construction of invariant whiskered tori by a parameterization method part ii quasi periodic and almost periodic breathers in coupled map lattices we construct quasi periodic and almost periodic solutions for coupled hamiltonian systems on an infinite lattice which is translation invariant the couplings can be long range provided that they decay moderately fast with respect to the distance for the solutions we construct most of the sites are moving in a neighborhood of a hyperbolic fixed point but there are oscillating sites clustered around a sequence of nodes the amplitude of these oscillations does not need to tend to zero in particular the almost periodic solutions do not decay at infinity we formulate an invariance equation solutions of this equation are embeddings of an invariant torus on which the motion is conjugate to a rotation we show that if there is an approximate solution of the invariance equation that satisfies some non degeneracy conditions there is a true solution close by the proof of this emph a posteriori theorem is based on a nash moser iteration which does not use transformation theory simpler versions of the scheme were developed in e fontich r de la llave y sire emph j differential equations bf 246 3136 2009 one technical tool important for our purposes is the use of weighted spaces that capture the idea that the maps under consideration are local interactions using these weighted spaces the estimates of iterative steps are similar to those in finite dimensional spaces in particular the estimates are independent of the number of nodes that get excited using these techniques given two breathers we can place them apart and obtain an approximate solution which leads to a true solution nearby by repeating the process infinitely often we can get solutions with infinitely many frequencies which do not tend to zero at infinity

two novel methods and multi mode periodic solutions for the fermi pasta ulam model we introduce two novel methods for studying periodic solutions of the fpu model both numerically and rigorously one is a variational approach based on the dual formulation of the problem and the other involves computer assisted proofs these methods are used e g to construct a new type of solutions whose energy is spread among several modes associated with closely spaced resonances

energy control of a pendulum with quantized feedback the problem of controlling a nonlinear system to an invariant manifold using quantized state feedback is considered by the example of controlling the pendulum s energy a feedback control law based on the speed gradient algorithm is chosen the main result consisting in precisely characterizing allowed quantization error bounds and resulting energy deviation bounds is presented

an iterative method for the calculation of narrow solitary excitations on atomic chains abstract we discuss the quasicontinuum approximation of collins and give a general condition for its validity for the solution of the difference differential equations of the full discrete model we develop an iteration procedure in fourier space the method is tested for solitary excitations with widths of the order of the lattice constant by a comparison with both exact results and computer simulations an additional result of the new mwthod is a simple derivation of the collins approximation which allows several generalizations

discrete lattice solitons properties and stability the authors present a detailed numerical study of the dynamics of high energy kink excitations in monatomic chains with a simple description using a microscopic model of the narrow kink core they present velocity amplitude and energy momentum curves for the kink which fit well the numerical results kink kink collisions are slightly inelastic depending on the velocities and point of collision within a lattice spacing

wave propagation and resonance in a one dimensional nonlinear discrete periodic medium we consider wave propagation in a nonlinear infinite diatomic chain of particles as a discrete model of propagation in a medium whose properties vary periodically in space the particles have alternating masses m1 and m2 and interact in accordance to a general nonlinear force f acting between the nearest neighbors their motion is described by the system of equations begin align ddot y n frac 1 m 1 f y n 1 y n f y n y n 1 ddot y n 1 frac 1 m 2 f y n y n 1 f y n 1 y n 2 end align where y n n infty infty is the position of the nth particle using fourier series methods and tools from bifurcation theory we show that for nonresonant wave numbers k this system admits nontrivial small amplitude traveling wave solutions g and h depending only on the linear combination z kn omega t we determine the nonlinear dispersion relation we also show that the system sustains binary oscillations with arbitrarily large amplitude

macroscopic behavior of microscopic oscillations in harmonic lattices via wigner husimi transforms we consider the dynamics of infinite harmonic lattices in the limit of the lattice distance tending to 0 we allow for general polyatomic crystals but assume exact periodicity such that the system can be solved in principle by fourier transform and linear algebra methods

