dynamics of circular arrays of josephson junctions and the discrete sine gordon equation abstract we analyze the damped driven discrete sine gordon equation with periodic boundary conditions and constant forcing analytical and numerical results are presented about the existence stability and bifurcations of traveling waves in this system these results are compared with experimental measurements of the current voltage i v characteristics of a ring of n 8 underdamped josephson junctions we find two types of traveling waves low velocity kinks and high velocity whirling modes the kinks excite small amplitude linear waves intheir wake at certain drive strengths the linear waves phase lock to the kink generating resonant steps in the i v curve steps also occur in the high velocity region due to parametric instabilities of the whirling mode we analyze the onset of these instabilities then numerically study the secondary bifurcations and complex spatiotemporal phenomena that occur past the onset in all cases the measured voltage locations of the resonant steps are in good agreement with the predictions

a note about positive solutions for an equation of kirchhoff type abstract a nonlinear boundary value problem related to an equation of kirchhoff is considered the existence of positive solutions is proved through alternative leray schauder s type combined with krasnoselskii s fixed point theorem numerical methods are presented and a result of local convergence is established

a symplectic integration algorithm for separable hamiltonian functions abstract we derive an algorithm to numerically integrate differential equations derivable from a separable hamiltonian function this symplectic algorithm is accurate to fourth order in the time step and preserves exactly the poincare cartan integral invariants associated with the topology of the phase flow we compare the efficiency and accuracy of this method to that of existing integrators both symplectic and non symplectic by integrating the equations of motion corresponding to a nonlinear pendulum a particle in the field of a standing wave and a harmonic oscillator perturbed by a plane wave

the cauchy problem of coupled elliptic sine gordon equations with noise analysis of a general kernel based regularization and reliable tools of computing abstract developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering in this paper we explore the regularization method for the coupled elliptic sine gordon equations along with cauchy data the system of equations originates from the static case of the coupled hyperbolic sine gordon equations modeling the coupled josephson junctions in superconductivity and so far it addresses the josephson junctions in general the cauchy problem is not well posed and herein the hadamard instability occurs drastically generalizing the kernel based regularization method we propose a stable approximate solution confirmed by the error estimate this solution strongly converges to the exact solution in l 2 norm the main concern of this paper is also with the way to compute the regularized solution formed by an alike integral equation we employ the proposed techniques that successfully approximated the highly oscillatory integral and apply the picard like iteration to organize an efficient and reliable tool of computations the results are viewed as the improvement as well as the generalization of many previous works the paper is also accompanied by a numerical example that demonstrates the potential of this idea

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

stationary breather modes of generalized nonlinear klein gordon lattices this paper studies the form of stationary breather modes in discrete generalized nonlinear klein gordon equations with symmetric and non symmetric potential energy functions in the case of static breathers the discrete nature of the spatial dimension has a much more subtle effect on the breather than the moving breather mode this effect is analysed using a variety of approximating partial differential equations whose solutions are found by using an extended multiple scales asymptotic approach to reduce the equation to a nonlinear schrodinger equation at leading order and more complex equations at higher order where secularity conditions are required to fully specify the solution as well as the much studied discrete sine gordon equation the methods are demonstrated on a discrete nonlinear klein gordon equation with second neighbour interactions and non symmetric on site potential new partial differential equations which approximate these lattice systems are also proposed and analysed

kinks in chains with on site bistable nondegenerate potential beyond traveling waves this paper revisits the well known transition fronts kinks in chains of coupled oscillators with nondegenerate on site potentials usually such transition fronts are considered in terms of traveling wave solutions we explore the loss of stability of such traveling waves generically it corresponds to one of the common scenarios for fixed points of discrete maps for example one can encounter the quasiperiodic kink propagation due to hopf bifurcation or the feigenbaum cascade of period doublings leading to a chaoticlike propagation pattern the aforementioned scenarios show up for instance for triparabolic and 4 on site potentials numeric evidence suggests that the loss of stability occurs due to resonances between the frequency associated with the kink propagation and the linear band gaps of the chain particular resonance mechanisms are model dependent for the classical atkinson cabrera model with a biparabolic on site potential the stability threshold is estimated by the simple means of linear algebra the loss of stability in this model occurs through hopf bifurcation the results are in good agreement with numerical simulations

