Showing papers on "Nonlinear system published in 1981"

Solitons and the Inverse Scattering Transform

Abstract: : Under appropriate conditions, ocean waves may be modeled by certain nonlinear evolution equations that admit soliton solutions and can be solved exactly by the inverse scattering transform (IST). The theory of these special equations is developed in five lectures. As physical models, these equations typically govern the evolution of narrow-band packets of small amplitude waves on a long (post-linear) time scale. This is demonstrated in Lecture I, using the Korteweg-deVries equation as an example. Lectures II and III develop the theory of IST on the infinite interval. The close connection of aspects of this theory to Fourier analysis, to canonical transformations of Hamiltonian systems, and to the theory of analytic functions is established. Typical solutions, including solitons and radiation, are discussed as well. With periodic boundary conditions, the Korteweg-deVries equation exhibits recurrence, as discussed in Lecture IV. The fifth lecture emphasizes the deep connection between evolution equations solvable by IST and Painleve transcendents, with an application to the Lorenz model.

Introduction to perturbation techniques

Abstract: Algebraic Equations. Integrals. The Duffing Equation. The Linear Damped Oscillator. Self-Excited Oscillators. Systems with Quadratic and Cubic Nonlinearities. General Weakly Nonlinear Systems. Forced Oscillations of the Duffing Equation. Multifrequency Excitations. The Mathieu Equation. Boundary-Layer Problems. Linear Equations with Variable Coefficients. Differential Equations with a Large Parameter. Solvability Conditions. Appendices. Bibliography. Index.

Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids

Abstract: Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto-fluid dynamics including new constructive local existence theorems for the time-singular limit equations. In particular, the authors give an entirely self-contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier-Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero.

Nonlinear regulation: The piecewise linear approach

Abstract: This paper approaches nonlinear control problems through the use of (discrete-time) piecewise linear systems. These are systems whose next-state and output maps are both described by PL maps, i.e., by maps which are affine on each of the components of a finite polyhedral partition. Various results on state and output feedback, observers, and inverses, standard for linear systems, are proved for PL systems. Many of these results are then used in the study of more general (both discrete- and continuous-time) systems, using suitable approximations.

A priori bounds for positive solutions of nonlinear elliptic equations

Abstract: We derive a priori bounds for positive solutions of the non-linear elliptic boundary value problem where Ω is a bounded domain in R n. Our proof is by contradiction and uses a scaling (“blow up”) argument reminiscent to that used in the theory of Minimal Surfaces. This procedure reduces the problem of a priori bounds to global results of Liouville type.

On the Existence of Positive Solutions of Semilinear Elliptic Equations

Abstract: In this paper we study the existence of positive solutions of semilinear elliptic equations. Various possible behaviors of nonlinearity are considered, and in each case nearly optimal multiplicity results are obtained. The results are also interpreted in terms of bifurcation diagrams.

Nonlinear decoupling via feedback: A differential geometric approach

Abstract: The paper deals with the nonlinear decoupling and noninteracting control problems. A complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems. The paper also includes algorithms concerned with the actual construction of the appropriate control laws.

Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria

Abstract: A nonlinear gyrokinetic formalism for low‐frequency (less than the cyclotron frequency) microscopic electromagnetic perturbations in general magnetic field configurations is developed. The nonlinear equations thus derived are valid in the strong‐turbulence regime and contain effects due to finite Larmor radius, plasma inhomogeneities, and magnetic field geometries. The specific case of axisymmetric tokamaks is then considered and a model nonlinear equation is derived for electrostatic drift waves. Also, applying the formalism to the shear Alfven wave heating scheme, it is found that nonlinear ion Landau damping of kinetic shear‐Alfven waves is modified, both qualitatively and quantitatively, by the diamagnetic drift effects. In particular, wave energy is found to cascade in wavenumber instead of frequency.

