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

discrete breathers in a two dimensional fermi pasta ulam lattice using asymptotic methods we investigate whether discrete breathers are supported by a two dimensional fermi pasta ulam lattice a scalar one component two dimensional fermi pasta ulam lattice is shown to model the charge stored within an electrical transmission lattice a third order multiple scale analysis in the semi discrete limit fails since at this order the lattice equations reduce to the 2 1 dimensional cubic nonlinear schrodinger nls equation which does not support stable soliton solutions for the breather envelope we therefore extend the analysis to higher order and find a generalised 2 1 dimensional nls equation which incorporates higher order dispersive and nonlinear terms as perturbations we find an ellipticity criterion for the wave numbers of the carrier wave numerical simulations suggest that both stationary and moving breathers are supported by the system calculations of the energy show the expected threshold behaviour whereby the energy of breathers does em not go to zero with the amplitude we find that the energy threshold is maximised by stationary breathers and becomes arbitrarily small as the boundary of the domain of ellipticity is approached

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

state local optimality of a bang bang trajectory a hamiltonian approach we give a sufficient condition for a bang bang extremal to be a strong local optimizer for the minimum time problem with fixed endpoints we underline that the conditions imply that the optimum is local with respect to the state and not necessarily to the final time moreover it is given through a finite dimensional minimization problem hence is suited for numerical verification a geometric interpretation through the projection of the hamiltonian flow on the state space is also given

tangent bifurcation of band edge plane waves dynamical symmetry breaking and vibrational localization abstract we study tangent bifurcation of band edge plane waves in nonlinear hamiltonian lattices the lattice is translationally invariant we argue for the breaking of permutational symmetry by the new bifurcated periodic orbits the case of two coupled oscillators is considered as an example for the perturbation analysis where the symmetry breaking can be traced using poincare maps next we consider a lattice and derive the dependence of the bifurcation energy on the parameters of the hamiltonian function in the limit of large system sizes a necessary condition for the occurence of the bifurcation is the repelling of the band edge plane wave from the linear spectrum with increasing energy we conclude that the bifurcated orbits will consequently exponentially localize in the configurational space

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

discrete breathers in honeycomb fermi pasta ulam lattices we consider the two dimensional fermi pasta ulam lattice with hexagonal honeycomb symmetry which is a hamiltonian system describing the evolution of a scalar valued quantity subject to nearest neighbour interactions using multiple scale analysis we reduce the governing lattice equations to a nonlinear schrodinger nls equation coupled to a second equation for an accompanying slow mode two cases in which the latter equation can be solved and so the system decoupled are considered in more detail firstly in the case of a symmetric potential we derive the form of moving breathers we find an ellipticity criterion for the wavenumbers of the carrier wave together with asymptotic estimates for the breather energy the minimum energy threshold depends on the wavenumber of the breather we find that this threshold is locally maximised by stationary breathers secondly for an asymmetric potential we find stationary breathers which even with a quadratic nonlinearity generate no second harmonic component in the breather plots of all our findings show clear hexagonal symmetry as we would expect from our lattice structure finally we compare the properties of stationary breathers in the square triangular and honeycomb lattices

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

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

first and second order necessary conditions for control problems with constraints second order necessary conditions are developed for an abstract nonsmooth control problem with mixed state control equality and inequality constraints as well as a constraint of the form g x u e if where f is a closed convex set of a banach space with nonempty interior the inequality constraints g s x u 0 depend on a parameter s belonging to a compact metric space s the equality constraints are split into two sets of equations k x u 0 and h x u 0 where the first equation is an abstract control equation and h is assumed to have a full rank property in u the objective function is maxtet f t x u where t is a compact metric space f is upper semicontinuous in t and lipschitz in x u the results are in terms of a function a that disappears when the parameter spaces t and s are discrete we apply these results to control problems governed by ordinary differential equations and having pure state inequality constraints and control state equality and inequality constraints thus we obtain a generalization and extension of the existing results on this problem