collective coordinates and length scale competition in spatially inhomogeneous soliton bearing equations perturbed one dimensional integrable i e soliton bearing equations arise in many applied contexts when trying to improve the usually highly idealized description of problems of interest in terms of the purely integrable equations in particular when the assumption of perfect homogeneity is dropped to account for unavoidable impurities or defects perturbations depending on the spatial coordinate must be added to the original equation in this review we use the one dimensional sine gordon sg equation perturbed by a spatially periodic term as a generic paradigm to discuss the main perturbative techniques available for the study of this class of problems to place the work in context we summarize the approaches developed to date and focus on the collective coordinate approach as one of the most useful tools we introduce several versions of this perturbative method and relate them to more involved procedures we analyze in detail the application to the sg equation but the procedure is very general to illustrate this other examples of the application of collective coordinates are briefly revisited in our case study this approach helps us identify perturbative and nonperturbative regimes yielding a very simple picture of the former beyond perturbative calculations the same example of the inhomogeneous sg equation allows us to introduce a phenomenon termed length scale competition which we show to be a rather general mechanism for the appearance of complex spatiotemporal behavior in perturbed integrable systems as other instances discussed in the review show such nonperturbative results are obtained by means of numerical simulations of the full perturbed problem numerical linear stability analysis is also used to clarify the origins of the instability originated by this competition to complement our description of the techniques employed in these studies computational details of our numerical simulations are also included finally the paper closes with a discussion of the above ideas and a speculative outlook on general questions concerning the interplay of nonlinearity with disorder

variational approximations to breather modes in the discrete sine gordon equation abstract long lived breather like modes have been observed in the numerical simulations of discrete sine gordon systems however little analytical work has been carried out on these systems we show how to use the variational form of nonlinear klein gordon equations to find corrections to breather modes caused by discreteness

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

the lorenz fermi pasta ulam experiment we consider a chain of lorenz 63 systems connected through a local nearest neighbour coupling we refer to the resulting system as the lorenz fermi pasta ulamlattice because of its similarity to the celebrated experiment conducted by fermi pasta and ulam at large coupling strengths the systems synchronize to a global chaotic orbit of the lorenz attractor for smaller coupling the synchronized state loses stability instead steady spatially structured equilibrium states are observed these steady states are related to the heteroclinic orbits of the system describing stationary solutions to the partial differential equation that emerges on taking the continuum limit of the lattice notably these orbits connect saddle foci suggesting the existence of a multitude of such equilibria in relatively wide systems on lowering the coupling strength yet further the steady states lose stability in what appear to be always subcritical hopf bifurcations this can lead to a variety of time dependent states with fixed time averaged spatial structure such solutions can be limit cycles tori or possibly chaotic attractors cluster states can also occur though with less regularity consisting of lattices in which the elements are partitioned into families of synchronized subsystems ultimately for very weak coupling the lattice loses its time averaged spatial structure at this stage the properties of the lattice are probably chaotic and approximately scale with the lattice size suggesting that the system is essentially an ensemble of elements that evolve largely independent of one another the weak interaction however is sufficient to induce widespread coherent phases these are ephemeral states in which the dynamics of one or more subsystems takes a more regular form we present measures of the complexity of these incoherent lattices and discuss the concept of a dynamical horizon that is the distance along the lattice that one subsystem can effectively influence another and error propagation how the introduction of a disturbance in one subsystem becomes spread throughout the lattice 2000 elsevier science 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

ultra analytic effect of cauchy problem for a class of kinetic equations abstract the smoothing effect of the cauchy problem for a class of kinetic equations is studied we firstly consider the spatially homogeneous nonlinear landau equation with maxwellian molecules and inhomogeneous linear fokker planck equation to show the ultra analytic effects of the cauchy problem those smoothing effect results are optimal and similar to heat equation in the second part we study a model of spatially inhomogeneous linear landau equation with maxwellian molecules and show the analytic effect of the cauchy problem

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

high order breather solutions to a discrete nonlinear klein gordon model abstract a general discrete nonlinear klein gordon model is studied with regards to the movability of breathers approximations to moving breather solutions are found by a semi discrete multiple scale perturbation expansion to 3rd order the higher order effects of discreteness have a remarkable connection to nonlinear fiber optics effects such as higher order dispersion self steepening of the pulse edge and self frequency shift are found modifying both the shape velocity and frequency of the solution here only their combined influence is studied discreteness is found to trap breathers with initial amplitudes higher than a critical value which depends on the internal state of the breather this shows that no strict definition of the peierls nabarro potential can be given for breathers furthermore we find that the envelope velocity of the breather cannot exceed the maximum group velocity which converges to zero as the degree of discreteness is increased thus no moving breathers exist in a highly discrete system

