A. C. Newell

• Professor at University of Arizona

Carcinogenesis is an evolutionary process whereby cells accumulate multiple mutations. Besides the 'driver mutations' that cause the disease, cells also accumulate a number of other mutations with seemingly no direct role in this evolutionary process. They are called passenger mutations. While it has been argued that passenger mutations render tumo...

Carcinogenesis is an evolutionary process whereby cells accumulate multiple mutations. Besides the “driver mutations” that cause the disease, cells also accumulate a number of other mutations with seemingly no direct role in this evolutionary process. They are called passenger mutations. While it has been argued that passenger mutations render tumo...

We connect the theories of the deformation of elastic surfaces and phase surfaces arising in the description of almost periodic patterns. In particular, we show striking parallels between expansions for the energy of elastic surfaces in powers of the thickness $h$ and the free energy averaged over a period of an almost periodic pattern expanded in...

The work we describe addresses the process of whitecapping. We first argue that, when the winds are strong enough, the ocean surface must develop an alternative means to dissipate energy when its flux from large to small scales becomes too large. We then show that the resulting Phillips' spectrum, which holds at small or meter length scales, is dom...

We introduce short pulse evolution equation (SPEE), first derived in a non-optical context in the 80s, the universal equation describing the propagation of short pulses in media which have weak dispersion in the propagation direction. We show how it connects with the first canonical examples of nonlinear wave propagation, the Korteweg–de Vries and...

This is a review article with a point of view. We summarize the long history of the subject and recent advances and suggest that almost all features of the architecture of shoot apical meristems can be captured by pattern-forming systems which model the biochemistry and biophysics of those regions on plants.

The following questions are addressed: How did the symmetries which lead to fractional invariants arise? Can one start with systems with much simpler symmetries, say space translation and rotation, and, by stressing such systems, give rise via phase transitions to objects with natural fractional invariants? Such systems are manifold in nature. They...

Using a partial differential equation model derived from the ideas of the Meyerowitz and Traas groups on the role of the growth hormone auxin and those of Green and his group on the role compressive stresses can play in plants, we demonstrate how all features of spiral phyllotaxis can be recovered by the passage of a pushed pattern forming front. T...

The instabilities of nonlinear waves with a square-root dispersion ω∼|k| are studied. We present a new type of instability that affects wavelengths of the order of the carrier wave. This instability can initiate the formation of collapses and of narrow pulses.

We demonstrate that the pattern forming partial differential equation derived from the auxin distribution model proposed by Meyerowitz, Traas and others gives rise to all spiral phyllotaxis properties observed on plants. We show how the advancing pushed pattern front chooses spiral families enumerated by Fibonacci sequences with all attendant self...

Ultrashort pulses can exhibit two distinctive types of singularity: self-focusing collapse and self-steepening shock. We examine various ultrashort pulse propagation models and their relative effectiveness in explaining these phenomena. In particular, the modified Kadomtsev-Petviashvilli equation of type 1 (MKP1) is examined in some detail. We show...

We report a surprising new result for wave turbulence which may have broader ramifications for general turbulence theories. Spatial homogeneity, the symmetry property that all statistical moments are functions only of the relative geometry of any configuration of points, can be spontaneously broken by the instability of the finite flux Kolmogorov-Z...

and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infini...

Disclinations, concave and convex, are the canonical point defects of two-dimensional planar patterns in systems with translational and rotational symmetries. From these, all other point defects (vortices, dislocations, targets, saddles and handles) can be built. Moreover, handles, coupled concave–convex disclination pairs arise as instabilities, s...

We examine the two types of singular behaviors of ultrashort pulses in a nonlinear medium, pulse steepening if the weak longitudinal dispersion is normal and collapse if it is anomalous. Connections with analogous behaviors of wave packets of almost monochromatic waves in strongly dispersive media are discussed.