a new barrier to the existence of moving kinks in frenkel kontorova lattices an explanation is offered for an observed lower bound on the wave speed of travelling kinks in frenkel kontorova lattices kinks exist at discrete wavespeeds within a parameter regime where there is a resonance with linear waves phonons however they fail to exist even in this codimension one sense whenever there is more than one phonon branch in the dispersion relation inside such bands only quasi kinks with nondecaying oscillatory tails are possible the results are presented for a discrete sine gordon lattice with an onsite potential that has a tunable amount of anharmonicity novel numerical methods are used to trace kinks with topological charge q 1 and 2 in three parameters representing the propagation speed lattice discreteness and anharmonicity although none of the analysis is presented as rigorous mathematics numerical results suggest that the bound on allowable wavespeeds is sharp the results also explain why the vanishing at discrete values of anharmonicity of the peierls nabarro barrier between stationary kinks as discovered by savin et al does not lead to bifurcation of kinks with small wave speed 2003 elsevier b v all rights reserved

existence theorem for solitary waves on lattices in this article we give an existence theorem for localized travelling wave solutions on one dimensional lattices with hamiltonian h sum limits n in mathbb z left tfrac 1 2 p n 2 v q n 1 q n right wherev is the potential energy due to nearest neighbour interactions until now apart from rare integrable lattices like the toda latticev ab 1 e b b 1 the only evidence for existence of such solutions has been numerical our result in particular recovers existence of solitary waves in the toda lattice establishes for the first time existence of solitary waves in the nonintegrable cubic and quartic latticesv 1 2 2 1 3a 3 v 1 2 2 1 4b 4 thereby confirming the numerical findings in 1 and shedding new light on the recurrence phenomena in these systems observed first by fermi pasta and ulam 2 and shows that contrary to widespread belief the presence of exact solitary waves is not a peculiarity of integrable systems but generic in this class of nonlinear lattices the approach presented here is new and quite general and should also be applicable to other forms of lattice equations the travelling waves are sought as minimisers of a naturally associated variational problem obtained via hamilton s principle and existence of minimisers is then established using modern methods in the calculus of variations the concentration compactness principle of p l lions 3

periodic and homoclinic travelling waves in infinite lattices abstract consider an infinite chain of particles subjected to a potential f where nearest neighbours are connected by nonlinear oscillators the nonlinear coupling between particles is given by a potential v the dynamics of the system is described by the infinite system of second order differential equations q j f q j v q j 1 q j v q j q j 1 j z we investigate the existence of travelling wave solutions two kinds of such solutions are studied periodic and homoclinic ones on one hand we prove under some growth conditions on f and v the existence of non constant periodic solutions of any given period t 0 and speed c c 0 where the constant c 0 depends on f 0 and v 0 on the other hand under very similar conditions we establish the existence of non trivial homoclinic solutions of any given speed c c 0 emanating from the origin moreover we prove that these homoclinics decay exponentially at infinity each homoclinic is obtained as a limit of periodic solutions when the period goes to infinity

atomic scale localization of high energy solitary waves on lattices one dimensional monatomic lattices with hamiltonian h p n2 z z 1 p 2 v qn 1 qn are known to carry localized travelling wave solutions for generic nonlinear potentials v fw94 in this paper we derive the asymptotic profile of these waves in the high energy limit h 1 for lennardjones type interactions the limit profile is proved to be a universal highly discrete piecewise linear wave concentrated on a single atomic spacing this shows that dispersionless energy transport in these systems is not confined to the long wave regime on which the theoretical literature has hitherto focused but also occurs at atomic scale localization

nanopteron solutions of diatomic fermi pasta ulam tsingou lattices with small mass ratio abstract consider an infinite chain of masses each connected to its nearest neighbors by a nonlinear spring this is a fermi pasta ulam tsingou lattice we prove the existence of traveling waves in the setting where the masses alternate in size in particular we address the limit where the mass ratio tends to zero the problem is inherently singular and we find that the traveling waves are not true solitary waves but rather nanopterons which is to say waves which are asymptotic at spatial infinity to very small amplitude periodic waves moreover we can only find solutions when the mass ratio lies in a certain open set the difficulties in the problem all revolve around understanding jost solutions of a nonlocal schrodinger operator in its semi classical limit