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

moving kinks and nanopterons in the nonlinear klein gordon lattice we study moving topological solitons kinks and antikinks in the nonlinear klein gordon chain these solitons are shown to exist with both monotonic non oscillating and oscillating asymptotics tails using the pseudo spectral method the anti kink solutions with oscillating background so called nanopterons are found as travelling waves of permanent profile propagating with constant velocity each of these solutions may be considered as a bound state of an anti kink with a background nonlinear periodic wave so that the wave pushes the anti kink over the peierls nabarro barrier the stability of these bound states is confirmed numerically travelling wave solutions of permanent profile are shown to exist depending on the convexity of the on site substrate potential the set of velocities at which the anti kinks with monotonic asymptotics propagate freely is calculated we also find moving non oscillating anti kink profiles with higher topological charges each of which appears to be the bound state of anti kinks with lower topological charge jq jd 1 2000 elsevier science b v all rights reserved

remarks on an elliptic equation of kirchhoff type abstract we present a short survey on the nonlocal elliptic boundary value problem m u 2 d x u f x u in u 0 on where is a smooth bounded domain of r n m is a positive function and f has subcritical growth

long distance wave computation using nonlinear solitary waves a recently developed method is described to propagate short wave equation pulses over indefinite distances and through regions of varying indices of refraction including multiple reflections the method wave confinement utilizes a newly developed nonlinear partial differential equation pde that propagates basis functions according to the wave equation these basis functions are generated as stable solitary waves where the discretized equation can be solved without any numerical dissipation the method can also be used to solve for harmonic waves in the high frequency eikonal limit including multiple arrivals the solution involves discretizing the wave equation on a uniform eulerian grid and adding a simple nonlinear confinement term this term does not change the amplitude integrated through each point on the pulse surface or the propagation velocity or arrival time and yet results in capturing the waves as thin surfaces that propagate as thin nonlinear solitary waves and remain 2 3 grid cells in thickness indefinitely with no numerical spreading a new feature described in this paper involves computing scattering of short pulses from complex objects such as complete aircraft a simple immersed surface approach is used that utilizes the same uniform grid as the propagation and avoids complex body fitted or adaptive grid schemes the new method should be useful in areas of wave propagation from radar scattering and long distance communications to cell phone transmission

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

solitary waves with damped oscillatory tails an analysis of the fifth order korteweg de vries equation abstract we construct oscillatory solitary wave solutions of a fifth order korteweg de vries equation where the oscillations decay at infinity these waves arise as a bifurcation from the linear dispersion curve at that wavenumber where the linear phase speed and group velocity coincide our approach is a wave packet analysis about this wavenumber which leads in the first instance to a higher order nonlinear schrodinger equation from which we then obtain the steady solitary wave solution we then describe a complementary normal form analysis which leads to the same result in addition we derive the nonlinear schrodinger equation for all wavenumbers and list all the various anomalous cases

a natural constraint approach to normalized solutions of nonlinear schrodinger equations and systems abstract the paper deals with the existence of normalized solutions to the system u 1 u 1 u 3 u v 2 in r 3 v 2 v 2 v 3 u 2 v in r 3 r 3 u 2 a 1 2 and r 3 v 2 a 2 2 for any 1 2 a 1 a 2 0 and 0 prescribed we present a new approach that is based on the introduction of a natural constraint associated to the problem we also show that as phase separation occurs for the solutions that we find our method can be adapted to scalar nonlinear schrodinger equations with normalization constraint and leads to alternative and simplified proofs to some results already available in the literature

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

stability of solitary waves in finite ablowitz ladik lattices we discuss the effects of different types of boundary conditions b c on the linear stability of a solitary wave in a finite length dynamical lattice described by the ablowitz ladik al model types of b c considered are fixed dirichlet no flux neumann and free as well as periodic b c the behaviour of eigenvalues around a stationary nonlinear wave consistent with several types of b c is studied analytically and numerically the translational eigenvalues are found to move away from the origin it is shown that for b c of the fixed type these eigenvalues bifurcate into a neutrally stable oscillatory kind while the no flux and free b c lead to an exponentially weak instability the rotational eigenmodes those related to the gauge invariance of the lattice strictly remain at the origin as gauge invariance is not broken by the linear homogeneous b c these numerical results are compared to the analytical and semi analytical predictions obtained within the approximation of the images generated by the solitary wave beyond the lattice edges a potential effect of continuous spectrum band edge bifurcations on stability is also evaluated with the conclusion that that these bifurcations cannot destabilize the soliton for periodic b c the translational invariance is preserved and so is the lax pair structure sustaining the integrable nature of the problem the al model was chosen so that the effects of boundary conditions are monitored in the absence of non integrable discreteness finally the possibility of the generation of multipulse configurations due to the interplay of finiteness effects with the exponential tail tail interaction of the pulses is examined

