Fluctuation Phenomena: Disorder and Nonlinearity

The following sections are included:

• PREFACE

• CONTENTS

• Group Picture

Fractal (correlation) dimension as an eigen parameter of time- or space-dependent process is considered. By concrete examples of electro magnetic wave transition through one-dimensional irregular medium, it is shown that fractal dimension of the scattering process can serve as a measure of ”irregularity” of the structure to be investigated and, therefore, can be an eigen parameter of the given scattering process.

Reported are our recent contributions to the study on motion of curves and surfaces and its applications in condensed matter physics. First, the Saffman-Taylor finger solution and its generalizations are given. Second, a level-set approach to the motion of hypersurfaces is shown. Third, the level-set formulation in a curved space is given, which may include ”disorder” into the geometrical models.

The action diffusion of an area preserving map, whose linear frequency is stochastically perturbed, is described by a Fokker-Planck equation with a coefficient obtained from the normal form. The local properties are investigated on the standard map and the angle is shown to have a fast relaxation. A comparison is made with the diffusive process determined by a slow periodic modulation of the frequency.

The Lie-group formalism is applied to deduce symmetries of the following equation

with n ≠ 0 and n ≠ 1. This equation serves as a simple matematical model for various physical problems. Perhaps its most common use is to describe the flow of liquids in porous media or the transport of thermal energy in plasma…

We shall be considering here mKP equation in the form:

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We have analyzed the role of intensity losses in the self-focusing and collapse of light beams propagating in cubic type nonlinear media. The nonlinear Schrödinger equation appropriate for this case is

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In a previous publication¹ we have described the electronic analog circuit for a model nonlinear oscillator

. In our case g=0.49, ω=1.195 and arbitrary A. In fig. (1) and (2) we show the experimental first return map (21022 points) with P₀ a third-order polynomial having slope close to minus one at the middle. A hole may be seen around the fixed point. We are going to show that this apparent hole does belong to the attractor with, however, vanishingly small probability…

Zimmermann & Velarde (1994), have proposed a forced BKdV equation which models the streamfunction response to a passing temperature wave in a strongly sheared fluid layer. If the passing temperature wave is sinusoidal, and if dissipation from turbulence is weak, then the nonlinear Helmholtz oscillator with forcing and damping is a good aproximation of the streamfunction dynamics. In this way they conjecture regimes of temperatute wave induced chaos, since the nonlinear oscillator has been proved to possess chaotic solutions. On the other hand Grinshaw & Tian, (1994) also propose a forced BKdV equation, from which a nonlinear Helmholtz oscillator after a reduction via steady solutions is obtained and they analize its chaotic behaviour.

In this work the dynamics of a parametrically driven reduction of the Burgers-Kortewegde Vries equation is studied numerically and analytically. It is assumed that certain information for the original perturbed equation could be obtained from the steady state dynamics. We carry out numerical experiments of the dynamics of the above mentioned equation. Period doubling bifurcations in the phase space, along with homoclinic crossings are observed and the Lyapunov attractor dimension is estimated. The analytical condition to be fulfilled for the homoclinic crossings to occur and thus for the appearance of chaos, is explicitely calculated, using the Melnikov method. This calculation is done for the general case in which we consider parametric drivings acting on the linear and nonlinear terms of the dynamical variable, besides the external forcing.

We sketch a new dynamic approach to statistical mechanics based on a procedure that reverses the standard approach to the Fokker-Planck equation (FPE). Rather than using statistical mechanics, derived from thermodynamics along the lines pioneered by Boltzmann, we derive a FPE for a set of variables of interest interacting with a booster, i.e., a dynamical system mimicking the action of an ideal thermostat with no need of ad hoc statistical assumptions.

We derive a mechanical expression for the temperature of the system of interest, which can be proved to be a generalization of the one proposed by Boltzmann; in the case of boosters with a limited number of degrees of freedom, it is shown that our definition depends also on dynamic properties, which are not accounted for by the “standard” approach. It turns out that a basic ingredient for our derivation is the applicability of a linear response theory for the response of the booster.

