse powers of a are explicitly given for n 0 1 2 3 knowledge of these expressions is fundamental e g in the zeta function regularization procedure

discrete breathers in nonlinear schrodinger hypercubic lattices with arbitrary power nonlinearity we study two specific features of onsite breathers in nonlinear schrodinger systems on d dimensional cubic lattices with arbitrary power nonlinearity i e arbitrary nonlinear exponent n their wavefunctions and energies close to the anti continuum limit small hopping limit and their excitation thresholds exact results are systematically compared to the predictions of the so called exponential ansatz ea and to the solution of the single nonlinear impurity model sni where all nonlinearities of the lattice but the central one where the breather is located have been removed in 1d the exponential ansatz is more accurate than the sni solution close to the anti continuum limit while the opposite result holds in higher dimensions the excitation thresholds predicted by the sni solution are in excellent agreement with the exact results but cannot be obtained analytically except in 1d an ea approach to the sni problem provides an approximate analytical solution that is asymptotically exact as n tends to infinity but the ea result degrades as the dimension d increases this is in contrast to the exact sni solution which improves as n and or d increase finally in our investigation of the sni problem we also prove a conjecture by bustamante and molina c a bustamante m i molina phys rev b 62 23 2000 15287 that the limiting value of the bound state energy is universal when n tends to infinity

multiple solutions for discrete periodic nonlinear schrodinger equations in this paper we study the existence of multibump solutions for discrete nonlinear schrodinger equations with periodic potentials we first reduce the existence of multibump homoclinic solutions to the existence of an isolated homoclinic solution with a nontrivial critical group then we study the existence of homoclinics with nontrivial critical groups for both superlinear and asymptotically linear discrete periodic nonlinear schrodinger equations and we provide simple sufficient conditions for the existence of homoclinics with nontrivial critical groups in the positive definite case as an application we get without any symmetry assumptions infinitely many geometrically distinct homoclinic solutions with exponential decay at infinity

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

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

on the zeta function regularization of a two dimensional series of epstein hurwitz type as a further step in the general program of zeta function regularization of multiseries expressions some original formulas are provided for the analytic continuation to any value of s of two dimensional series of epstein hurwitz type namely n1 n2 0 a1 n1 c1 2 a2 n2 c2 2 s where the aj are positive reals and the cj are not simultaneously nonpositive integers they come out from a generalization to hurwitz functions of the zeta function regularization theorem of the author and romeo phys rev d 40 436 1989 for ordinary zeta functions for s k 0 2 with k 1 2 3 the final results are in fact expressed in terms of hurwitz zeta functions only for general s they also involve bessel functions a partial numerical investigation of the different terms of the exact algebraic equations is also carried out as a by product the series n 0exp a n c 2 a c 0 is conveniently calculated in terms of them

simplified vacuum energy expressions for radial backgrounds and domain walls we extend our previous results of simplified expressions for functional determinants for radial schrodinger operators to the computation of vacuum energy or mass corrections for static but spatially radial backgrounds and for domain wall configurations our method is based on the zeta function approach to the gel fand yaglom theorem suitably extended to higher dimensional systems on separable manifolds we find new expressions that are easy to implement numerically for both zero and non zero temperatures

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

smoothing property for schrodinger equations with potential superquadratic at infinity we prove a smoothing property for one dimensional time dependent schrodinger equations with potentials which satisfy at infinity k 2 as an application we show that the initial value problem for certain nonlinear schrodinger equations with such potentials is l2 well posed we also prove a sharp asymptotic estimate of the lp norm of the normalized eigenfunctions of h v for large energy

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

existence and stability of quasi periodic breathers in networks of ginzburg landau oscillators abstract it is shown that there is an attracting quasi periodic breather in networks of ginzburg landau oscillators

special set and solutions of coupled nonlinear schrodinger equations a special set of n coupled nonlinear schrodinger equations which consists of 2n interaction types each of which is characterized by a specific array of interaction parameters is presented it is shown that every lame function of order n n is a solution for one or more components for one or more interaction types of this special set simple rules that relate interaction and solution types and interesting features of these relationships are presented

breather solutions of the discrete nonlinear schrodinger equations with sign changing nonlinearitya in this paper we investigate the breather solutions of the discrete nonlinear schrodinger equation with the unbounded potential and nonlinearity containing both self focusing and defocusing sites which means the coefficients of nonlinearity change sign by using the generalized nehari manifold method we obtain the existence of nontrivial exponential decay breather solutions

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

correspondence theory on p fock spaces with applications to toeplitz algebras abstract we prove several results concerning the theory of toeplitz algebras over p fock spaces using a correspondence theory of translation invariant symbol and operator spaces the most notable results are the full toeplitz algebra is the norm closure of all toeplitz operators with bounded uniformly continuous symbols this generalizes a result obtained by j xia 25 in the case p 2 which was proven by different methods further we prove that every toeplitz algebra which has a translation invariant c subalgebra of the bounded uniformly continuous functions as its set of symbols is linearly generated by toeplitz operators with the same space of symbols