regularization of cauchy abstract problem for a coupled system for nonlinear elliptic equations abstract in this paper we consider a cauchy problem of coupled elliptic equations we show that the problem is ill posed in the sense of hadamard and a regularization is in order under some weak a priori assumptions on the exact solution we propose a new regularization method to stabilize the problem when the source term is a global or local lipschitz function furthermore we also obtain error estimates between the regularized solution and the sought solution

upper bound on the products of particle interactions in cellular automata abstract particle like objects are observed to propagate and interact in many spatially extended dynamical systems for one of the simplest classes of such systems one dimensional cellular automata we establish a rigorous upper bound on the number of distinct products that these interactions can generate the upper bound is controlled by the structural complexity of the interacting particles a quantity which is defined here and which measures the amount of spatio temporal information that a particle stores along the way we establish a number of properties of domains and particles that follow from the computational mechanics analysis of cellular automata thereby elucidating why that approach is of general utility the upper bound is tested against several relatively complex domain particle cellular automata and found to be tight

exceptional discretizations of the sine gordon equation the method of one dimensional maps was recently introduced as a means of generating exceptional discretizations of the phi 4 theory i e discrete phi 4 models which support kinks centered at a continuous range of positions relative to the lattice in this paper we employ this method to obtain exceptional discretizations of the sine gordon equation i e exceptional frenkel kontorova chains we also use one dimensional maps to construct a discrete sine gordon equation supporting kinks which move with arbitrary velocities without emitting radiation

a new general filter regularization method for cauchy problems for elliptic equations with a locally lipschitz nonlinear source up to now studies on the semi linear cauchy problem for elliptic partial differential equations needed to assume that the source term present in the governing equation is a global lipschitz function the current paper is the first investigation to not only the more general but also the more practical case of interest when the source term is only a local lipschitz function in such a situation the methods of solution from the previous studies with a global lipschitz source term are not directly applicable and therefore novel ideas and techniques need to be developed to tackle the local lipschitz nonlinearity this locally lipschitz source arises in many applications of great physical interest governed by for example the sine gordon lane emden allen cahn and liouville equations the inverse problem is severely ill posed in the sense of hadamard by violating the continuous dependence upon the input cauchy data therefore in order to obtain a stable solution we consider theoretical aspects of regularization of the problem by a new generalized filter method under some priori assumptions on the exact solution we prove and obtain rigorously convergence estimates

quasi continuum approximations for travelling kinks in diatomic lattices we extend the quasi continuum method of approximation to the case of a diatomic lattice this is illustrated by a lattice in which both small and large atoms interact with first and second nearest neighbours we show that highly accurate quasi continuum techniques may be generalized to determine the shape of nontopological kinks many previous analyses of such systems have found travelling wave solutions only for one particular speed using a second order continuum theory we show i how wave shapes can be found for arbitrary speeds and ii how solution profiles can be calculated from fourth order partial differential equations which approximate the lattice alongside the theoretical analysis we also present numerical simulations of the lattice which demonstrate propagation of the predicted waves through the lattice we show that the particular speed for which solutions have been found in previous studies is a special speed for waves in the lattice but waves can travel for long periods of time at faster or slower speeds whilst slowly relaxing to this critical speed

a fixed point operator for a nonlinear boundary value problem we study a semilinear second order equation with a nonlinear boundary condition for the axial deformation of a nonlinear elastic beam in the presence of friction under appropriate conditions we define a fixed point operator in order to obtain solutions for this equation

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

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 an inverse boundary value problem of a nonlinear elliptic equation in three dimensions abstract this work considers an inverse boundary value problem for a 3d nonlinear elliptic partial differential equation in a bounded domain in general the problem is severely ill posed the formal solution can be written as a hyperbolic cosine function in terms of the 2d elliptic operator via its eigenfunction expansion and it is shown that the solution is stabilized or regularized if the large eigenvalues are cut off in a theoretical framework a truncation approach is developed to approximate the solution of the ill posed problem in a regularization manner under some assumptions on regularity of the exact solution we obtain several explicit error estimates including an error estimate of holder type a local lipschitz case of source term for this nonlinear problem is obtained for numerical illustration two examples on the elliptic sine gordon and elliptic allen cahn equations are constructed to demonstrate the feasibility and efficiency of the proposed methods

oscillatory wave fronts in chains of coupled nonlinear oscillators wave front pinning and propagation in damped chains of coupled oscillators are studied there are two important thresholds for an applied constant stress f for f f cs static peierls stress there are only stable moving wave fronts for piecewise linear models extending an exact method of atkinson and cabrera s to chains with damped dynamics corroborates this description for smooth nonlinearities an approximate analytical description is found by means of the active point theory generically for small or zero damping stable wave front profiles are nonmonotone and become wavy oscillatory in one of their tails