The oxacyanine Scheibe aggregate is modelled by a two-dimensional cubic nonlinear Schrödinger equation with multiplicative noise, accounting for thermal fluctuations. For a possible choice of the physical parameters collapse of the initial state modelling the Möbius-Kuhn coherent exciton excited by impinging light may occur. The collapse time then increases with temperature. Prom experimental observations Möbius and Kuhn find a life time of the exciton which also increase with temperature.

We introduce smooth and generalized nonlinear functional of Gaussian white noise through series expansions and apply this technique to selfintersection local times of Brownian motion and to Burgers’ equation.

We investigate a generalization of the oscillator model of a particle interacting with a thermal reservoir, in which arbitrary couplings, nonlinear in the particle coordinates, are present. The equilibrium positions of the heat bath oscillators are promoted to space-time functions, which are shown to represent a modulation of the internal noise by the external forces. A classical generalized Langevin equation with nonlinear multiplicative noise and a position-dependent fluctuationdissipation theorem are obtained. The corresponding classical Fokker-Planck equation presents interesting non perturbative effects. The model is quantized to obtain the non homogenous influence functional and master equation for the reduced density matrix of the Brownian particle. When time-modulation of the noise is present, a new term appears in the quantum evolution equations, which may have important consequences for the decoherence process. Non-linear effects and finite-size effects of the environment are briefly discussed.

We consider a forced Burgers equation with a forcing of distribution type. Analytical solutions are compared to the results coming from numerical simulation.

We consider the escape problem for systems driven by dichotomous noise by splitting the contribution of trajectories escaping through each boundary, and presenting exact results for the escape probabilities and first passage times. Different boundary conditions are also analyzed.

Three model systems described by partial differential Langevin equations with external noise are studied. They present phase transitions controlled by the parameters of the noise. These nonequilibrium transitions are characterized by studying the behaviour of the linear relaxation time in the vicinity of the transition.

We present a process to achieve the solution of the two dimensional nonlinear Schrödinger equation using a multigrid technique on a distributed memory machine. Some features about the multigrid technique as its good convergence and parallel properties are explained in this paper. This makes multigrid method the optimal one to solve the systems of equations arising at each time step from an implicit numerical scheme. We give some experimental results about the parallel numerical simulation of this equation on a message passing parallel machine.

The lagrangian formulation is of great relevance in many fields of Physics. The idea is to construct numerical schemes in finite differences that integrate the Euler-Lagrange equations and have a discrete counterpart of the conservation/variation law of the energy. We consider a general lagrangian with explicit time dependence, and the associated physical quantities such as the energy and momenta. The idea from the numerical point of view is to consider a discrete equivalent of the variation law for the energy, and to deduce from there a suitable numerical representation of the derivative of the positions, the momenta and the Euler-Lagrange equation which define the numerical scheme. All the schemes that can be obtained in this way have, by construction, discrete counterparts of the variation law of the energy and all of them are conservatives whenever the lagrangian is time-independent. In general it is possible to obtain several schemes for the same lagrangian: further considerations on other symmetries may help to decide which one is best suited for a specific problem. There are families of conservative schemes that are widely and succesfully used [1-4]. Nevertheless, these schemes are usually only applicable to problems with hamiltonians or lagrangians sepparable into a kinetic and a potential parts. The schemes we present here can be applied to more general problems…

When looking for new algorithms to carry out the parallel numerical simulation of physics systems. it is necessary to find fast and accurate algorithms with good parallel properties. Our goal is the parallel numerical simulation of large multidimensional nonlinear Schrödinger systems (Eq. 1).