Pattern patterns, or phyllotaxis, the arrangements of phylla (flowers, leaves, bracts, florets) in the neighborhood of growth tips, have intrigued natural scientists for over four hundred years. Prominent amongst the observed features is the fact that phylla lie on families of alternately oriented spirals and that the numbers in these families belo...

In this article, we state and review the premises on which a successful asymptotic closure of the moment equations of wave turbulence is based, describe how and why this closure obtains, and examine the nature of solutions of the kinetic equation. We discuss obstacles that limit the theory's validity and suggest how the theory might then be modifie...

Wave turbulence concerns itself with the non-equilibrium statistical mechanics of ensembles of nonlinearly interacting dispersive waves in the presence of external forcing and damping. In the limit of weakly interacting waves, one may consistently derive a kinetic equation describing the evolution of the wave spectrum. The stationary solution of th...

We present a systematic study of the dynamical scaling process leading to the establishment of the Kolmogorov-Zakharov (KZ) spectrum in weak three-wave turbulence. In the finite-capacity case, in which the transient spectrum reaches infinite frequency in finite time, the dynamical scaling exponent is anomalous in the sense that it cannot be determi...

We propose a new mechanism for turbulent transport in systems which support radiating nonlinear solitary wave packets or pulses. The direct energy cascade is provided by adiabatically evolving pulses, whose widths and carrier wavelengths decrease. The inverse cascade is due to the excitation of radiation. The spectrum is steeper than the Kolmogorov...

Weak turbulence of shear-Alfvén waves is considered in the limit of strongly anisotropic pulsations that are elongated along the external magnetic field. The kinetic equation thus derived agrees with the Galtier et al. formulation of the full three-dimensional helical case when taking the proper limit. This new approach allows for significant simpl...

It has been known for centuries that the configurations of phylla (flowers, bracts) on plants have special properties. The difference in angular positions of successive phylla is locally constant (called the divergence angle) and is close to the golden angle; the radial positions are related by the locally constant plastichrone ratio. The phylla th...

Phyllotaxis, the arrangement of a plant's phylla (flowers, bracts, stickers) near its shoot apical meristem (SAM), has intrigued natural scientists for centuries. Even today, the reasons for the observed patterns and their special properties, the physical and chemical mechanisms which give rise to strikingly similar configurations in a wide variety...

We suggest the generalized Phillips' spectrum, which we define as that spectrum for which the statistical properties of wave turbulence inherit the symmetries of the original governing equations, is, in many circumstances, the spectrum which obtains in those regions of wavenumber space in which the Kolmogorov–Zakharov (KZ) spectra are no longer val...

Current theories and models of the formation of phyllotactic patterns at plant apical meristems center on either transport of the growth hormone auxin or the mechanical buckling of the plant tunica. By deriving a continuum approximation of an existing discrete biochemical model and comparing it with a mechanical model, we show that the model partia...

We wish to transmit messages to and from a hypersonic vehicle around which a plasma sheath has formed. For long distance transmission, the signal carrying these messages must be necessarily low frequency, typically 2 GHz, to which the plasma sheath is opaque. The idea is to use the plasma properties to make the plasma sheath appear transparent.

We wish to transmit messages to and from a hypersonic vehicle around which a plasma sheath has formed. For long distance transmission, the signal carrying these messages must be necessarily low frequency, typically 2 GHz, to which the plasma sheath is opaque. The idea is to use the plasma properties to make the plasma sheath appear transparent. Num...

DOI:https://doi.org/10.1103/PhysRevE.75.019901

Without Abstract

We study conical refraction in crystals where both diffraction and nonlinearity are present. We develop a new set of evolution equations. We find that nonlinearity induces a modulational instability when it is defocussing as well as focussing. We also examine the evolution of incident beams which contain analytic singularities, and in particular op...

We prove the finite-time collapse of a system of N classical fields, which are described by N coupled nonlinear Schrödinger equations. We derive the conditions under which all of the fields experiences this finite-time collapse. Finally, for two-dimensional systems, we derive constraints on the number of particles associated with each field that ar...