lyapunov schmidt reduction algorithm for three dimensional discrete vortices abstract we address the persistence and stability of three dimensional vortex configurations in the discrete nonlinear schrodinger equation and develop a symbolic package based on wolfram s mathematica for computations of the lyapunov schmidt reduction method the lyapunov schmidt reduction method is a theoretical tool which enables us to study continuations and terminations of the discrete vortices for small coupling between lattice nodes as well as the spectral stability of the persistent configurations the method was developed earlier in the context of the two dimensional lattice and applied to the onsite and offsite configurations called the vortex cross and the vortex cell by using semianalytical computations d e pelinovsky p g kevrekidis d frantzeskakis physica d 212 2005 20 53 p g kevrekidis d e pelinovsky proc r soc a 462 2006 2671 2694 the present treatment develops a full symbolic computational package which takes a desired waveform at the anticontinuum limit of uncoupled sites performs a required number of lyapunov schmidt reductions and outputs the predictions on whether the configuration persists for finite coupling in the three dimensional lattice and whether it is stable or unstable it also provides approximations for the eigenvalues of the linearized stability problem we report a number of applications of the algorithm to important multisite three dimensional configurations such as the simple cube the double cross and the diamond for each configuration we identify exactly one solution which is stable for small coupling between lattice nodes

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

extended jacobi sufficiency criterion for optimal control in this paper we present sufficient conditions for strong local minimality in optimal control we do not require the dynamic f to be linear nor the control set u to be open moreover these conditions extend a known sufficiency criterion in the calculus of variations involving the jacobi condition

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

the conjugate point condition for smooth control sets abstract in this paper we present for optimal control problems with smooth control constraints second order necessary conditions stated in terms of conjugate points and a riccati type equation we show explicitly how to construct a set of feasible directions and how to derive the corresponding accessory problem

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

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

time scale symplectic systems without normality abstract we present a theory of the definiteness nonnegativity and positivity of a quadratic functional f over a bounded time scale the results are given in terms of a time scale symplectic system s which is a time scale linear system that generalizes and unifies the linear hamiltonian differential system and discrete symplectic system the novelty of this paper resides in removing the assumption of normality in the characterization of the positivity of f and in establishing equivalent conditions for the nonnegativity of f without any normality assumption to reach this goal a new notion of generalized focal points for conjoined bases x u of s is introduced results on the piecewise constant kernel of x t are obtained and various picone type identities are derived under the piecewise constant kernel condition the results of this paper generalize and unify recent ones in each of the discrete and continuous time setting and constitute a keystone for further development in this theory

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

entire functions on banach spaces with a separable dual abstract let e and f be complex banach spaces we show that if e has a separable dual then every holomorphic function from e into f which is bounded on weakly compact sets is bounded on bounded sets

discrete breathers in a two dimensional spring mass lattice we consider a two dimensional spring mass lattice with square symmetry in which each particle experiences a nonlinear onsite potential and nonlinear nearest neighbour interactions at equilibrium the particles are equally spaced in both the horizontal and vertical directions and all springs are unextended motivated by the work of marin et al 1998 phys lett a 248 225 9 2001 phys lett a 281 21 5 we seek a solution in which most of the breather s energy is focused along three chains we construct an asymptotic approximation to the breather using the method of multiple scales to describe the coherent oscillations in the three main chains that constitute the discrete breather we reduce the equation of motion to a nonlinear schrodinger equation for the leading order term and find a family of solutions which encompasses both stationary and moving bright soliton solutions we use numerical simulations of the lattice to verify the shape and velocity of breathers and find that while stationary breathers are found to persist for long times moving breathers decay by radiating energy in the direction perpendicular to their motion

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

construction of quasi periodic breathers via kam technique by developing a kam theorem which involves an infinitely multiple normal frequency it is shown that there are plenty of breathers quasi periodic in time and super exponentially localized in space for the networks of weakly coupled oscillators this answers an open problem by aubry a2 in case the linearized system has no continuous spectrum

entire functions on c0 abstract it is shown that every holomorphic function on c 0 which is bounded on weakly compact sets is bounded on bounded sets