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There is a common framework where many different connectionist systems (neural networks, classifier systems, immune networks, autocatalytic reactions) may be treated in a unified way. The general system in which they may all be mapped is a network where not only the connection strengths but also the node parameters have learning plasticity. In this paper we discuss Hebbian-type and competitive learning algorithms for networks with such node parameters. We find improvements on the convergence rate for small eigenvalues in principal component analysis (PCA). Improved discrimination or robustness is also found for competitive learning and feature mapping. For non-Gaussian signals, PCA algorithms or higher order generalisations provide incomplete or misleading information on the statistical properties of the data. An algorithm is proposed for on-line learning of the characteristic function as an unsupervised learning scheme.

After some considerations concerning the nonlinear behaviour of some biological systems, ranging from cardiology to psychiatry, an optical processing element is reported with a chaotic behaviour. The employed cell was reported previously as the basic building block for the performing of logic operations. It is employed in this paper as the basis for the modelling of the mammalian retina.

Our study adresses the problem of multistability in a network of weakly coupled bistable elements [1]. A bistable element is a dynamical system which has simultaneously two stable steady states. We study conditions under which a network consisting of such coupled elements admits a family of stationary states which are in one-to-one correspondence with arbitrary sequences of binary digits. This question is important in the design of ‘optical memories’ based on a mechanism of optical bistability, or in the study of ‘propagation failure’ in a network of biological cells…

A simple model of DNA is developed, in which the base-ribose moieties are represented by rigid bodies. The dynamical fluctuations of this statically irregular system are similar to those observed in all-atom molecular dynamics simulations, but multiple timescales for different types of dynamical disorder are observed for times beyond the scope of normal molecular dynamics studies.

Thermal effects on nonlinear structures such as the Davydov soliton are studied with the help of two distinct approaches. One, a nonequilibrium approach based on Brownian motion methods, is shown to result in the prediction of limit cycles and bifurcations. The other, an equilibrium approach consisting of a Gibbs technique of the evaluation of the partition function at arbitrary temperature, is shown to predict a dual tendency of a thermal reservoir: to enhance the nonlinear structure at low tempetaures and to destroy it at high temperatures. Experiments designed to probe into some of the effects are described.

In this paper we present recent studies on a two-dimensional anharmonic model, related to the lifetime of the open states precursors to full denaturation, in inhomogeneous ring-shaped DNA molecules. DNA sequences of the following types are considered XXXX … XXYYY … YYYXX … XXXX, where we use X to denote W (weak base pairs), S (strong base pairs) or M (medium base pairs) type, and YYY … YYY represents alternances of the following types: SWW, SSWW and randoms. These studies show that the presence of inhomogeneities enhances the hydrogen bond breaking and that the kind of inhomogeneity is not very important.

In a nonlinear system, disorder can cooperate with nonlinearity to localize energy. This may be relevant for some biological phenomena at the molecular scale and the thermal denaturation of DNA, which is a phase transition in a quasi-one dimensional system, provides an interesting example. Nonlinear energy localization plays an important role in the initiation of the transition, but the “disorder” of the base sequence, i.e. the genetic code, can modify significantly its dynamics. This effect, which is observed experimentally, is studied in the framework of a simple model of DNA. Disorder can enhance the effect of nonlinearity by providing specific sites for nonlinear energy localization as well as an additional pathway for the separation of the two strands.

A new integral equation approach, analagous to that used by Evans and Ford [1] to investigate the properties of the normal solitary wave, is presented for Internal Solitary waves. A two fluid-layer model is employed where potential flow is assumed in both layers with the pressure continuous across the fluid interface. This implies a discontinuity of velocity in the two fluids along the interface streamline so that the latter becomes a line of non-vanishing vorticity. The resulting coupled non-linear integral equations describe the various wave profiles and interface fluid velocities and can be solved by the multi-variable Newton-Raphson method. New solutions are presented and their properties examined for the “constrained flat-topped” internal waves, which are argued to be virtually identical to the “free-surface”, oceanic internal waves where the layer densities are nearly equal. Comparison with the observed properties of internal waves (as seen, for example, in the Andaman Sea) is discussed.

