Michael Tabor

Bio: Michael Tabor is an academic researcher from University of Arizona. The author has contributed to research in topics: Nonlinear system & Kirchhoff equations. The author has an hindex of 32, co-authored 56 publications receiving 6275 citations. Previous affiliations of Michael Tabor include University of Bristol & Columbia University.

The Painlevé property for partial differential equations

Abstract: In this paper we define the Painleve property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Backlund transforms, the linearizing transforms, and the Lax pairs of three well‐known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painleve property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.

Level Clustering in the Regular Spectrum

Abstract: In the regular spectrum of an f -dimensional system each energy level can be labelled with f quantum numbers originating in f constants of the classical motion. Levels with very different quantum numbers can have similar energies. We study the classical limit of the distribution P(S) of spacings between adjacent levels, using a scaling transformation to remove the irrelevant effects of the varying local mean level density. For generic regular systems P(S) = e -s , characteristic of a Poisson process with levels distributed at random. But for systems of harmonic oscillators, which possess the non-generic property that the ‘energy contours’ in action space are flat, P(S) does not exist if the oscillator frequencies are commensurable, and is peaked about a non-zero value of S if the frequencies are incommensurable, indicating some regularity in the level distribution; the precise form of P(S) depends on the arithmetic nature of the irrational frequency ratios. Numerical experiments on simple two-dimensional systems support these theoretical conclusions.

Closed Orbits and the Regular Bound Spectrum

Abstract: The energy levels of systems whose classical motion is multiply periodic are accurately given by the quantum conditions of Einstein, Brillouin & Keller (E. B. K.). We transform the E. B. K. conditions into a representation of the spectrum in terms of a ‘topological sum’ involving only the closed classical orbits; the theory applies equally to separable and non-separable systems; stability parameters are not involved. Significant contributions come from complex closed orbits which however have real constants of the motion. Clustering of levels on different scales is demonstrated by smoothing the spectrum using the formal device, due to Balian & Bloch, of adding a variable imaginary part to the energy. The topological sum is shown to agree very well with exactly-computed spectra for circular and spherical potential wells with repulsive core.

Analytic structure of the Henon–Heiles Hamiltonian in integrable and nonintegrable regimes

Abstract: The solutions of the Henon–Heiles Hamiltonian are investigated in the complex time plane. The use of the ’’Painleve property,’’ i.e., the property that the only movable singularities exhibited by the solution are poles, enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable. Special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes. These boundaries have a remarkable self‐similar structure whose form changes as a function of the nonlinear coupling.

Pattern formation outside of equilibrium

Abstract: A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

Differential Equations and Dynamical Systems

Abstract: Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

The Painlevé property for partial differential equations

Abstract: In this paper we define the Painleve property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Backlund transforms, the linearizing transforms, and the Lax pairs of three well‐known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painleve property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.