multiple periodic solutions for lattice dynamical systems with superquadratic potentials abstract in this paper we consider one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction the autonomous dynamical system is described by the following infinite system of second order differential equations q i i 1 q i 1 q i i q i q i 1 i z where i denotes the interaction potential between two neighboring particles and q i t is the state of the i th particle supposing i is superquadratic at infinity for all t 0 we obtain a nonzero t periodic solution of finite energy which may be nonconstant in some range of period if in addition i x is even in x we also obtain infinitely many geometrically distinct solutions for any period t 0 in particular a prescribed number of geometrically distinct nonconstant periodic solutions is obtained for some range of period since the functional associated to the above system is invariant under the actions of the non compact group z and the continuous compact group s 1 under our assumptions in order to prove our results we need to extend the abstract critical point theorem about strongly indefinite functional developed by bartsch and ding math nachr 279 2006 1267 1288 to a more general class of symmetry

quasi continuum approximations to lattice equations arising from the discrete nonlinear telegraph equation we show how quasi continuum methods can be used to construct approximate solutions of nonlinear differential delay equations derived from symmetry reductions of the discrete nonlinear telegraph equation travelling wave solutions are proven to exist and the existence of solutions to two other symmetry reductions are studied two of the less familiar reductions are studied the first supports both a one parameter family of single pulse solitary wave type solutions and a one parameter family of periodic waves the size and shape of these waves are examined using the quasi continuum technique this approximates the differential difference equation with a higher order differential equation which is integrated and analysed using phase plane techniques in the large amplitude limit the shape of the pulse approaches a limiting form which has a corner at its peak the manner of this approach is elucidated using matched asymptotic expansions the second reduction though differing only by the addition of a single term appears not to support the solitary wave type of solution even in the limit where the additional term is premultiplied by an asymptotically small constant

nontrivial periodic motions for resonant type asymptotically linear lattice dynamical systems abstract in this paper we consider the following one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction q i t i 1 t q i 1 t q i t i t q i t q i 1 t i z where i t x i 2 x 2 v i t x is t periodic in t for t 0 and satisfies i n i for some n n q i t is the state of the i th particle assume that i 0 for some i z and v i t x denoting the derivative of v i respect to x is asymptotically linear with x both at origin and at infinity we would like to point out that this system is resonant both at origin and at infinity and not studied up to now based on some new results concerning the precise computations of the critical groups for a given m z we obtain the existence of nontrivial periodic solutions satisfying q i m n t t q i t for all t r and i z under some additional conditions

travelling wave solutions for the discrete sine gordon equation with nonlinear pair interaction the focus of study is the nonlinear discrete sine gordon equation where the nonlinearity refers to a nonlinear interaction of neighbouring atoms the existence of travelling heteroclinic homoclinic and periodic waves is shown the asymptotic states are chosen such that the action functional is finite the proofs employ variational methods in particular a suitable concentration compactness lemma combined with direct minimisation and mountain pass arguments

travelling breathers with exponentially small tails in a chain of nonlinear oscillators we study the existence of travelling breathers in klein gordon chains which consist of one dimensional networks of nonlinear oscillators in an anharmonic on site potential linearly coupled to their nearest neighbors travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity approximate solutions of this type have been constructed in the form of modulated plane waves whose envelopes satisfy the nonlinear schrodinger equation m remoissenet phys rev b33 n 4 2386 1986 j giannoulis and a mielke nonlinearity17 p 551 565 2004 in the case of travelling waves where the phase velocity of the plane wave equals the group velocity of the wave packet the existence of nearby exact solutions has been proved by iooss and kirchgassner who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail g iooss k kirchgassner commun math phys 211 439 464 2000 however a rigorous existence result has been lacking in the more general case when phase and group velocities are different this situation is examined in the present paper in a case when the breather period and the inverse of its velocity are commensurate we show that the center manifold reduction method introduced by iooss and kirchgassner is still applicable when the problem is formulated in an appropriate way this allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations whose principal part admits homoclinic solutions to quasi periodic orbits under general conditions on the potential for an even potential using the additional symmetry of the system we obtain homoclinic orbits to small periodic ones for the full reduced system for the oscillator chain these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail their principal part excluding the tail coincides at leading order with the nonlinear schrodinger approximation