Pinning effects are present in different pattern forming systems. Perfect patterns can be analyzed by means of simple amplitude equations obtained after some adiabatic elimination techniques. However, perfect structures are rarely obtained in experiments, which always contain defects, domain walls,… The study of these structures within the amplitude equations formalism is not complete, and when the short spatial scale is taken into account some effects not considered in this formalism appear. Two examples of pinning effect are analyzed in this paper: the pinning in a front between hexagons and squares and the pinning of a defect in a system under rotation.

A simple model is chosen to point at fluctuation induced anomalous plasma transport effects, both analytically and numerically. For drift-wave fluctuations it is shown that nonlinear spatially coherent modes do play a very important role in the particle transport beyond a critical magnetic field strength. The transition from collisional to convective transport is investigated. With increasing control parameter spatially coherent but timechaotic states appear. The signatures of chaos are: (i) a cross-over in the magnetic-field dependencies of the particle diffusion coefficients and (ii) different algebraic k-dependencies (for large wave-numbers k) in the spectra of the density fluctuations.

A brief account is provided here of recent analytical, numerical and experimental work establishing the existence of (‘imperfect’, hence ‘aging’) dissipative solitons as a consequence of instability and thus when no energy is conserved but there is, however, appropriate (steady) balance between energy supply usually at long wavelengths and dissipation at shorter ones by viscosity. Created and maintained following an instability threshold, these propagating localized structures (van der Waals-like particles) in the form of solitary waves/pulses or crested wave trains, feature elastic and inelastic bead-on and oblique collisions, wall reflections with and without formation of Mach stems, bound states and chaos (erratic versus periodic wave trains).

We investigate the stability of breathers, in systems of N inductively coupled long Josephson junctions (LJJ). We perform our analysis on a dc-driven system of coupled sine-Gordon equations, using the perturbative approach of collective coordinates. Performing numerical simulations, we find very good qualitative and quantitative agreement between our ODE predictions and the numerical solutions of the corresponding system of PDEs.

Recent investigations of soliton dynamics in inductively coupled Josephson transmission lines are reviewed. Experimentally, the most efficient way to couple long Josephson junctions is to stack them vertically. Solitons (Josephson vortices) moving in neighboring junctions interact with each other via their screening currents flowing in a thin superconducting layer between the junctions. The effects of the interaction between solitons are observed in the current-voltage characteristics of these structures. Two dynamic regimes characterized by different soliton propagation velocities were found in two-fold stacks. In the lower velocity (anti-phase) mode solitons located in different lines repel each other. In contrast, the high velocity (in-phase) mode corresponds to an attraction between unipolar solitons which form bound states between neighboring lines. This behavior is justified by direct numerical simulations of coupled sine-Gordon equations which models the dynamics of the system. Both the anti-phase and the in-phase locking of moving soliton chains have been observed experimentally.

Josephson junctions (JJ) arrays have been widely used to modelize the properties of type II superconductors. They also provide an excellent theoretical and experimental system to find the complex behaviour which characterizes the non-linear problems…

The study of the morphology of growing surfaces is very important both from the fundamental and the applied viewpoint [1]. A key to progress is to achieve a complete understanding of roughening and its dynamical consequences, as roughening transitions play significant roles in three-dimensional crystal growth. Equilibrium roughening transitions are by now well understood and most of them belong to the Kosterlitz-Thouless class. The nature of the transition in the presence of external driving (corresponding, e.g., to the chemical potential difference between surface and vapor or to liquid undercooling when growing from melts) is much less well understood. Therefore, we concern ourselves with a study of non-equilibrium roughening transitions in surface growth models…

A discrete drift model of resonant tunneling transport in weakly-coupled GaAs quantum-well structures under laser illumination is introduced and analyzed. Results include explanations (via formation of stationary electric field domains) of the oscillatory shape of the I – V diagram leading to multistability and hysteresis for both doped and undoped superlattices under strong laser illumination. Moreover, the dynamics of negatively charged domain walls (formation, motion, annihilation and regeneration) accounts for damped and undamped time-dependent oscillations of the current in a dc voltage bias situation, with the laser photoexcitation acting as a damping factor.

