Showing papers on "Nonlinear system published in 1997"

Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations

Abstract: Preface.- Basic notations.- Outline of the main ideas on a model problem.- Continuous viscosity solutions of Hamilton-Jacobi equations.- Optimal control problems with continuous value functions: unrestricted state space.- Optimal control problems with continuous value functions: restricted state space.- Discontinuous viscosity solutions and applications.- Approximation and perturbation problems.- Asymptotic problems.- Differential Games.- Numerical solution of Dynamic Programming.- Nonlinear H-infinity control by Pierpaolo Soravia.- Bibliography.- Index

Long-scale evolution of thin liquid films

Abstract: Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. In this review a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations. As a result of this long-wave theory, a mathematical system is obtained that does not have the mathematical complexity of the original free-boundary problem but does preserve many of the important features of its physics. The basics of the long-wave theory are explained. If, in addition, the Reynolds number of the flow is not too large, the analogy with Reynolds's theory of lubrication can be drawn. A general nonlinear evolution equation or equations are then derived and various particular cases are considered. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolution of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlinear evolution equation are first examined. These include: (a) films with constant interfacial shear stress and constant surface tension, (b) films with constant surface tension and gravity only, (c) films with van der Waals (long-range molecular) forces and constant surface tension only, (d) films with thermocapillarity, surface tension, and body force only, (e) films with temperature-dependent physical properties, (f) evaporating/condensing films, (g) films on a thick substrate, (h) films on a horizontal cylinder, and (i) films on a rotating disc. The dynamics of the films with a spatial dependence of the base-state solution are then studied. These include the examples of nonuniform temperature or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those include (a) the dynamics of free liquid films, (b) bounded films with interfacial viscosity, and (c) dynamics of soluble and insoluble surfactants in bounded and free films. The spreading of drops on a solid surface and moving contact lines, including effects of heat and mass transport and van der Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also briefly discussed. The results discussed give motivation for the development of careful experiments which can be used to test the theories and exhibit new phenomena.

Coherence Resonance in a Noise-Driven Excitable System

Abstract: We study the dynamics of the excitable Fitz Hugh ‐ Nagumo system under external noisy driving. Noise activates the system producing a sequence of pulses. The coherence of these noise-induced oscillations is shown to be maximal for a certain noise amplitude. This new effect of coherence resonance is explained by different noise dependencies of the activation and the excursion times. A simple one-dimensional model based on the Langevin dynamics is proposed for the quantitative description of this phenomenon. [S0031-9007(97)02349-1] The response of dynamical systems to noise has attracted large attention recently. There are many examples demonstrating that noise can lead to more order in the dynamics. To be mentioned here are the effects of noiseinduced order in chaotic dynamics [1], synchronization by external noise [2], and stochastic resonance [3‐5]. Also, noise has been shown to play a stabilizing role in ensembles of coupled oscillators and maps [6]. Especially interesting is the phenomenon of stochastic resonance, which appears when a nonlinear system is simultaneously driven by noise and a periodic signal. At a certain noise amplitude the periodic response is maximal; this has been confirmed by numerous experimental studies (cf. [7,8]). In this paper we study the effect of noise on the autonomous excitable oscillator—the famous Fitz Hugh ‐ Nagumo system. We demonstrate that a characteristic correlation time of the noise-excited oscillations has a maximum for a certain noise amplitude, and present a theory of this effect. Contrary to the usual setup of stochastic resonance, no external periodic driving is assumed, so the coherence appears as a nonlinear response to purely noisy excitation. The phenomenon considered is also different from stochastic resonance without periodic force reported recently in Ref. [9], where the effect of noise on a limit cycle at a bifurcation point was studied. The Fitz Hugh‐Nagumo model is a simple but representative example of excitable systems that occur in different fields of application ranging from kinetics of chemical reactions and solid-state physics to biological processes [10]. Originally it was suggested for the description of nerve pulses [11]; it was also widely used for modeling of spiral waves in a two-dimensional excitable medium. Different aspects of the dynamics of this and similar excitable models in the presence of noise have been discussed in Refs. [12‐16]. The equations of motion are

Solitons : nonlinear pulses and beams

Abstract: Basic equations. The nonlinear Schroedinger equation. Exact solutions. Non-Kerr-law nonlinearities. Normal dispersion regime. Multiple-port linear devices made from solitons. Nonlinear pulses in birefringent media. Pulses in nonlinear couplers. Multi-core nonlinear fiber arrays. The influence of higher-order dispersion on solitons. Beam dynamics. Planar nonlinear guided waves. Nonlinear pulses in presence of gain, loss and spectral filtering. References. Index.