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

sufficient conditions for the generalized problem of bolza this paper presents sufficient conditions for strong local optimality in the generalized problem of bolza these conditions represent a unification in the sense that they can be applied to both the calculus of variations and to optimal control problems as well as problems with nonsmooth data also this paper brings to light a new point of view concerning the jacobi condition in the classical calculus of variations showing that it can be considered as a condition which guarantees the existence of a canonical transformation which transforms the original hamiltonian to a locally concave convex hamiltonian

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

new second order optimality conditions for variational problems with c 2 hamiltonians the generalized problem of bolza with a c2 hamiltonian is considered necessary and sufficient conditions are obtained respectively in terms of the accessory problem the existence of conjoined basis and the existence of a solution to a riccati equation with boundary conditions no strengthened legendre clebsch condition is required when applied to a general optimal control problem these results include and generalize known results

weakly continuous mappings on banach spaces abstract it is shown that every n homogeneous continuous polynomial on a banach space e which is weakly continuous on the unit ball of e is weakly uniformly continuous on the unit ball of e applications of the result to spaces of polynomials and holomorphic mappings on e are given

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

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

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

bound states of two dimensional solitons in the discrete nonlinear schrodinger equation the existence and stability of bound states bss of solitary excitations in the two dimensional discrete nonlinear schrodinger lattice are considered stable bss of two or three solitons are categorized according to their vorticity s s 0 corresponds to ordinary pulses while s 1 corresponds to discrete vortex solitons interactions of s 0 solitons are found to have clear particle like characteristics which can be very accurately predicted in the framework of the perturbation theory that we develop triangular bound states of three solitons are also constructed and particle like characteristics in their behaviour are followed in numerical experiments finally stable bound states of two vortex solitons and s 0 1 stable complexes are constructed for the latter state we find a weak dependence of the interaction generated stability eigenvalue on the distance between the two solitons

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

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

conjugate points and shocks in nonlinear optimal control in this paper the authors use the method of characteristics to extend the jacobi conjugate points theory to the bolza problem arising in nonlinear optimal control this yields necessary and sufficient optimality conditions for weak and strong local minima stated in terms of the existence of a solution to a corresponding matrix riccati differential equation the same approach allows to investigate as well smoothness of the value function

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

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

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

factorization of holomorphic mappings in infinite dimensions this note is concerned with a generalization of the first identity to the non linear setting of holomorphic i e complex analytic functions on e in fact our interest in this generalization arose from a study made by the second author 15 of a suitable generalization of the second identity to the holomorphic setting however our results and methods have little in common with 15 in somewhat simplistic terms our results provide further evidence that in many situations one is justified in regarding the space h e of holomorphic functions on e as the holomorphic dual of e which shares many properties with the standard linear dual of e see for example 6 a brief outline of this paper follows after which we recall some basic results about holomorphic mappings between infinite dimensional locally convex spaces in section 2 we investigate when a holomorphic function f u c on a balanced open subset u of e admits a factorization through u a characterization of factorizable functions is given in terms of the derivative o f f our basic technique is

analytic approximations of uniformly continuous functions in real banach spaces abstract it is shown that if a separable real banach space x admits a separating analytic function with an additional condition property k concerning uniform behaviour of radii of convergence then every uniformly continuous mapping on x into any real banach space y can be approximated by analytic operators in particular the result applies to c0

weakly continuous holomorphic functions on banach spaces with a shrinking and unconditional basis let e and f be complex banach spaces and let u be an open ball in e we show that if e has a shrinking and unconditional basis then every holomorphic function f u f that is weakly continuous on u bounded sets is weakly uniformly continuous on u bounded sets