We have studied the propagation of non-linear waves across a random medium, using the nonlinear Schrdinger equation with a random potential as a model. By simulating a scattering experiment, we show that non-linearity leads to an improvement of the transmission only when it contributes to create pulses. The propagation properties of these pulses ca...

We first present the phase diffusion and mean drift equation which describe convective patterns in large aspect ratio containers and for arbitrary Rayleigh and Prandtl numbers. Some applications are presented such as the prediction of the selected wavenumber or the instability of foci. We propose in a second step a regularized form of the phase dif...

The method, developed by Gardner, Greene, Kruskal and Miura, for solving nonlinear evolution equations is looked at from three perspectives; (i) as a nonlinear Fourier transform, (ii) as a completely integrable Hamiltonian system, and (iii) from the operator theoretic standpoint. The interrelation between the three points of view is discussed and,...

The universality of many features of plant patterns and phyllotaxis has mystified and intrigued natural scientists for at least four hundred years. It is remarkable that, to date, there is no widely accepted theory to explain the observations. We hope that the ideas explained below lead towards increased understanding

We demonstrate how phyllotaxis (the arrangement of leaves on plants) and the ribbed, hexagonal, or parallelogram planforms on plants can be understood as the energy-minimizing buckling pattern of a compressed sheet (the plant's tunica) on an elastic foundation. The key idea is that the elastic energy is minimized by configurations consisting of spe...

Fingerprints (epidermal ridges) have been used as a means of identifications for more than 2000 years. They have also been extensively studied scientifically by anthropologists and biologists. However, despite all the empirical and experimental knowledge, no widely accepted explanation for the development of epidermal ridges on fingers, palms and s...

In this note, we introduce the equations for the order parameters describing the buckling of thin, elastic sheets. What is new is the realization that mean creep, namely in-plane displacements, are soft (Goldstone) modes which can be driven by variations in the pattern intensity and which, in turn, affect how the buckling pattern develops. The orde...

We define, for wave turbulence, probability density functions rho (pdfs) on a suitably chosen phase space. We derive the Liouville equation for their evolution and identify their long time behaviours corresponding to equipartition and finite flux Kolmogorov-Zakharov ( KZ) spectra. We demonstrate that, even in nonisolated systems, entropy production...

The uniqueness of fingerprints (epidermal ridges) has been recognized for over two thousand years. They have been studied scientifically for more than two hundred years. Yet, in spite of the accumulation of a wealth of empirical and experimental knowledge, no widely accepted explanation for the development of epidermal ridges on fingers, palms and...

We demonstrate how phyllotaxis (the arrangement of leaves on plants) and the deformation configurations seen on plant surfaces may be understood as the energy-minimizing buckling pattern of a compressed shell (the plant's tunica) on an elastic foundation. The key new idea is that the strain energy is minimized by configurations consisting of specia...

The turbulent flows that transfer energy from a stirring range at large scales to the dissipation range at small scales were investigated. It was shown that a single cascade of weakly interacting waves that transport particles from a source to a link leads to a steady loss of energy in the system. It was observed that coherent structures are necess...

Intermittent high-amplitude structures emerge in a damped and driven discrete nonlinear Schroedinger equation whose solutions transport both energy and particles from sources to sinks. These coherent structures are necessary for any solution that has statistically stationary transport properties.

In the study of weakly turbulent wave systems possessing incomplete self-similarity, it is possible to use dimensional arguments to derive the scaling exponents of the Kolmogorov–Zakharov spectra, provided the order of the resonant wave interactions responsible for nonlinear energy transfer is known. Furthermore, one can easily derive conditions fo...

We study the irreversible dynamics of nonlinear, nonintegrable Hamiltonian oscillator chains approaching their statistical asymptotic states. In systems constrained by more than one conserved quantity, the partitioning of the conserved quantities leads naturally to localized and coherent structures. If the phase space is compact, the final equilibr...

