Michael Tabor

The Painlevé property for partial differential equations

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

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.

Painlevé expansions for nonintegrable evolution equations

Abstract The WTC method is applied to a variety of nonintegrable nonlinear evolution equations and shown to provide a systematic procedure for obtaining special solutions. Some of the special techniques developed for integrable systems are also shown to be applicable to nonintegrable equations.

Closed orbits and the regular bound spectrum

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.

Experimental study of Lagrangian turbulence in a Stokes flow

Experimental studies are made of chaotic particle motion in a simple Stokes flow system. Surfaces-of-section that exhibit a generic mixture of regular and chaotic particle motions in good agreement with computer simulations are constructed experimentally. Deformations of line elements exhibit ‘ whorl ’ and ‘ tendril ’ structures of great complexity that can be correlated directly with the underlying particle dynamics and that agree well with numerical computations. In some cases, the laboratory studies are able to resolve dynamical features more accurately than the computer studies. Experiments demonstrating that the flows exhibit poor time reversal in régimes of chaotic particle motion are also performed.

A Cascade Theory of Drag Reduction

We describe homogeneous, isotropic, three-dimensional turbulence in a dilute solution of neutral, flexible chains. Energy flows by the usual cascade down to a scale r* such that the shear rates U(r*)/r* (U being the velocity) become equal to the relaxation rate of one coil. At small scales r, the molecules follow affinely the deformation of a local volume element. At a certain smaller scale r** the elastic stresses in the coils become comparable to the Reynolds stresses. The polymer truncates the cascade when r** becomes larger than the usual Kolmogorov limit rk. This defines a critical concentration, which depends on both polymer and flow parameters.

A unified approach to Painleve´ expansions

Abstract The Painleve test for partial differential equations developed by Weiss, Tabor and Carnevale (WTC) is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations. A simple modification of the WTC procedure used for partial differential equations enables us to determine the Lax pairs, Hirota equations and auto-Backlund transformations for ordinary differential equations, including a new Lax pair for an integrable case of the Henon-Heiles system. A detailed study of the KdV hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained. The role of the singular branches is examined in detail and important new insights obtained. In particular we find that each singular branch is simply a re-expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce. A parallel analysis is carried out for the AKNS hierarchy.

Nonlinear dynamics of filaments I. Dynamical instabilities

Abstract The Kirchhoff model provides a well-established mathematical framework to study, both computationaly and theoretically, the dynamics of thin filaments within the approximations of linear elasticity theory. The study of static solutions to these equations has a long history and the usual approach to describing their instabilities is to study the time-dependent version of the Kirchhoff model in the Euler angle frame. Here we study the linear stability of the full, time-independent, equations by introducing a new are length preserving perturbation scheme. As an application, we consider the instabilities of various stationary solutions, such as the planar ring and straight rod, subjected to twisting perturbations. This scheme gives a direct proof of the existence of dynamical instabilities and provides the selection mechanism for the shape of unstable filaments.

Biomechanical models of hyphal growth in actinomycetes

The tip growth of filamentary actinomycetes is investigated within the framework of large deformation membrane theory in which the cell wall is represented as a growing elastic membrane with geometry-dependent elastic properties. The model exhibits realistic hyphal shapes and indicates a self-similar tip growth mechanism consistent with that observed in experiments. It also demonstrates a simple mechanism for hyphal swelling and beading that is observed in the presence of a lysing agent.

The Nonlinear Dynamics of Filaments

The Kirchhoff equations provide a well-established framework tostudy the statics and dynamics of thin elastic filaments. The study ofstatic solutions to these equations has a long history and provides thebasis for many investigations, both past and present, of theconfigurations taken by filaments subject to various external forces andboundary conditions. Here we review recently developed techniquesinvolving linear and nonlinear analyses that enable one to study, insome detail, the actual dynamics of filament instabilities and thelocalized structures that can ensue. By introducing a novel arc-lengthpreserving perturbation scheme a linear stability analysis can beperformed which, in turn, leads to dispersion relations that provide theselection mechanism for the shape of an unstable filament. Thesedispersion relations provide the starting point for nonlinear analysisand the derivation of new amplitude equations which describe thefilament dynamics above the instability threshold. Here we will mainlybe concerned with the analysis of rods of circular cross-sections andsurvey the behavior of rings, rods, helices and show how these resultslead to a complete dynamical description of filament buckling.