One-dimensional Nonlinear Lattices Research Articles

Identifying Diffusion Processes in One-Dimensional Lattices in Thermal Equilibrium

In this Letter, I propose that a properly rescaled spatiotemporal correlation function of the energy density fluctuations may be applied to characterize the equilibrium diffusion processes in lattice systems with finite temperature. Applying this function, the diffusion processes in three one-dimensional nonlinear lattices are studied. The diffusion exponent is shown to be related to the diverging exponent of the thermal conductivity of a lattice through the relation , as has been proved based on the Lévy walk assumption. The diffusion behavior is explained in terms of solitons and phonons.

Observation of composite gap solitons in optically induced nonlinear lattices in LiNbO3:Fe crystal

We observe experimentally, the first time to our knowledge, two types of composite gap solitons in optically induced one-dimensional nonlinear lattice in LiNbO(3) crystal. We observe the staggered bright composite gap soliton when a single Gauss probe beam is incident at Bragg angle as well as a dipole probe beam is incident at normal incidence. When a single Gauss beam is at normal incidence, the in-phase bright composite gap solitons are observed.

Discrete light localization in one-dimensional nonlinear lattices with arbitrary nonlocality

We model discrete spatial solitons in a periodic nonlinear medium encompassing any degree of transverse nonlocality. Making a convenient reference to a widely used material--nematic liquid crystals--we derive a form of the discrete nonlinear Schrödinger equation and find a family of discrete solitons. Such self-localized solutions in optical lattices can exist with an arbitrary degree of imprinted chirp and have breathing character. We verify numerically that both local and nonlocal discrete light propagation and solitons can be observed in liquid crystalline arrays.

Fermi–Pasta–Ulam, solitons and the fabric of nonlinear and computational science: History, synergetics, and visiometrics

This paper is mostly a history of the early years of nonlinear and computational physics and mathematics. I trace how the counterintuitive result of near-recurrence to an initial condition in the first scientific digital computer simulation led to the discovery of the soliton in a later computer simulation. The 1955 report by Fermi, Pasta, and Ulam (FPU) described their simulation of a one-dimensional nonlinear lattice which did not show energy equipartition. The 1965 paper by Zabusky and Kruskalshowed that the Korteweg-de Vries (KdV) nonlinear partial differential equation, a long wavelength model of the alpha-lattice (or cubic nonlinearity), derived by Kruskal, gave quantitatively the same results obtained by FPU. In 1967, Zabusky and Deem showed that a localized short wavelength initial excitation (then called an "optical" and now a "zone-boundary mode" excitation ) of the alpha-lattice revealed "n-curve" coherent states. If the initial amplitude was sufficiently large energy equipartition followed in a short time. The work of Kruskal and Miura (KM), Gardner and Greene (GG), and myself led to the appreciation of the infinity of denumerable invariants (conservation laws) for Hamiltonian systems and to a procedure by GGKM in 1967 for solving KdV exactly. The nonlinear science field exponentiated in diversity of linkages (as described in Appendix A). Included were pure and applied mathematics and all branches of basic and applied physics, including the first nonhydrodynamic application to optical solitons, as described in a brief essay (Appendix B) by Hasegawa. The growth was also manifest in the number of meetings held and institutes founded, as described briefly in Appendix D. Physicists and mathematicians in Japan, USA, and USSR (in the latter two, people associated with plasma physics) contributed to the diversification of the nonlinear paradigm which continues worldwide to the present. The last part of the paper (and Appendix C) discuss visiometrics: the visualization and quantification of simulation data, e.g., projection to lower dimensions, to facilitate understanding of nonlinear phenomena for modeling and prediction (or design). Finally, I present some recent developments that are linked to my early work by: Dritschel (vortex dynamics via contour dynamics/surgery in two and three dimensions); Friedland (pattern formation by synchronization in Hamiltonian nonlinear wave, vortex, plasma, systems, etc.); and the author ("n-curve" states and energy equipartition in a FPU lattice).