An analysis of temporal-difference learning with function approximation

Abstract: We discuss the temporal-difference learning algorithm, as applied to approximating the cost-to-go function of an infinite-horizon discounted Markov chain. The algorithm we analyze updates parameters of a linear function approximator online during a single endless trajectory of an irreducible aperiodic Markov chain with a finite or infinite state space. We present a proof of convergence (with probability one), a characterization of the limit of convergence, and a bound on the resulting approximation error. Furthermore, our analysis is based on a new line of reasoning that provides new intuition about the dynamics of temporal-difference learning. In addition to proving new and stronger positive results than those previously available, we identify the significance of online updating and potential hazards associated with the use of nonlinear function approximators. First, we prove that divergence may occur when updates are not based on trajectories of the Markov chain. This fact reconciles positive and negative results that have been discussed in the literature, regarding the soundness of temporal-difference learning. Second, we present an example illustrating the possibility of divergence when temporal difference learning is used in the presence of a nonlinear function approximator.

Discrete Breathers

Abstract: Nonlinear classical Hamiltonian lattices exhibit generic solutions in the form of discrete breathers. These solutions are time-periodic and (typically exponentially) localized in space. The lattices exhibit discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. Necessary ingredients for their occurence are the existence of upper bounds on the phonon spectrum (of small fluctuations around the groundstate) of the system as well as the nonlinearity in the differential equations. We will present existence proofs, formulate necessary existence conditions, and discuss structural stability of discrete breathers. The following results will be also discussed: the creation of breathers through tangent bifurcation of band edge plane waves; dynamical stability; details of the spatial decay; numerical methods of obtaining breathers; interaction of breathers with phonons and electrons; movability; influence of the lattice dimension on discrete breather properties; quantum lattices - quantum breathers. Finally we will formulate a new conceptual aproach capable of predicting whether discrete breather exist for a given system or not, without actually solving for the breather. We discuss potential applications in lattice dynamics of solids (especially molecular crystals), selective bond excitations in large molecules, dynamical properties of coupled arrays of Josephson junctions, and localization of electromagnetic waves in photonic crystals with nonlinear response.

First-principles responses of solids to atomic displacements and homogeneous electric fields: Implementation of a conjugate-gradient algorithm

Abstract: The changes in density, wave functions, and self-consistent potentials of solids, in response to small atomic displacements or infinitesimal homogeneous electric fields, are considered in the framework of the density-functional theory. A variational: principle for second-order derivatives of the energy provides a basis for efficient algorithmic approaches to these linear responses, such as the state-by-state conjugate-gradient algorithm presented here in detail. The phase of incommensurate perturbations of periodic systems, that are, like phonons, characterized by some wave vector, can be factorized: the incommensurate problem is mapped on an equivalent one presenting the periodicity of the unperturbed ground state. The singularity of the potential change associated with an homogeneous field is treated by the long-wave method. The efficient implementation of these theoretical ideas using plane waves, separable pseudopotentials, and a nonlinear exchange-correlation core correction is described in detail, as well as other technical issues.

Transactions costs and nonlinear adjustment in real exchange rates: an empirical investigation

Abstract: Eolquilibrium models of real exchange rate determination in the presence of transactions costs imply a nonlinear adjustment process toward purchasing power parity (PPP). Conventional cointegration tests, Which ignore the effect of transactions costs, may be biased against the long‐run PPP hypothesis. Our results, using both monthly data for the interwar period and annual data spanning two centuries, clearly reject the linear framework in favor of an exponential smooth transition autoregressive process. The systematic pattern in the estimates of the nonlinear models provides strong evidence of mean‐reverting behavior for PPP deviations and helps explain the mixed results of previous studies

Lectures on Nonlinear Hyperbolic Differential Equations

Abstract: In this introductory textbook, a revised and extended version of well-known lectures by L. Hormander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Four chapters are devoted to microanalysis of the singularities of the solutions. This part assumes some familiarity with pseudodifferential operators which are standard in the theory of linear differential operators, but the extension to the more exotic classes of opertors needed in the nonlinear theory is presented in complete detail.

Nonlinear Partial Differential Equations for Scientists and Engineers

Abstract: Preface to the Third Edition.- Preface.- Linear Partial Differential Equations.- Nonlinear Model Equations and Variational Principles.- First-Order, Quasi-Linear Equations and Method of Characteristics.- First-Order Nonlinear Equations and Their Applications.- Conservation Laws and Shock Waves.- Kinematic Waves and Real-World Nonlinear Problems.- Nonlinear Dispersive Waves and Whitham's Equations.- Nonlinear Diffusion-Reaction Phenomena.- Solitons and the Inverse Scattering Transform.- The Nonlinear Schroedinger Equation and Solitary Waves.- Nonlinear Klein--Gordon and Sine-Gordon Equations.- Asymptotic Methods and Nonlinear Evolution Equations.- Tables of Integral Transforms.- Answers and Hints to Selected Exercises.- Bibliography.- Index.