The evolution of the Kolmogorov–Zakharov (K–Z) spectrum of weak turbulence is studied in the limit of strongly local interactions where the usual kinetic equation, describing the time evolution of the spectral wave-action density, can be approximated by a PDE. If the wave action is initially compactly supported in frequency space, it is then redist...

The Cross–Newell phase diffusion equation , and its regularization describe patterns and defects far from onset in large aspect ratio systems with translational and rotational symmetry. In this paper we show how director field solutions of this equation can be used to describe features of global patterns. The ideas are illustrated in the context of...

We study the structure functions of wave turbulence at small separations. We show that the criteria for breakdown obtained previously by examining the uniform validity of the asymptotic closure govern how close wave turbulence stays to joint Gaussianity. A new result in the case of small separations in that the system behavior is organized by a spe...

We study the structure functions and cumulants of wave turbulence at small and large separations, respectively. We show that the criteria for breakdown obtained previously by examining the uniform validity of the asymptotic closure govern how close wave turbulence stays to joint Gaussianity. The results for large separations are new and include the...

The Cross-Newell phase diffusion equation т(k)ΘT = – ▽ · kB(k), k = ▽ Ω, |k| = k, and its regularization describe patterns and defects far from onset in large aspect ratio systems with translational and rotational symmetry. In this paper we show how director field solutions of this equation can be used to describe features of global patterns. The i...

In the early sixties, it was established that the stochastic initial value problem for weakly coupled wave systems has a natural asymptotic closure induced by the dispersive properties of the waves and the large separation of linear and nonlinear time scales. One is thereby led to kinetic equations for the redistribution of spectral densities via t...

A 2D numerical investigation is presented of shock wave propagation into a gas whose density is modulated in the transverse direction across the width of a shock tube. These density modulations represent temperature distributions in which low density corresponds to high temperature gas and high density corresponds to low temperature gas. This work...

The asymptotic expansions for (1) the slow changes in particle number/energy density; namely, the kinetic equation, (2) frequency renormalization; and (3) the Nth-order structure functions for wave turbulence systems are almost always nonuniform at either small or large length scales. The manifestation of this nonuniformity is fully nonlinear behav...

Many studies have shown that nonintegrable systems with modulational instabilities constrained by more than one conservation law exhibit universal long time behavior involving large coherent structures in a sea of small fluctuations. We show how this behavior can be explained in detail by simple thermodynamic arguments.

In the early 1960s, it was established that the stochastic initial value problem for weakly coupled wave systems has a natural asymptotic closure induced by the dispersive properties of the waves and the large separation of linear and nonlinear time scales. One is thereby led to kinetic equations for the redistribution of spectral densities via thr...

We consider an example of strongly non-local interaction in incompressible magnetohydrodynamic (MHD) turbulence which corresponds to the case where the Alfvén waves travelling in the opposite directions have essentially different charac- teristic wavelengths. We use two approaches to the dynamics of turbulent Alfvénic wavepackets: the first is a ge...

We describe the fundamental differences between weak (wave) turbulence in incompressible and weakly compressible MHD at the level of three-wave interactions. The main difference is in the structure of the resonant manifolds and the mechanisms of redistribution of spectral densities along the applied magnetic field B ₀ . Similar to pure a...

Starting from the Maxwell-Bloch equations for a 3D ring-cavity laser, we analyze stability of the nonlasing state and demonstrate that, at the instability threshold, the wave vectors of the critical perturbations belong to a paraboloid in the 3D space. Next, we derive a system of nonlinear evolution equations above the threshold. The nonlinearity i...

This Letter demonstrates that the kinetic equations for wave turbulence, the long time statistical behavior of a sea of weakly coupled, dispersive waves, will almost always develop solutions for which the theory fails due to strongly nonlinear and intermittent events either at small or large scales.