multipeak solutions for an elliptic system of fitzhugh nagumo type we consider the dirichlet problem for an elliptic system of fitzhugh nagumo type we prove that the problem has solutions with arbitrary many sharp peaks near the boundary but not on the boundary

simple theories of complex lattices abstract while the theory of solitons has been very successful for continuous systems very few nonlinear discrete lattices are amenable to an exact analytical treatment in these complex lattices discreteness can be hostile to the solitons preventing them to move due to the lack of translational invariance or even to exist as localized excitations on the other hand lattice discreteness can sometimes be very helpful it can stabilize solutions that otherwise would split apart as in the discrete sine gordon lattice or even allow the existence of localized oscillatory modes as exact solutions in systems where they would decay in the continuum limit it is interesting that many of these phenomena can be understood qualitatively and sometimes quantitatively with very simple theories that rely on the usual concepts of linear wave propagation resonances linear stability of waves for instance there are however phenomena specific to discrete nonlinear lattices which allow the build up of large amplitude localized excitations sometimes out of thermal fluctuations which are more resistant to simple approaches and could deserve further interest because they may be relevant for various physical systems

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

modulational and numerical solutions for the steady discrete sine gordon equation in two space dimensions abstract we develop a modulation theory based on a suitably averaged lagrangian for steady solutions of the sine gordon equation in a two dimensional lattice these lump solutions are nonzero in circular or polygonal regions and zero elsewhere the modulation theory gives approximate solutions away from small perturbations of the exact anti continuum solutions for both radial and polygonal solutions we show how the peierls nabarro potential determines the shape of the boundary between excited sites and the zero solution these solutions are compared with the corresponding numerical solutions and significant agreement is found moreover we show that solutions with a large radius more than sixteen lattice sites can be explained using a continuous trial function for the averaged lagrangian while smaller polygonal solutions can be constructed using a trial function which takes into account the angular variation of the boundary imposed by the lattice finally the ideas of equivariant bifurcation theory are used to obtain a full numerical description of the solution branches as functions of the coupling parameter between neighboring sites the results of this work can be used to study steady solutions for other types of lattice 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

kink dynamics in the highly discrete sine gordon system abstract we study kink dynamics in a very discrete sine gordon system where the kink width is of the order of the lattice spacing numerical simulations exhibit new properties of kinks in this case they lose the memory of their initial velocity and propagate preferentially at well defined velocities which correspond to quasi steady states while a kink moving at other velocities suffers relatively high rates of radiation of small amplitude oscillations when a small external driving force is applied to the system the same velocities appear as plateus in the strongly nonlinear mobility of the kink the energy radiated by the kink is calculated for a simple model that preserves the discrete character of the system and the preferential velocities for the kink are obtained to good accuracy similar results may be expected to be valid for other discrete systems manifesting topological solitons the numerical simulations reveal also new stable multiple kink excitations which can propagate almost freely in extremely discrete systems where ordinary simple kinks are pinned to the lattice by discreteness the stability of the multiple kinks is discussed

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

chaos transition of soliton motion on a microstructured lattice abstract soliton motion on a damped driven one dimensional lattice is proposed the lattice model is made of a one dimensional chain equipped at each point with a rotatory molecule the set of coupled difference differential equations for the molecule rotation is obtained for the discrete system using a semi discrete kink solution for which its center of mass is considered as a collective variable we derive the equation of motion for the kink location we show that the kink location is then governed by the equation of the damped sinusoidally driven pendulum when the first order corrections are dropped by means of numerical simulations carried out directly on the lattice model the complex dynamics of the kink location is investigated and compared to that of the well known forced pendulum for some ranges of the control parameters the numerical simulations exhibit a cascade of odd subharmonic components as well as strange attractors which characterizes the chaotic behavior of the kink motion these results agree fairly well with the main features of the transition to chaos for the damped driven pendulum

discrete klein gordon models with static kinks free of the peierls nabarro potential for the nonlinear klein gordon type models we describe a general method of discretization in which the static kink can be placed anywhere with respect to the lattice these discrete models are therefore free of the peierls nabarro potential previously reported models of this type are shown to belong to a wider class of models derived by means of the proposed method a relevant physical consequence of our findings is the existence of a wide class of discrete klein gordon models where slow kinks practically do not experience the action of the peierls nabarro potential such kinks are not trapped by the lattice and they can be accelerated by even weak external fields