Parabolized stability equations

Abstract: Parabolized stability equations (PSE) have opened new avenues to the analysis of the streamwise growth of linear and nonlinear disturbances in slowly varying shear flows such as boundary layers, jets, and far wakes. Growth mechanisms include both algebraic transient growth and exponential growth through primary and higher instabilities. In contrast to the eigensolutions of traditional linear stability equations, PSE solutions incorporate inhomogeneous initial and boundary conditions as do numerical solutions of the Navier-Stokes equations, but they can be obtained at modest computational expense. PSE codes have developed into a convenient tool to analyze basic mechanisms in boundary-layer flows. The most important area of application, however, is the use of the PSE approach for transition analysis in aerodynamic design. Together with the adjoint linear problem, PSE methods promise improved design capabilities for laminar flow control systems.

Blow-up solutions of some nonlinear elliptic problems

Abstract: posed in a baunded domain fi of R~ with smooth boundary 8 SI wiih Dirichiel dala u100 = O, and a continuous, positive, increasing and convex funetion f an [O,oc) such thai f(s)/s — 00 as a —. oc>. Under Ihese conditions Ihere is a maximal or extremal value of the parameter A > O such thaI ihe problem has a solution. We invesligate Ihe exisience and properties of Ihe corresponding extrema) solutions when Ihe>’ are unhaunded (i.e., singular or blow-up solutions). We characterize ihe singular Hí extremal solulions and ihe extrenial value by a eriterioncansisiing of twa condiiions: (i) they musí be ener~’ saluiions, mal in L~; (Ii) tite>’ musí sntisfy a Hardy inequalil>’ which transíates Ihe fact thai ihe firsí eigenvalue of the linearized aperator is nannegalive. In arder to apply ibis characlerizatiat to ihe typical examples arising in Ihe lilerature we need an improved version of ihe cíassical Hardy inequalil>’ wiih besí constaní. We aiablish such a resulí as a simultaneous generalization of Hardy’s and Poincaré’s inequaUiies for ah dirnensions n > 2. A striking prapert>’ of sorne examples of unbaunded exiremal solutiona is the fact thai tite hinearization of ihe prablem araund them happens to be formalí>’ invertible and nevertheless tite npplication of ihe Inverse and Imphicii Funetion iheorems falís to produce Ihe usual exislence ar continuation resulís. We consider ihis question and explain tite pitenomenon as a lack of appropriate funetional setting.

A Review of Nonlinear Diffusion Filtering

Abstract: This paper gives an overview of scale-space and image enhancement techniques which are based on parabolic partial differential equations in divergence form. In the nonlinear setting this filter class allows to integrate a-priori knowledge into the evolution. We sketch basic ideas behind the different filter models, discuss their theoretical foundations and scale-space properties, discrete aspects, suitable algorithms, generalizations, and applications.

Nonlinear scanning laser microscopy by third harmonic generation

Abstract: Third harmonic generation near the focal point of a tightly focused beam is used to probe microscopical structures of transparent samples. It is shown that this method can resolve interfaces and inhomogeneities with axial resolution comparable to the confocal length of the beam. Using 120 fs pulses at 1.5 μm, we were able to resolve interfaces with a resolution of 1.2 μm. Two-dimensional cross-sectional images have also been produced.

Stable knot-like structures in classical field theory

Abstract: In 1867, Lord Kelvin proposed that atoms—then considered to be elementary particles—could be described as knotted vortex tubes in either1. For almost two decades, this idea motivated an extensive study of the mathematical properties of knots, and the results obtained at that time by Tait2 remain central to mathematical knot theory3,4. But despite the clear relevance of knots to a large number of physical, chemical and biological systems, the physical properties of knot-like structures have not been much investigated. This is largely due to the absence of a theoretical means for generating stable knots in the nonlinear field equations that can be used to describe such systems. Here we show that knot-like structures can emerge as stable, finite-energy solutions in one such class of equations—local, three-dimensional langrangian field-theory models. Our results point to several experimental and theoretical situations where such structures may be relevant, ranging from defects in liquid crystals and vortices in superfluid helium to the structure-forming role of cosmic strings in the early Universe.

A unified transform method for solving linear and certain nonlinear PDEs

Abstract: A new transform method for solving initial boundary value problems for linear and for integrable nonlinear PDEs in two independent variables is introduced. This unified method is based on the fact that linear and integrable nonlinear equations have the distinguished property that they possess a Lax pair formulation. The implementation of this method involves performing a simultaneous spectral analysis of both parts of the Lax pair and solving a Riemann–Hilbert problem. In addition to a unification in the method of solution, there also exists a unification in the representation of the solution. The sine–Gordon equation in light–cone coordinates, the nonlinear Schrodinger equation and their linearized versions are used as illustrative examples. It is also shown that appropriate deformations of the Lax pairs of linear equations can be used to construct Lax pairs for integrable nonlinear equations. As an example, a new Lax pair of the nonlinear Schrodinger equation is derived.