approximations to solitary waves on lattices iii the monatomic lattice with second neighbour interactions we find new behaviour in a lattice with second neighbour interactions first we find solitary waves with oscillatory spatial decay this is something that cannot occur in lattices with nearest neighbour interactions alone this is found by standard asymptotic analysis which leads to an exact curve in parameter space where this behaviour starts a number of highly accurate quasi continuum approximations are derived and solved one of these suggests a possible method for subsonic solitary waves to cease existing an alternative method of approximation elegantly reveals the differences in shape between subsonic and supersonic solitary waves this is based on the weak form of the differential delay equation for the travelling wave and is derived using the calculus of variations

on the impact of fpu lattice discreteness upon solitary wave interactions abstract this work is concerned with the motions of an infinite one dimensional lattice with nearest neighbour interactions governed by a generic potential the hamiltonian of such a system may be written h i 1 2 p i 2 v q i 1 q i in terms of the momenta p i and the displacements q i of the lattice sites all sites are assumed to be of equal mass certain generic conditions are placed on the potential v the particular results of this paper concern the way in which lattice discreteness impacts upon the solitary wave interaction process which is known to be well approximated in the long wave continuum limit by kdv soliton interaction an evolution equation for discreteness effects on the lattice is proposed and is found to be strikingly similar to corresponding equations known in both the theories of shallow water waves and ion acoustic waves it is proved by standard techniques to have a unique classical solution whose profile is explicitly obtained by numerical simulation the most important outcome is the observation that lattice discreteness imposes an effective repulsion between post collision solitary waves relative to their unperturbed pure kdv positions

breathers on diatomic fermi pasta ulam lattices we prove the existence of breathers spatially localized and time periodic oscillations in diatomic fermi pasta ulam fpu chains with arbitrary mass ratio this completes an existence result by livi spicci and mackay valid for large mass ratio the problem is formulated as a mapping in a loop space and analyzed via a discrete spatial centre manifold reduction plane wave solutions of the linearized system have frequencies in a higher optic band or a lower acoustic band for frequencies close to band edges all small amplitude solutions of the nonlinear system lie on a finite dimensional centre manifold which reduces the problem locally to the study of a finite dimensional mapping for good parameter values the map admits homoclinic orbits to 0 corresponding to discrete breathers when the fpu interaction potential satisfies a hardening condition we find breathers with frequencies slightly above the optic band or in the gap slightly above the acoustic band for a potential satisfying the opposite softening condition we obtain breathers with frequencies in the gap slightly below the optic band 2004 elsevier b v all rights reserved

travelling waves for a frenkel kontorova chain abstract in this article the frenkel kontorova model for dislocation dynamics is considered where the on site potential consists of quadratic wells joined by small arcs which can be spinodal concave as commonly assumed in physics the existence of heteroclinic waves making a transition from one well of the on site potential to another is proved by means of a schauder fixed point argument the setting developed here is general enough to treat such a frenkel kontorova chain with smooth c 2 on site potential it is shown that the method can also establish the existence of two transition waves for a piecewise quadratic on site potential

cocycle superrigidity and bounded cohomology for negatively curved spaces we introduce new techniques to extend superrigidity theory beyond the scope of lie or algebraic groups we construct a cohomological invariant which accounts for and generalizes all known superrigidity results for actions on negatively curved spaces together with a new vanishing result and the machinery of bounded cohomology this enables us to prove a general superrigidity theorem for actions of irreducible lattices on spaces of negative curvature we also prove a cocycle version a la zimmer

asymptotic stability of lattice solitons in the energy space orbital and asymptotic stability for 1 soliton solutions of the toda lattice equations as well as for small solitary waves of the fpu lattice equations are established in the energy space unlike analogous hamiltonian pdes the lattice equations do not conserve the adjoint momentum in fact the toda lattice equation is a bidirectional model that does not fit in with the existing theory for the hamiltonian systems by grillakis shatah and strauss to prove stability of 1 soliton solutions we split a solution around a 1 soliton into a small solution that moves more slowly than the main solitary wave and an exponentially localized part we apply a decay estimate for solutions to a linearized toda equation which has been recently proved by mizumachi and pego to estimate the localized part we improve the asymptotic stability results for fpu lattices in a weighted space obtained by friesecke and pego