New examples of nonlinear dynamics for protons in solids studied with inelastic neutron-scattering spectroscopy are presented. Firstly, collective rotational dynamics of methyl groups are modeled with the quantum sine-Gordon equation and, secondly, proton transfer dynamics in infinite chains of hydrogen bonded peptide groups are represented with chains of coupled double minimum potentials.

We present here a summaiy of relevants results on the dissipative dynamics of a discrete elastic medium on a periodic pinning potential (Frenkel-Kontorova model) under external time-periodic forces. For commensurate structures, the macroscopic mean velocity shows always mode locking as a function of the average force. The mechanism of the locking-unlocking transition is clarified in terms of the occurrence of instantons, a type of intermittencies carrying a topological charge. For incommensurate structures, a dynamical Aubry transition is observed, separating two dfferent dynamical regimes.

We considered the effect of a harmonic long-range interatomic potential in a monatomic anharmonic chain. The existence of two characteristic scales in the problem leads to two types of solitons with characteristically different widths. Thus care is required in taking the quasicontinuum approximation. The low velocity branch exists up to a maximum critical velocity in accordance with an effective potential description. The high velocity branch is separated by a gap where no stable solitons are found. The kink amplitude is in agreement with the results of numerical simulations.

The thermodynamic properties of the one-dimensional sine-lattice model in an external field are calculated in a form of a series expansion in terms of the modified Bessel function of integer order. The free energy, internal energy, order parameter and pair correlation function are derived based on n-site expansion.

We consider a square lattice with nonlinear interactions between nearest and next-nearest neighbours, where the potentials are expanded to fourth order in the relative displacements. We search for solitons depending only on one space variable and obtain two coupled nonlinear partial differential equations in the continuum limit. These can be reduced to one nonlinear Schrödinger equation for one of the displacement fields. We get localized solutions, which are used as initial conditions for computer simulations. For the case of Morse interaction potentials we find solitary waves which are a combination of a pulse and an envelope soliton for the longitudinal and transversal displacements, respectively.

It is shown that localized nonlinear modes associated with a gap in the frequency spectrum of linear waves, i.e. gap solitons, can exist in nonlinear systems with purely quadratic nonlinearities. As an example we consider the discrete Klein-Gordon type diatomic chain. Using the asymptotic expansion technique, a system of coupled nonlinear equations describing interaction between two conter-propagating waves of the same frequency is derived and its spatially localized solutions are found in an explicit form.

The effective equation method is developed for description of disordered lattices generated by the discretized Hirota equation. A small parameter of the problem is connected with a correlation radius of inhomogeneities rather than with dispersion of fluctuations. Dynamics of a soliton of the stochastic discrete nonlinear Schrödinger equation is considered in details.

Intrinsic collapse to self localized states is studied. We have investigated a one dimensional electrical lattice where the dynamics of modulated waves can be modeled by a discrete nonlinear Schrödinger equation which interpolates between the Ablowitz-Ladik and Discrete-self-trapping equations. Experimentally, we have observed that modulational instability can develop for continuous waves with frequencies higher than the linear cut-off frequency of the lattice. These results are confirmed by the observation of “staggered” localized modes. Unlike to envelope-solitons which can be observed close to the zerodispersion point, the staggered modes experience strong lattice effects. Second, we have investigated, theoretically and numerically, the dynamical behavior of a two-dimensional Sine-Gordon system. We have shown that, via modulational instability, an initial-lowamplitude plane wave can evolve spontaneously into a spiral wave with large amplitude. These localized spirals, with dimensions depending on the characteristic wavelengths of the instability, behave like breathing solitary waves.