Coupled nonlinear waves in two-dimensional lattice

For one-dimensional nonlinear lattices, such as Toda lattice, it has been extensively studied. By considering the nonlinear effects of two-dimensional lattice, we set up the equation of motion for each particles (atoms, molecules or ions). For small amplitude and long wavelength nonlinear waves in this system, both the linear dispersion relation and the coupled Korteweg de Vries (KdV) equation are obtained. The simple soliton solution is obtained. If the nonlinear lattice is symmetric in the x and y directions, It is noted that there are two kinds of solitons. one is that propagates in either x or y directions, (1, 0) or (0, 1), the other is that propagates in the direction of (1, 1). It is in agreements with that of one-dimensional lattice. The different properties are investigated for different nonlinear interacting potentials, such as Toda potential, Morse potential and LJ potential.

Existence and Stability of Localized Oscillations in 1-Dimensional Lattices With Soft-Spring and Hard-Spring Potentials

In this paper, we use the method of homoclinic orbits to study the existence and stability of discrete breathers, i.e., spatially localized and time-periodic oscillations of a class of one-dimensional (1D) nonlinear lattices. The localization can be at one or several sites and the 1D lattices we investigate here have linear interaction between nearest neighbors and a quartic on-site potential Vu=12Ku2±14u4, where the (+) sign corresponds to “hard spring” and (−) to “soft spring” interactions. These localized oscillations—when they are stable under small perturbations—are very important for physical systems because they seriously affect the energy transport properties of the lattice. Discrete breathers have recently been created and observed in many experiments, as, e.g., in the Josephson junction arrays, optical waveguides, and low-dimensional surfaces. After showing how to construct them, we use Floquet theory to analyze their linear (local) stability, along certain curves in parameter space (α,ω), where α is the coupling constant and ω the frequency of the breather. We then apply the Smaller Alignment Index method (SALI) to investigate more globally their stability properties in phase space. Comparing our results for the ± cases of Vu, we find that the regions of existence and stability of breathers of the “hard spring” lattice are considerably larger than those of the “soft spring” system. This is mainly due to the fact that the conditions for resonances between breathers and linear modes are much less restrictive in the former than the latter case. Furthermore, the bifurcation properties are quite different in the two cases: For example, the phenomenon of complex instability, observed only for the “soft spring” system, destabilizes breathers without giving rise to new ones, while the system with “hard springs” exhibits curves in parameter space along which the number of monodromy matrix eigenvalues on the unit circle is constant and hence breather solutions preserve their stability character.

Coupled nonlinear waves in two-dimensional lattice

For one-dimensional nonlinear lattices, such as Toda lattice, it has been extensively studied. By considering the nonlinear effects of two-dimensional lattice, we set up the equation of motion for each particles (atoms, molecules or ions). For small amplitude and long wavelength nonlinear waves in this system, both the linear dispersion relation and the coupled Korteweg de Vries (KdV) equation are obtained. The simple soliton solution is obtained. If the nonlinear lattice is symmetric in the x and y directions, It is noted that there are two kinds of solitons. one is that propagates in either x or y directions, (1, 0) or (0, 1), the other is that propagates in the direction of (1, 1). It is in agreements with that of one-dimensional lattice. The different properties are investigated for different nonlinear interacting potentials, such as Toda potential, Morse potential and LJ potential.

Study of one-dimensional nonlinear lattice

In this article a brief review of the theory of one-dimensional nonlinear lattice is presented. Special attension is paid for the lattice of particles with exponential interaction between nearest neighbors (the Toda lattice). The historical exposition of findings of the model system, basic equations of motion, special solutions, and the general method of solutions are given as chronologically as possible. Some reference to the Korteweg-de Vries equation is also given. The article consists of three parts. Firstly, the idea of dual system is presented. It is shown that the roles of masses and springs of a harmonic linear chain can be exchanged under certain condition without changing the eigenfrequencies. Secondly, the idea is applied to the anharmonic lattice and an integrable lattice with exponential interaction force between adjacent particles is obtained. Special solutions to the equations of motion and general method of solution are shown. In the last part, some studies on the Yang-Yang's thermodynamic formalism is given.