Non-linear oscillations

Abstract: Part 1 The mathematical pendulum as an illustration of linear and non-linear oscillations - systems which are similar to a simple linear oscillator: Undamped free oscillations of the pendulum damped free oscillations forced oscillations. Part 2 Liapounov stability theory and bifurcations: The concept of Liapounov stability the direct method of Liapounov stability by the first approximation the Poincare map the critical case of a conjugate pair of eigenvalues simple bifurcation of equilibria and the Hopf bifurcation. Part 3: Self-excited oscillations in mechanical and electrical systems analytical approximation methods for the computation of self-excited oscillations analytical criteria for the existence of limit cycles forced oscillations in self-excited systems self-excited oscillations in systems with several degrees of freedom Part 4 Hamiltonian systems: Hamiltonian differential equations in mechanics canonical transformations the Hamilton-Jacobi differential equation canonical transformations and the motion perturbation theory Part 5 Introduction to the theory of optimal control: Control problems, controllability the Pontryagin maximum principle transversality conditions and problems with target sets canonical perturbation theory in optimal control.

Modelling nonlinear random vibrations using an amplitude-dependent autoregressive time series model

Abstract: SUMMARY The behaviour of nonlinear deterministic vibrations has been studied by many authors, and may typically include such features as jump phenomena and limit cycles. Nonlinear random vibrations in continuous time have also been studied and these may commonly give rise to the phenomenon of amplitude-dependent frequency. A discrete time series model is introduced, which may be demonstrated to have properties similar to those of nonlinear random vibrations. This model is of autoregressive form with amplitudedependent coefficients and may be estimated using an extension of a method for estimating linear time series models. The model is fitted to the Canadian lynx data and demonstrates that it may be possible to regard the periodic behaviour of this series as being generated by some underlying self-exciting mechanism.

Large-Amplitude Quasi-Solitons in Superfluid Films

Abstract: Nonlinear time evolution of the condensate wave function in superfluid films is studied on the basis of a Schrodinger equation, which incorporates van der Waals potential due to substrate in its fully nonlinear form, and a surface tension term. In the weak nonlinearity limit, our equation reduces to the ordinary (cubic) nonlinear Schrodinger equation for which exact soliton solutions are known. It is demonstrated by numerical analysis that even under strong nonlinearity , where our equation is far different from cubic Schrodinger equation, there exist quite stable composite “quasi-solitons" . These quasi-solitons are bound states of localized excitations of amplitude and phase of the condensate (superfluid thickness and superfluid velocity, in more physical terms). Thus the present work shows the persistence of the solitonic behavior of superfluid films in the fully nonlinear situation.

Sensitivity theory for nonlinear systems. I. Nonlinear functional analysis approach

Abstract: Concepts of nonlinear functional analysis are employed to investigate the mathematical foundations underlying sensitivity theory. This makes it possible not only to ascertain the limitations inherent in existing analytical approaches to sensitivity analysis, but also to rigorously formulate a considerably more general sensitivity theory for physical problems characterized by systems of nonlinear equations and by nonlinear functionals as responses. Two alternative formalisms, labeled the ’’forward sensitivity formalism’’ and the ’’adjoint sensitivity formalism,’’ are developed in order to evaluate the sensitivity of the response to variations in the system parameters. The forward sensitivity formalism is formulated in normed linear spaces, and the existence of the Gâteaux differentials of the operators appearing in the problem is shown to be both necessary and sufficient for its validity. This formalism is conceptually straightforward and can be advantageously used to assess the effects of relatively few parameter alterations on many responses. On the other hand, for problems involving many parameter alterations or a large data base and comparatively few functional‐type responses, the alternative adjoint sensitivity formalism is computationally more economical. However, it is shown that this formalism can be developed only under conditions that are more restrictive than those underlying the validity of the forward sensitivity formalism. In particular, the requirement that operators acting on the state vector and on the system parameters must admit densely defined Gâteaux derivatives is shown to be of fundamental importance for the validity of this formalism. The present analysis significantly extends the scope of sensitivity theory and provides a basis for still further generalizations.