The following sections are included:

• Introduction

• Approximnte and Exact Analytical Solutions to the d-Dimensional Sine-Lattice Equations for the Ordered dSC Lattice

• Nonlinear Surface Modes

• Stationary Nonlinear Localized Modes and Vortices in Disordered Lattices

• References

We investigate electronic propagation in a one-dimensional nonlinear random binary alloy in the tight-binding formalism. We find absence of electronic localization below certain nonlinearity parameter threshold. Above threshold, the electron is partially trapped while the rest propagates away to infinity in a ballistic manner. The presence of disorder is completely overcome by the nonlinear terms leading to ballistic propagation of the untrapped electronic fraction. The existence of disorder in the model is manifested in the power-law decay of the transmissivity of plane waves through the medium as a function of the system size.

We consider nonlinear excitations in an elastic three-atomic one-dimentional chain with symmetric anharmonic potential of nearest-neighbor interaction and small mass difference. In a three-atomic chain there are two gaps in the spectrum of linear excitations. Gap solitons in these two gaps are essentially identical to one another and differ only in a period of the carrier wave. In the main approximation the structure of a gap soliton in three-atomic chain does not depend on the type of anharmonicity and for any relationship between interatomic nonlinear interaction and nonlinear on-site potential this structure is identical with this of Bragge solitons in nonlinear modulated optical media.

An solvable integrable nonlinear 2D field model is proposed, which is shown to be in close connection with the nonlinear Schrodinger chain. The conservation laws and soliton solutions for the correspondent field equations are obtained. Analogous relation between the 2D Toda lattice (2DTL) and the Ablowitz-Ladik hierarchy is established which is used to reinvestigate the inverse scattering transform for the 2DTL.

The 1-D antiferromagnetic (AFM) model is analyzed permitting the generalization of the Frenkel-Kontorova model to a system of two fields: the atomic displacement field and that of the atomic spin orientation. It is assumed that the equilibrium ordering of atomic masses caused by the mechanical interatomic interaction, corresponds to the fully frustrated spin ordering. However, the magnetoelastic interaction gives rise to the spontaneous uniform deformation which results either in two-fold degenerated or non - degenerated AFM main state. In both cases, the magnetic domains can exist which are separated by domain boundary (DB) representing a soliton solution of the non-linear equation for the displacement field. It is shown that the DBs may be of two types. The DB of the first type retains the uniform spin distribution in the chain and is associated only with the kink (large and small). This DB includes vacancy or crowdion, i.e. causes the non-elastic deformation of the displacement field. The DB of the second type is the 2π-kink of the spin orientation field and gives rise to the deformation of the displacement field not altering the atomic ordering in the chain.

The existence of intrinsic localized modes in classical, anharmonic lattice models has been stablished in the works of Sievers and Takeno^[l]. By ‘intrinsic’ it is meant that the localization of the vibration is due only to the anharmonicity of the potentials involved, and no disorder or impurities are needed to sustain these modes. In analogy to the continuum models, these type of solutions have been called breathers. In the original references a lattice Green’s function approach is used to calculate breathers in different model chains…

We present a study made on a generalized Frenkel Kontorova model in which the standard cosine-type substrate potential has been modified by the insertion of a new term. The different length scales induced by both terms are chosen to be incommensurate related one each other (irrational α). On the other hand, the interaction potential is kept to be convex, so many properties discovered in the past in relation with other FK models can be applied…

We report interesting effects of the interplay of quantum phases and nonlinearity in small systems characterized by a strong interaction between a quasiparticle (an excitation or an electron) and lattice vibrations, and described by the Discrete Nonlinear Schrödinger Equation.¹ The fundamental issue under analysis is the effect of the nature of initial placement of a quasiparticle on the process of self-trapping. We ask the following questions: (a) Is self-trapping assisted or harmed by having the quasiparticle occupy a few versus many sites at the initial time? (b) Does the initial spread of electronic excitation created by the absorption of light over a large number of molecular sites help or deter excitation trapping?2 (c) In the context of photosynthesis, could one design more efficient photosynthetic antennae by controlling the phases of the absorbed excitation?…