. A simple model describing the interaction between flow and bed sediment in a straight channel is considered. The model allows for a uniform current over a flat bottom, and this solution becomes unstable when the width-to-depth ratio exceeds some finite critical value. Linear theory gives a narrow spectrum of perturbations which grow at near-criti...

. The phenomenon of self-induced transparency (SIT) is reinterpreted in the context of competition between randomness, non-linearity and dispersion, and furthermore the problem is recast to show that it is isomorphic to a problem of the non-linear Schroedinger (NLS) type with a random potential in which the randomness is manifested spatially. It is...

Sand banks and sand waves are two types of sand structures that are commonly observed on an off-shore sea bed. We describe the formation of these features using the equations of the fluid motion coupled with the mass conservation law for the sediment transport. The bottom features are a result of an instability due to tide–bottom interactions. Ther...

We derive a weak turbulence formalism for incompressible magnetohydrodynamics. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. The kinetic equations conserve energy in all wavevector planes normal to the applied magnetic field B0ê[parallel R: parallel]. Numerically and analytically, w...

We propose and illustrate in the context of the semiconductor laser that, in nonequilibrium fermionic systems with sources and sinks, the family of finite flux stationary solutions of the quantum Boltzmann equation is central and more important then the zero flux Fermi-Dirac spectrum. We present the quantum analog of the finite flux Kolmogorov spec...

The phenomenon of self-induced transparency (SIT) is reinterpreted in the context of competition between randomness, nonlinearity and dispersion, and furthermore the problem is recast to show that it is isomorphic to a problem of the nonlinear Schroedinger (NLS) type with a random potential in which the randomness is manifested spatially. It is sho...

The authors suggest and develop a method for following the dynamics of systems whose long-time behaviour is confined to an attractor or invariant manifold A of potentially large dimension. The idea is to embed A in a set of local coverings. The dynamics of the phase point P on A in each local ball is then approximated by the dynamics of its project...

The authors investigate a modification of the Weiss-Tabor-Carnevale procedure that enables one to obtain Lax pairs and Backlund transformations for systems of ordinary differential equations. This method can yield both auto-Backlund transformations and, where necessary, Backlund transformations between different equations. In the latter case they i...

We derive a weak turbulence formalism for incompressible MHD. Three-wave interactions leadto a system of kinetic equations for the spectral densities of energy and helicity. The kinetic equations conserve energy in all wavevector planes normal to the appliedmagnetic field B 0 ê∥. Numerically andanalytically, we find energy spectra E ± ∼ k ⊥n ±, suc...

Self Organised Criticality, Emergent Behaviour in Physical and Biological Systems. By H. J. Jensen. Cambridge University Press, 1998. 153 pp. ISBN 0521 48371, £18.95. - - Volume 377 - A. C. Newell

In this paper, recent results on the behavior of roll patterns in a class of problems typified by high Prandtl number convection are presented. A key finding is that the Gaussian curvature of the “crumpled” phase surface, which consists of patches with an almost constant wave number, line defects on which most of the free energy is stored and point...

Based on a model Hamiltonian appropriate for the description of fermionic systems such as semiconductor lasers, we describe a natural asymptotic closure of the BBGKY hierarchy in complete analogy with that derived for classical weak turbulence. The main features of the interaction Hamiltonian are the inclusion of full Fermi statistics containing Pa...

Patterns with a local roll structure arise in many diverse physical systems which have little in common at the microscopic level. In this paper we construct an algorithm based on the wavelet transform that can be used as a diagnostic tool to extract from such patterns macroscopic information like the local director field, the local amplitude away f...

We suggest that the transport properties and dissipation rates of a wide class of turbulent flows are determined by the random occurrence of coherent events which correspond to certain orbits which we call homoclinic excursions in the high dimensional strange attractor. Homoclinic excursions are trajectories in the noncompact phase space that are a...