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

nonlinear localized periodic solutions in a coupled map lattice abstract we prove the existence of nonlinear localized time periodic solutions in a chain of symplectic mappings with nearest neighbour coupling this is a class of systems whose behaviour can be seen as a representation of a lattice of pendula the effect of discrete time changes the mathematical as well as the numerical procedures applying the discrete version of floquet theory eases and clarifies the procedure of proving the existence of the localized time periodic solutions as an extension of the concept of rotobreathers one can produce solutions which rotate at every site of the lattice to consider these we use a general definition of localization

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

local smoothing property and strichartz inequality for schrodinger equations with potentials superquadratic at infinity abstract we study smoothing properties for time dependent schrodinger equations i u t 1 2 u v x u u 0 l 2 r n with potentials which satisfy v x o x m at infinity m 2 we show that the solution u t x is 1 m times differentiable with respect to x at almost all t r and explain that this is the result of the fact that the sojourn time of classical particles with energy in arbitrary compact set is less than ct 1 m during 0 t when is very large we also show strichartz s inequality with derivative loss for such potentials and give its application to nonlinear schrodinger equations

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

homoclinic solutions of discrete nonlinear schrodinger equations with asymptotically or super linear terms abstract in this paper we study the following periodic discrete nonlinear schrodinger equation u n n u n n g n u n n z where the nonlinearity g n is asymptotically or super linear the sufficient conditions on the existence and on the nonexistence of nontrivial solitons was established by using critical point theory

zeta functions formulas and applications abstract the existence conditions of the zeta function of a pseudodifferential operator and the definition of determinant thereby obtained are reviewed as well as the concept of multiplicative anomaly associated with the determinant and its calculation by means of the wodzicki residue exponentially fast convergent formulas valid in the whole of the complex plane and yielding the pole positions and residua that extend the ones by chowla and selberg for the epstein zeta function quadratic form and by barnes affine form are then given after briefly recalling the zeta function regularization procedure in quantum field theory some applications of these expressions in physics are described

discrete solitons for periodic discrete nonlinear schrodinger equations in this paper we study a class of periodic discrete nonlinear schrodinger equations with superlinear nonlinearities and periodic potential by using the critical point theory we establish results on the existence of discrete solitons our results improve some recent ones

excitation threshold for the discrete coupled schrodinger lattice system this paper is concerned with the excitation threshold for the ground state in the coupled discrete nonlinear schrodinger lattice system excitation threshold is characterized by the variational methods we establish the existence of the excitation threshold connected with the dimensionality d of the lattice we prove that if d 2 then the excitation threshold exists and the ground state exists if and only if the total power is greater than the excitation threshold the compactness of the minimizing sequence follows by the concentration compactness principle we also prove the upper estimates on the excitation threshold and the frequency of the ground state

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

on a class of discretizations of hamiltonian nonlinear partial differential equations abstract we present a new class of discretizations of partial differential equations pdes that preserve a momentum like integral of the motion this results in an effective translational invariance for the dynamical problem and the absence of a peierls nabarro barrier that is usually present in discretizations of hamiltonian pdes a general method to construct such discretizations for any nonlinearity is given and the properties of the resulting differential difference equations are analyzed in a number of different cases for nonlinear klein gordon as well for nonlinear schrodinger type systems while for the former nonintegrability of the dynamical problem is evident in the latter case numerical evidence suggests that the behavior is close to the one of integrable systems we also show how static solutions of the equations can be constructed for these discretizations and discuss the similarities and differences of the method with previously reported ones

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

global existence and compact attractors for the discrete nonlinear schrodinger equation abstract we study the asymptotic behavior of solutions of discrete nonlinear schrodinger type dnls equations for a conservative system we consider the global in time solvability and the question of existence of standing wave solutions similarities and differences with the continuous counterpart nls partial differential equation are pointed out for a dissipative system we prove existence of a global attractor and its stability under finite dimensional approximations similar questions are treated in a weighted phase space finally we propose possible extensions for various types of dnls equations

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

analysis of an inhomogeneous generalized epstein hurwitz zeta function with physical applications the inhomogeneous generalized epstein hurwitz like multidimensional series em s a1 am c1 cm c2 n1 nm 0 a1 n1 c1 2 am nm cm 2 c2 s is investigated by means of a nontrivial asymptotic recurrence it is reduced to the one dimensional case f s a b2 n 0 n a 2 b2 s which is then studied in full detail in particular asymptotic expansions for f and its derivatives f s and f a together with analytical continuations of the same in the variable s are explicitly obtained using zeta function techniques several plots and tables of the numerical results are given some explicit applications to the regularization by means of hurwitz zeta functions of different problems that have appeared recently in the physical literature are considered

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

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

an extension of the chowla selberg formula useful in quantizing with the wheeler de witt equation the two dimensional inhomogeneous zeta function series with homogeneous part of the most general epstein type sigma m n in z am2 bmn cn2 q s is analytically continued in the variable s using zeta function techniques a simple formula is obtained which extends the chowla selberg formula to inhomogeneous epstein zeta functions the new expression is then applied to solve the problem of computing the determinant of the basic differential operator that appears in an attempt at quantizing gravity by using the wheeler de witt equation in 2 1 dimensional spacetime with the torus topology