The discrete nonlinear Schrödinger equation (DNLSE) of the self-trapping type was derived by Davydov from an exciton-phonon Hamiltonian, by assuming that: i) the state vector is a product of a single excitation state by a coherent phonon state; ii) the parameters of the state vector can be treated as classical variables driven by the expectation value of the quantum Hamiltonian; iii) the phonon vibrations can be adiabatically eliminated. Kenkre and Wu¹ released the adiabatic assumption and also introduced damping of the vibrations. Then they focused on the two-site case and studied the so-called nonlinear nonadiabatic dimer, which turns out to be isomorphic to the model of a classical dipole interacting with a classical damped oscillator…

When a uniform magnetic field is applied to a single electron in 2 and 3D continuous and elastically deformable medium, a polaronic self localisation occurs at any small coupling although it does not without magnetic field. In 3D there is also a first order transition as a function of the magnetic field corresponding to the collapse of the large polaron into a small polaron. The discrete lattice effect were numerically investigated in 2D for completing this picture. It is suggested that strong anisotropies in real systems could brought these phenomena in the physically observable range.

Using examples of the perturbed (1+1) dimensional sine-Gordon, the continuous and discrete nonlinear Schrödinger systems, and a three-site quantum polaron problem, we briefly review some phenomena related to the fascinating interplays between nonlinearity, disorder, noise, nonadiabaticity, and lattice discreteness. The concept of competing length-scales and time-scales is emphasized as they pertain to the common concept of solitons and polarons behaving as “particles”.

We consider non-interacting electrons in two dimensions subject to a perpendicular magnetic field and interacting with point impurities with random strength located on the sites of a square lattice. For very strong magnetic field (such that there are more than n+1 flux quanta per plaque), we obtain (in closed analytic form) extended solutions, which are independent on the strength of disorder, for each Landau level with a number smaller or equal to n. If the rational magnetic flux per plaque is less than one unit, we construct dispersion laws and Hofstadter-like butterfly for all denominators less than ten.

The following sections are included:

• Introduction

• Numerical experiments

• Linear stability analysis and discussion

• Acknowledgments

• References

The dynamics of both the ionic and orientational defects in hydrogen fluoride has been studied on the basis of multi-component soliton models. The mobility of the positive ionic defects has been shown to essentially exceed the mobility of the negative ionic defects. On the contrary, the positive (negative) orientational defects have been shown to be immobile (mobile) objects. The role of cooperativity of hydrogen bonding has been studied in ice on the basis of a dynamical square ice lattice model. Ice as a dichotomously branching random system has been shown not to support the free soliton dynamics, even in the case of strong interbond interactions.

A brief review of our recent works on solitary waves in polyethylene (PE) crystals is presented. A soliton model with three variables is derived from realistic inter and intramolecular potentials. Several types of topolgkali solitons that describe twisting and elongation of the PE chain are obtained. It is shown that these solitons can propagate smoothly along the chain and that their interactions are inelastic.

Many basic properties of solids depend crucially on the interactions between electrons and lattice vibrations which give rise to phenomena such as polaron formation [1]. Most of our knowledge on electron-phonon systems is based on many-body theory treatments of the ground state (or low-lying excited states) problem for periodic lattices. Disorder on the other hand modifies the electronic properties of a system in profound ways: the extended Bloch states develop phase incoherence and amplitude fluctuations, the sharp band edges disappear and tails of localized states emerge. Over the past two decades, our understanding in the field of localization has been greatly advanced [2]. Much less has been done to understand the combined effects of disorder and elph interactions, effects that are important in any real solid. It has been argued, for example, that even vanishing small el-ph coupling strongly enhances localization and polaron formation at the so-called mobility edge in disordered systems [3], or that impurities play the role of nucleation centers in the polaron formation [4].