One-sided difference approximations for nonlinear conservation laws

Abstract: We analyze one-sided or upwind finite difference approximations to hyperbolic partial differential equations and, in particular, nonlinear conservation laws. Second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions. We show that no such stable approximation of order higher than two is possible. These one-sided schemes have desirable properties for shock calculations. We show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance. New and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles, existence, uniqueness, and stability of discrete shocks. Numerical examples are given.

Nonlinear system modeling based on the Wiener theory

Abstract: This paper is a tutorial of nonlinear system modeling methods which are based on the Wiener theory of nonlinear systems. The basic concepts that underlie the Wiener theory are discussed and illustrated. Various modeling methods are presented by which a non-linear system can be modeled using either white Gaussian, nonwhite Gaussian, or certain non-Gaussian inputs. The experimental error in determining the Wiener model is discussed in terms of a new concept called measurement stability. Since attempts are being made to apply these modeling methods to diverse areas of study, this paper is written to be comprehensible by nonspecialists in system theory

Solvability of Nonlinear Equations and Boundary Value Problems

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Global approximate Newton methods

Abstract: We derive a class of globally and quadratically converging algorithms for a system of nonlinear equations,g(u)=0, whereg is a sufficiently smooth homeomorphism. Particular attention is directed to key parameters which control the iteration. Several examples are given that have been successful in solving the coupled nonlinear PDEs which arise in semiconductor device modelling.

Finite dimensional approximation of nonlinear problems

Abstract: In the first two papers of this series [4, 5], we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a Banach space. We derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point. The abstract theory is applied to the Galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von Karman equations.

Nonlinear Parabolic Equations Involving Measures as Initial Conditions

Abstract: : The Cauchy problem is considered for certain equations with a boundary condition and an initial condition. A solution of the equations exists if and only if O p n+2/n. This paper deals with the question of existence (and uniqueness) when the initial data is a measure, for example a Dirac mass. Physically this corresponds to the important case when the initial temperature (or initial density etc. ..) is extremely high near one point. The main novelty of this paper is to show that a solution exists only under some severe restrictions on the parameter P (or m); namely P must be less than n+2/n (mn+2/n). For example, one striking conclusion reached is the fact that an equation possesses no solution in any dimension n or = 1 and on any time interval (O,T). This result pinpoints the sharp contrast between linear and nonlinear equations from the point of view of existence.

A modified algorithm for invertibility in nonlinear systems

Abstract: The algorithm of Hirschorn [1] for the construction of inverses of nonlinear systems is modified, and new sufficient conditions for invertibility and inverse system for a larger class of systems are obtained. For the systems which do not satisfy the invertibility, conditions, a characterization of input space on which the input-output map is injective is derived.

The nonlinear wave equation for higher harmonics in free-electron lasers

Abstract: The nonlinear wave equation and self-consistent pendulum equation are generalized to describe free-electron laser operation in higher harmonics; this can significantly extend their tunable range to shorter wavelengths. The dynamics of the laser field's amplitude and phase are explored for a wide range of parameters using families of normalized gain curves applicable to both the fundamental and harmonics. The electron phase-space displays the fundamental physics driving the wave, and we use this picture to distinguish between the effects of high gain and Coulomb forces.

Nonlinear Evolution Equations - Global Behavior of Solutions

Abstract: Generalities and local theory.- The global existence problem.- Theory of monotone operators and applications.- Smoothing effect for some nonlinear evolution equations.- Schrodinger and wave equations with a logarithmic nonlinearity.- The linear case: Hilbertian theory and applications.- Some nonlinear monotone cases.- Some nonlinear, non monotone cases.- Autonomous dissipative systems.- General results for quasi-autonomous periodic systems.- More on asymptotic behavior for solutions of the nonlinear dissipative forced wave equation.- Boundedness of trajectories for quasi-autonomous dissipative systems.- Almost-periodic quasi-autonomous dissipative systems in a Hilbert space.