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

solitary waves with prescribed speed on infinite lattices using a variant of the mountain pass theorem we prove the existence of solitary waves with prescribed speed on infinite lattices of particles with nearest neighbor interaction the problem is to solve a second order forward backward differential difference equation c 1997 academic press

asymptotic stability of n solitary waves of the fpu lattices we study stability of n solitary wave solutions of the fermi pasta ulam fpu lattice equation solitary wave solutions of the fpu lattice equation cannot be characterized as critical points of conservation laws due to the lack of infinitesimal invariance in the spatial variable in place of standard variational arguments for hamiltonian systems we use an exponential stability property of the linearized fpu equation in a weighted space which is biased in the direction of motion the dispersion of the linearized fpu equation balances the potential term for low frequencies whereas the dispersion is superior for high frequencies we approximate the low frequency part of a solution of the linearized fpu equation by a solution to the linearized korteweg de vries kdv equation around an n soliton solution we prove an exponential stability property of the linearized kdv equation around n solitons by using the linearized backlund transformation and use the result to analyze the linearized fpu equation

approximations to solitary waves on lattices ii quasi continuum methods for fast and slow waves a number of highly accurate analytic approximations to solitary waves on lattices are detailed and the results of numerical tests presented two types of approximation are considered the first set of approximations are generalizations of the continuum approximation based around various pade expansions of the operator these are convergent to the exact solution in the limit of wavespeed c to 1 or a dense lattice the author also considers ways of constructing approximations to fast solitary waves from a continuum type theory the two types are closely related in both cases significant improvements are made on existing approximations a symplectic hamiltonian integration scheme is used to perform computer simulations a new method of measuring the accuracy of a predicted waveform is also used on new and old approximations alike to compare the various methods considered

the becker doring cluster equations basic properties and asymptotic behaviour of solutions existence and uniqueness results are established for solutions to the becker doring cluster equations the density is shown to be a conserved quantity under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined denoting the set of equilibrium solutions byc 0 s the principal result is that if the initial density 0 s then the solution converges strongly toc o while if 0 s the solution converges weak toc s in the latter case the excess density 0 s corresponds to the formation of larger and larger clusters i e condensation the main tools for studying the asymptotic behaviour are the use of a lyapunov function with desirable continuity properties obtained from a known lyapunov function by the addition of a special multiple of the density and a maximum principle for solutions

a study of a new class of discrete nonlinear schrodinger equations a new class of 1d discrete nonlinear schrodinger hamiltonians with tunable nonlinearities is introduced which includes the integrable ablowitz ladik system as a limit a new subset of equations which are derived from these hamiltonians using a generalized definition of poisson brackets and collectively referred to as the n al equation is studied the symmetry properties of the equation are discussed these equations are shown to possess propagating localized solutions having the continuous translational symmetry of the one soliton solution of the ablowitz ladik nonlinear schrodinger equation the n al systems are shown to be suitable for studying the combined effect of the dynamical imbalance of nonlinearity and dispersion and the peierls nabarro potential arising from the lattice discreteness on the propagating solitary wave like profiles a perturbative analysis shows that the n al systems can have discrete breather solutions due to the presence of saddle centre bifurcations in phase portraits the unstaggered localized states are shown to have positive effective mass on the other hand large width but small amplitude staggered localized states have negative effective mass the collision dynamics of two colliding solitary wave profiles is studied numerically notwithstanding colliding solitary wave profiles that are seen to exhibit nontrivial nonsolitonic interactions certain universal features are observed in the collision dynamics the future scope of this work and possible applications of the n al systems are discussed