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

toeplitz quantization on fock space for toeplitz operators t f t acting on the weighted fock space h t 2 we consider the semi commutator t f t t g t t fg t where t 0 is a certain weight parameter that may be interpreted as planck s constant hbar in rieffel s deformation quantization in particular we are interested in the semi classical limit begin equation tag lim limits t to 0 t f t t g t t fg t t end equation it is well known that t f t t g t t fg t t tends to 0 under certain smoothness assumptions imposed on f and g this result was recently extended to f g in textup buc c n by bauer and coburn we now further generalize to not necessarily bounded uniformly continuous functions and symbols in the algebra textup vmo cap l infty of bounded functions having vanishing mean oscillation on mathbb c n our approach is based on the algebraic identity t f t t g t t fg t h bar f t h g t where h g t denotes the hankel operator corresponding to the symbol g and norm estimates in terms of the weighted heat transform as a consequence only f or likewise only g has to be contained in one of the above classes for to vanish for g we only have to impose limsup t to 0 h g t t infty e g g in l infty c n we prove that the set of all symbols f in l infty mathbb c n with the property that lim t rightarrow 0 t t ft t g t t fg t lim t rightarrow 0 t g t t f t t gf t t 0 for all g in l infty mathbb c n coincides with textup vmo cap l infty additionally we show that lim limits t to 0 t f t t f infty holds for all f in l infty c n finally we present new examples including bounded smooth functions where does not vanish

zeta function regularization is uniquely defined and well hawking s zeta function regularization procedure 1977 is shown to be rigorously and uniquely defined thus putting an end to the spreading lore about different difficulties associated with it basic misconceptions misunderstandings and errors which keep appearing in important scientific journals when dealing with this beautiful regularization method and other analytical procedures are clarified and corrected

multiple harmonic oscillator zeta functions the partition function for a quantum mechanical system can at least formally be expressed as a high temperature series in terms of the zeta function of the system s hamiltonian this is done explicitly for a system of non interacting harmonic oscillators the hamilton zeta function belongs to a class of zeta functions sigma f ni n1 omega 1 n2 omega 2 nn omega n a s summed over non negative integers ni very little is known about these zeta functions and an investigation of some of their properties is presented

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

complete determination of the singularity structure of zeta functions series of extended epstein type provide examples of non trivial zeta functions with important physical applications the regular part of their analytic continuation is seen to be a convergent or an asymptotic series their singularity structure is completely determined in terms of the wodzicki residue in its generalized form which is proven to yield the residua of all the poles of the zeta function and not just that of the rightmost pole obtainable from the dixmier trace the calculation is a very down to earth application of these powerful functional analytical methods in physics

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

discrete breathers in disordered media abstract we describe some simple physical models where discrete breathers nonlinear localised modes in a lattice exist together with spatial disorder as the models are translation invariant both spatial localisation and spatial disorder are only due to the interplay of nonlinearity and discreteness this result is obtained as a straightforward application of a general result of existence of discrete breathers j a sepulchre r s mackay localised oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators nonlinearity 10 1997 1 35

the cauchy problem in local spaces for the complex ginzburg landau equation i compactness methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces

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

general moment theorems for nondistinct unrestricted partitions a well known result from hardy and ramanujan aysmptotic formulae in combinatory analysis proc lond math soc 17 75 1918 gives an asymptotic expression for the number of possible ways to express an integer as the sum of smaller integers in this vein we consider the general partitioning problem of writing an integer n as a sum of summands from a given sequence of nondecreasing integers under suitable assumptions on the sequence we obtain results using associated zeta functions and saddle point techniques we also calculate higher moments of the sequence as well as the expected number of summands applications are made to various sequences including those of barnes and epstein types these results are of potential interest in statistical mechanics in the context of bose einstein condensation

discrete schrodinger equations with sign changing nonlinearities infinitely many homoclinic solutions abstract we obtain infinitely many homoclinic solutions for a class of discrete nonlinear schrodinger equations where nonlinearities are superlinear at infinity and primitive functions of nonlinearities are allowed to be sign changing by using some weaker conditions our results extend and improve some existed results in the literature besides some examples are given to illuminate our results

solitons in one dimensional nonlinear schrodinger lattices with a local inhomogeneity in this paper we analyze the existence stability dynamical formation and mobility properties of localized solutions in a one dimensional system described by the discrete nonlinear schrodinger equation with a linear point defect we consider both attractive and repulsive defects in a focusing lattice among our main findings are a the destabilization of the on site mode centered at the defect in the repulsive case b the disappearance of localized modes in the vicinity of the defect due to saddle node bifurcations for sufficiently strong defects of either type c the decrease of the amplitude formation threshold for attractive and its increase for repulsive defects and d the detailed elucidation as a function of initial speed and defect strength of the different regimes trapping trapping and reflection pure reflection and pure transmission of interaction of a moving localized mode with the defect

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