Wave scattering by discrete breathers.

We present a theoretical study of linear wave scattering in one-dimensional nonlinear lattices by intrinsic spatially localized dynamic excitations or discrete breathers. These states appear in various nonlinear systems and present a time-periodic localized scattering potential for plane waves. We consider the case of elastic one-channel scattering, when the frequencies of incoming and transmitted waves coincide, but the breather provides with additional spatially localized ac channels whose presence may lead to various interference patterns. The dependence of the transmission coefficient on the wave number q and the breather frequency Omega(b) is studied for different types of breathers: acoustic and optical breathers, and rotobreathers. We identify several typical scattering setups where the internal time dependence of the breather is of crucial importance for the observed transmission properties.

Heat conduction in a one-dimensional aperiodic system.

We study the energy transport properties of one-dimensional nonlinear aperiodic lattice models in this paper. It is found that, compared with their periodic and disordered counterparts, the quasiperiodic and fractal models exhibit critical macroscopic behavior in the low-temperature region, while in the high-temperature region all the models show the same property of energy transport. The relationship between the observed macroscopic behaviors and the localization theory is discussed.

Homoclinic orbits of invertible maps

In this paper, we present a systematic method for finding all homoclinic orbits of invertible maps in any finite dimension. One advantage of this method is that it can also be used to order and classify all the homoclinic orbits, using symbolic dynamics, if a certain criterion is satisfied. We also present a more direct scheme, which quickly locates homoclinic orbits without, however, being able to order and classify them. Our work represents an extension of a method introduced in an earlier paper, with which one could only find homoclinic orbits possessing a certain symmetry. Thus, asymmetric homoclinic orbits can now be as easily computed. One application of our results is the explicit construction of breather (and multibreather) solutions of a class of one-dimensional nonlinear lattices.

Resonance effects in nonlinear lattices

We study a class of one-dimensional nonlinear lattices with nearest-neighbour interactions described by a potential of the binomial type. This potential contains a free parameter which can be chosen to reproduce a variety of models, such as the Toda, the Fermi-Pasta-Ulam and the Coulomb-like lattices. Carrying out essentially numerical experiments, the effects of soliton propagation on a lattice with defects are investigated. In particular, the properties of the localized mode, generated by the propagation of the soliton through the defect, are discussed with respect to the defect mass and the potential parameter, in the light of a simple theoretical model. Furthermore, an interesting phenomenon is observed: the amplitude of the speed of the mass defect shows a sequel of resonance peaks in terms of the mass defect. The positions of these peaks appear to be independent of the potential parameter.

Oscillatory instabilities of standing waves in one-dimensional nonlinear lattices.

In one-dimensional anharmonic lattices, we construct nonlinear standing waves (SWs) reducing to harmonic SWs at small amplitude. For SWs with spatial periodicity incommensurate with the lattice period, a transition by breaking of analyticity versus wave amplitude is observed. As a consequence of the discreteness, oscillatory linear instabilities, persisting for arbitrarily small amplitude in infinite lattices, appear for all wave numbers Q not equal 0,pi. Incommensurate analytic SWs with |Q|>pi/2 may however appear as "quasistable," as their instability growth rate is of higher order.

Nonlinear Inter-Particle Potentials Sustaining Exact Steady Motions in One-Dimensional Lattices

A method is presented to get a sustaining nonlinear inter-particle potential for an arbitrarily given steady motion in a one-dimensional nonlinear lattice. Here each particle is simply assumed to interact only with its nearest neighbors. This scheme is carried with an extensive use of formal series expansions and manipulations. In the latter half of this paper, this method is applied to some simple cases.

Breathers and multibreathers in a periodically driven damped discrete nonlinear Schrödinger equation.