Traveling Waves in a Convolution Model for Phase Transitions

Abstract: The existence, uniqueness, stability and regularity properties of traveling-wave solutions of a bistable nonlinear integrodifferential equation are established, as well as their global asymptotic stability in the case of zero-velocity continuous waves. This equation is a direct analog of the more familiar bistable nonlinear diffusion equation, and shares many of its properties. It governs gradient flows for free-energy functionals with general nonlocal interaction integrals penalizing spatial nonuniformity.

A new Volterra predistorter based on the indirect learning architecture

Abstract: Nonlinear compensation techniques are becoming increasingly important. We present a new Volterra-based predistorter, which utilizes the indirect learning architecture to circumvent a classical problem associated with predistorters, namely that the desired output is not known in advance. We utilize the indirect learning architecture and the recursive least square (RLS) algorithm. Specifically, we propose an indirect Volterra series model predistorter which is independent of a specific nonlinear model for the system to be compensated. Both 16-phase shift keying (PSK) and 16-quadrature amplitude modulation (QAM) are used to demonstrate the efficacy of the new approach.

Stochastic Burgers and KPZ equations from particle systems

Abstract: We consider two strictly related models: a solid on solid interface growth model and the weakly asymmetric exclusion process, both on the one dimensional lattice. It has been proven that, in the diffusive scaling limit, the density field of the weakly asymmetric exclusion process evolves according to the Burgers equation and the fluctuation field converges to a generalized Ornstein-Uhlenbeck process. We analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the (non linear) Burgers equation with a random noise on the density current. For the solid on solid model, we prove that the fluctuation field of the interface profile, if suitably rescaled, converges to the Kardar–Parisi–Zhang equation. This provides a microscopic justification of the so called kinetic roughening, i.e. the non Gaussian fluctuations in some non-equilibrium processes. Our main tool is the Cole-Hopf transformation and its microscopic version. We also develop a mathematical theory for the macroscopic equations.

Semiclassical States of Nonlinear Schrödinger Equations

Abstract: The existence of semiclassical states to some nonlinear Schrodinger equations that concentrate near the critical points of the potential V is studied by means of a local approach, variational in nature. We also discuss stability and necessary conditions for concentration. The same method is used to find multiple homoclinic orbits to a class of second‐order Hamiltonian systems.

Blind separation of mixture of independent sources through a quasi-maximum likelihood approach

Abstract: We propose two methods for separating mixture of independent sources without any precise knowledge of their probability distribution. They are obtained by considering a maximum likelihood (ML) solution corresponding to some given distributions of the sources and relaxing this assumption afterward. The first method is specially adapted to temporally independent non-Gaussian sources and is based on the use of nonlinear separating functions. The second method is specially adapted to correlated sources with distinct spectra and is based on the use of linear separating filters. A theoretical analysis of the performance of the methods has been made. A simple procedure for optimally choosing the separating functions is proposed. Further, in the second method, a simple implementation based on the simultaneous diagonalization of two symmetric matrices is provided. Finally, some numerical and simulation results are given, illustrating the performance of the method and the good agreement between the experiments and the theory.

Nonlinear self-excited oscillations of a ducted flame

Abstract: Self-excited oscillations of a confined flame, burning in the wake of a bluff-body flame-holder, are considered. These oscillations occur due to interaction between unsteady combustion and acoustic waves. According to linear theory, flow disturbances grow exponentially with time. A theory for nonlinear oscillations is developed, exploiting the fact that the main nonlinearity is in the heat release rate, which essentially ‘saturates’. The amplitudes of the pressure fluctuations are sufficiently small that the acoustic waves remain linear. The time evolution of the oscillations is determined by numerical integration and inclusion of nonlinear effects is found to lead to limit cycles of finite amplitude. The predicted limit cycles are compared with results from experiments and from linear theory. The amplitudes and spectra of the limit-cycle oscillations are in reasonable agreement with experiment. Linear theory is found to predict the frequency and mode shape of the nonlinear oscillations remarkably well. Moreover, we find that, for this type of nonlinearity, describing function analysis enables a good estimate of the limit-cycle amplitude to be obtained from linear theory.Active control has been successfully applied to eliminate these oscillations. We demonstrate the same effect by adding a feedback control system to our nonlinear model. This theory is used to explain why any linear controller capable of stabilizing the linear flow disturbances is also able to stabilize finite-amplitude oscillations in the nonlinear limit cycles.