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

discrete breathers in a two dimensional hexagonal fermi pasta ulam lattice we consider a two dimensional fermi pasta ulam fpu lattice with hexagonal symmetry using asymptotic methods based on small amplitude ansatz at third order we obtain a eduction to a cubic nonlinear schr o dinger equation nls for the breather envelope however this does not support stable soliton solutions so we pursue a higher order analysis yielding a generalised nls which includes known stabilising terms we present numerical results which suggest that long lived stationary and moving breathers are supported by the lattice we find breather solutions which move in an arbitrary direction an ellipticity criterion for the wavenumbers of the carrier wave symptotic estimates for the breather energy and a minimum threshold energy below which breathers cannot be found this energy threshold is maximised for stationary breathers and becomes vanishingly small near the boundary of the elliptic domain where breathers attain a maximum speed several of the results obtained are similar to those obtained for the square fpu lattice butt wattis em j phys a bf 39 4955 2006 though we find that the square and hexagonal lattices exhibit different properties in regard to the generation of harmonics and the isotropy of the generalised nls equation

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

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

equivalent subgradient versions of hamiltonian and euler llagrange equations in variational analysis much effort in recent years has gone into generalizing the classical hamiltonian and euler lagrange equations of the calculus of variations so as to encompass problems in optimal control and a greater variety of integrands and constraints these generalizations in which nonsmoothness abounds and gradients are systematically replaced by subgradients have succeeded in furnishing necessary conditions for optimality that reduce to the classical ones in the classical setting but important issues have remained unsettled especially concerning the exact relationship of the subgradient versions of the hamiltonian equations versus those of the euler lagrange equations here it is shown that new tighter subgradient versions of these equations are actually equivalent to each other the theory of epi convergence of convex functions provides the technical basis for this development

hamiltonian analysis of the generalized problem of bolza on obtient des conditions necessaires en forme hamiltonienne sous des hypotheses moins exigeantes qu anterieurement en particulier sans qualification sur les contraintes on developpe le lien avec les problemes de commande optimale ainsi que l apport de ces conditions a la theorie de la regularite de la solution

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

oscillation theorems and rayleigh principle for linear hamiltonian and symplectic systems with general boundary conditions the aim of this paper is to establish the oscillation theorems rayleigh principle and coercivity results for linear hamiltonian and symplectic systems with general boundary conditions i e for the case of separated and jointly varying endpoints and with no controllability normality and strong observability assumptions our method is to consider the time interval as a time scale and apply suitable time scales techniques to reduce the problem with separated endpoints into a problem with dirichlet boundary conditions and the problem with jointly varying endpoints into a problem with separated endpoints these more general results on time scales then provide new results for the continuous time linear hamiltonian systems as well as for the discrete symplectic systems this paper also solves an open problem of deriving the oscillation theorem for problems with periodic boundary conditions furthermore the present work demonstrates the utility and power of the analysis on time scales in obtaining new results especially in the classical continuous and discrete time theories

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

solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity we study the existence and stability of localized states in the discrete nonlinear schrodinger equation on two dimensional nonsquare lattices the model includes both the nearest neighbor and long range interactions for the fundamental strongly localized soliton the results depend on the coordination number i e on the particular type of lattice the long range interactions additionally destabilize the discrete soliton or make it more stable if the sign of the interaction is respectively the same as or opposite to the sign of the short range interaction we also explore more complicated solutions such as twisted localized modes and solutions carrying multiple topological charge vortices that are specific to the triangular and honeycomb lattices in the cases when such vortices are unstable direct simulations demonstrate that they typically turn into zero vorticity fundamental solitons

discrete gap breathers in a diatomic klein gordon chain stability and mobility a one dimensional diatomic chain with harmonic intersite potential and nonlinear external potential is considered the klein gordon model localized solutions of the corresponding nonlinear differential equations with frequencies inside the gap of the linear wave spectrum gap breathers are studied numerically the linear stability analysis for these solutions is performed while changing the system parameters from the anticontinuous to the continuous limit two different types of solutions are considered symmetric centered at a heavy atom and antisymmetric centered at a light atom respectively different mechanisms of instability oscillatory as well as nonoscillatory of the gap breathers are studied and the influence of the instabilities on the breather solutions is investigated in the dynamics simulations in particular the presence of an inversion of stability regime with simultaneous nonoscillatory instabilities of symmetric and antisymmetric solutions with respect to antisymmetric perturbations is found yielding practically radiationless mobility