Complex pattern formation or destruction is commonly observed in spatially extended, continuous, dissipative systems, such as convection in ordinary and binary mixture fluids and in liquid crystals, Taylor-Coullet flow, chemical reaction, and directional solidification, etc. [NPL]. Rayleigh-Benard convection has been a canonical system in developin...

An introductory review of pattern formation in extended dissipative systems is presented. Examples from many areas of physics are introduced, and the mathematical analysis of the patterns formed by these systems is outlined, for patterns near and far from onset. The wavelet transform is introduced as a useful tool for the extraction of order parame...

In this article, we make a prima facie case that there could be distinct advantages to exploiting a new class of finite flux equilibrium solutions of the Quantum Boltzmann equation in semiconductor lasers. Comment: lvov.tex - plain tex file lvov_fig1.ps and lvov_fig2.ps plain ps fi les to be attached to paper - do not called from lvov.tex

The main goal of the research was to perform a systematic investigation and evaluation of the utility of envelope equations. In particular, the nonlinear Schroedinger equation (NLS) and its generalizations, for modeling self-focusing (SF) collapse phenomena in transparent dielectric media. The work was successfully carried to completion. We (1) dev...

We develop expressions for the nonlinear wave damping and frequency correction of a field of random, spatially homogeneous, acoustic waves. The implications for the nature of the equilibrium spectral energy distribution are discussed. S1063-651X9702606-8 PACS numbers: 47.27.i

A comprehensive model is developed for focused pulse propagation in water. The model incorporates self-focusing, group velocity dispersion, and laser-induced breakdown in which an electron plasma is generated via cascade and multiphoton ionization processes. The laser-induced breakdown is studied first without considering self-focusing to give a br...

We show that defects are weak solutions of the phase diffusion equation for the macroscopic order parameter for natural patterns. Further, by exploring a new class of nontrivial solutions for which the graph of the phase function has vanishing Gaussian curvature (in 3D, all sectional curvatures) excetp at points, we are able to derive explicit expr...

As part of a research program to understand and model eye damage produced by exposure to subnanosecond laser pulses, an effort is currently being made to model and analyze ultrashort pulse propagation from the cornea to the retina. Both analytical models and numerical simulations are being used to analyze the effects of self-focusing, laser-induced...

Labyrinthic patterns are observed both in systems where the uniform states are metastable, as a result of a front instability, and in systems displaying a cellular instability, when the band of excited Fourier modes is wide enough to support resonant interactions between modes lying on different shells. We show that the phase formalism is a suitabl...

The resonant absorption of a short pulse of electromagnetic radiation in a plasma slab is considered in order to examine the effectiveness of using short pulses for information transfer through the plasma sheaths surrounding reentry vehicles. The amplitudes of the Raman scattering (RS) on the Langmuir oscillations associated with resonant absorptio...

We explore parallels between Whitham theory (nonlinear geometrical optics) applied to gradient systems, such as high Prandtl number pattern forming convection layers, and Hamiltonian systems, such as superfluids at zero temperature and thin elastic shells. In particular, we discuss certain universal features such as the canonical nature of the aver...

We present numerical studies of nonlinear propagation for picosecond pulses focused in water. Depending on the pulse duration and focusing conditions, for some input powers self-focusing may precede laser-induced breakdown and vice versa. We derive a criterion that predicts the relative roles of laser-induced breakdown and self-focusing.

Complex order parameter descriptions of large aspect ratio, single longitudinal mode, two-level lasers with flat end reflectors, valid near onset of lasing and for small detunings of the laser from the peak gain, are given in terms of a complex Swift-Hohenberg equation for Class A and C lasers and by a complex Swift-Hohenberg equation coupled to a...

The resonant transformation of a Gaussian pulse of electromagnetic radiation into a Langmuir wave in an inhomogeneous plasma is studied. It is shown that if the pulse duration is smaller than the Langmuir wave (plasmon) lifetime, then substantial changes in the Langmuir field profile occur. Namely, the width of this profile becomes greater and its...