The problem of nonlinear wave propagation in media with time dependent second-order dispersion and nonlinearity is considered. The modulational instability of electromagnetic waves in periodically and randomly modulated optical fibers is investigated and the new regions of modulational instability are predicted. The envelope soli ton propagation in media with periodically and randomly modulated dispersion is studied. The law of radiative decay of soliton is found. The chirped pulse dynamics in fiber with a random dispersion is analyzed by means of variational approach. The stochastic decay length is estimated and compared with the numerical simulations.

The following sections are included:

• Introduction

• Characteristic equations for pulse dynamics

• Almost adiabatic soliton compression

• References

We study the use of nonlinear amplifying loop mirrors as elements of high dispersion fiber transmission systems to reshape soliton pulses in order to overcome broadening. We approach this problem as a mapping problem of input pulse to output pulse, for a segment of transmission fiber followed by a combination of linear and nonlinear amplification. For a wide range of amplifier spacings, we find that an optimal soliton input pulse can be found which is well reproduced at output. These optimal input pulses produce only about three percent continuous radiation in the output pulse.

Conventional optical fiber presents group dispersion and nonlinearity for light waves having a reasonable power (a few miliwatts) and a pulse width (a few tens of picoseconds). As a result, the optical soliton is most suitable as the mean of transmitting ultra-high speed signal in a fiber because it provides a precise input-output relationship (integrability) as well as possibility of reducing undesirable effects during the transmission (controllability).

A linearly implicit finite difference scheme for the Maxwell-Bloch equations has been developed and implemented using a multigrid technique. To illustrate the behavior we simulate some conventional and symmetry breaking physical examples and find very good behavior of the scheme.

Generation of spatio-temporal Brillouin solitons in a cw-pumped fiber-ring cavity is controlled by the only feedback parameter R. The nonlinear dynamics of stimulated Brillouin scattering (SBS) is governed by the 1-D coherent three-wave model, which resonantly couples two optical waves (pump and backscattered Stokes) with an acoustic wave. Stability analysis around the steady solution presents a regular Hopf bifurcation. For R < R_crit the unstable amplitude modes merge together into a localized pulse, giving rise to soliton morphogenesis. Numerical simulation of the dynamical model starting from any initial conditions (material noise or steady state), and experiments in an optical fiber-ring cavity confirm this bifurcation and the generation of an asymptotically stable train of soliton-like pulses. They may be associated to the asymptotic three-wave soliton in an unbounded line, and in the case of strongly damped material wave to a simple traveling wave envelope solution of the two optical intensity equations.

The transmission of solitons generated by using as a pulse source a gain-switched and optically filtered laser diode is analyzed. The pulse spreading observed in transmission experiments is shown to be due to the pulse-to-pulse frequency jitter originated in the gain-switched laser diode. This spreading is suppressed with the use of periodically spaced filters. When biasing the laser slightly below threshold pseudorandom sequences of solitons are obtained by direct modulation. A notable reduction of the pulse spreading during transmission is observed for this bias current.

The interplay between the electric field polarization and transverse effects in lasers is analyzed through amplitude equations, phase equations and numerical stability analysis. Linearly polarized, transverse traveling waves are found, whose stability near threshold is restricted by a phase instability associated with the direction of polarization. Far above threshold the phase instability is replaced by amplitude instabilities which significantly reduce the second lasing threshold.

New mechanism of optical pulse compression is examined. Two examples of collapse dynamics in nonlinear optical systems are considered. First, we analyze a pulse generation in the laser systems employing passive mode locking by Kerreffect or additive-pulse mode locking. Basing on a phenomenological laser model for homogeneously broadened systems we study the efect that collapse has on the process of a pulse generation. Second, we discuss a pulse propagation in onedimensional fiber array. Quasi-collapse effect leads to the localization of all energy initially dispersed in array into few fibers. Rate of compression is estimated for Gaussian input pulses.