We study an integrable discretization of the nonlinear Schrödinger equation (NLS) under the effects of damping and periodic driving, from the point of view of spatially localized solutions oscillating in time with the driver's frequency. We locate the equilibrium states of the discretized (DNLS) system in the plane of its dissipation gamma and forcing amplitude H parameters and use a shooting algorithm to construct the desired solutions psi(n)(t)=phi(n) exp(it) as homoclinic orbits of a four-dimensional symplectic map in the complex phi(n),phi(n+1) space, for -infinity<n<infinity. We derive, in the gamma=0 case, closed form expressions for two fundamental such solutions having a single hump in n, psi(n)+, and psi(n)-, and determine analytically their threshold of existence in the (gamma,H) plane using Mel'nikov's theory. Then, we demonstrate numerically that above this threshold a remarkable variety of multihump structures appear, whose complexity in terms of their spatial extrema grows with increasing H. All these solutions are numerically found to be unstable in time, except for psi(n)-, which is seen to be stable over a certain region in the (gamma,H) plane. In the continuum limit our results are in close agreement with recent studies on the NLS equation. From a more general perspective, we view these DNLS multihump solutions as homoclinic orbits of a higher-dimensional map thereby providing a possible mechanism for explaining the occurrence of similar structures called discrete (multi-) breathers found in a wide variety of one-dimensional nonlinear lattices.

Coupled wave method in the theory of a diatomic anharmonic lattice

We use the coupled wave method to investigate a one-dimensional nonlinear diatomic lattice. We consider the case where the two different lattice atoms have similar masses. This narrows the forbidden frequency gap between the longitudinal acoustic and optic phonon branches. As a result, the nonlinear shift of the gap's borders becomes comparable with the width of the gap itself. We find the exact solution, in the form of only two waves propagating towards each other. This is an important first step towards a full three-dimensional solution. We calculate the ratio of the two waves' amplitudes, and analyse the dispersive properties of the diatomic chain. Of special interest is how stable localized modes emerge in the forbidden frequency gap, both stationary and slowly propagating along the chain. We show that the envelope of their amplitudes has the form of solitary waves. What is crucial for such localized modes to appear is the spatial phase modulation of the particles' vibrations. This is because it makes the chain transparent at the margins of the forbidden gap, due to the self-action of the wave (optical switching).

Quantum breathers in a nonlinear lattice.

We study nonlinear phonon excitations in a one-dimensional quantum nonlinear lattice model using numerical exact diagonalization. We find that multi-phonon bound states exist as eigenstates which are natural counterparts of breather solutions of classical nonlinear systems. In a translationally invariant system, these quantum breather states form particle-like bands and are characterized by a finite correlation length. The dynamic structure factor has significant intensity for the breather states, with a corresponding quenching of the neighboring bands of multi-phonon extended states.

Soliton dynamics and interaction in a deterministic aperiodic nonlinear lattice.

We perform a numerical study of the dynamical properties of solitons in a one-dimensional deterministic aperiodic nonlinear lattice. The lattice is described by a discrete nonlinear Schr\"odinger equation, where the aperiodicity enters through an on-site potential chosen according to the substitutionally generated Thue-Morse sequence. Rapidly moving solitons are shown to propagate with small dispersion and radiation if they have a phase-modulation wavelength corresponding to regions of small amplitudes in the Fourier spectrum of the on-site potential. Such solitons also remain essentially unchanged after collision. Furthermore, we show that, for slowly moving solitons, an effective particle model is useful in order to understand some of the dynamic behavior, e.g., trapping of the solitons around points where they experience a local extremum in energy. The relation between the width and energy of the solitons is also discussed.

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, 836 (1994)]. Thus we confirm earlier suggestions about the generic property of Hamiltonian lattices to exhibit localized excitations independent of the dimensionality of the lattice.

Galois extensions in Kowalevski exponents and nonintegrability of nonlinear lattices

Nonintegrability of some one-dimensional nonlinear lattices with many degrees of freedom is proved by using the Ziglin-Yoshida test for integrability. The key of the present proof is to consider the degree of algebraic extensions with respect to Kowalevski exponents over Q .