| 书目名称 | Nonlinear Phenomena in Physics and Biology | | 编辑 | Richard H. Enns (Director),Billy L. Jones (Co-dire | | 视频video | | | 丛书名称 | NATO Science Series B: | | 图书封面 |  | | 描述 | The Advanced Study Institute (ASI) on Nonlinear Phenomena-in Physics and Biology was held at the Banff Centre, Banff, Alberta, Canada, from 17 - 29 August, 1980. The Institute was made possible through funding by the North Atlantic Treaty Organization (who sup plied the major portion of the financial aid), the National Research and Engineering Council of Canada, and Simon Fraser University. The availability of the Banff Centre was made possible through the co sponsorship (with NATO) of the ASI by the Canadian Association of Physicists. 12 invited lecturers and 82 other participants attended the Institute. Except for two lectures on nonlinear waves by Norman Zabusky, which were omitted because it was felt that they already had been exhaustively treated in the available literature, this volume contains the entire text of the invited lectures. In addition, short reports on some of the contributed talks have also been included. The rationale for the ASI and this resulting volume was that many of the hardest problems and most interesting phenomena being studied by scientists today ar.e nonlinear in nature. The nonlinear models involved often span several different disciplines, °a simp | | 出版日期 | Book 1981 | | 关键词 | Potential; animals; behavior; bifurcation; biological; biology; differential equation; dynamics; linear diff | | 版次 | 1 | | doi | https://doi.org/10.1007/978-1-4684-4106-2 | | isbn_softcover | 978-1-4684-4108-6 | | isbn_ebook | 978-1-4684-4106-2Series ISSN 0258-1221 | | issn_series | 0258-1221 | | copyright | Springer Science+Business Media New York 1981 |
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Front Matter |
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Abstract
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| 2 |
Computation and Innovation in the Nonlinear Sciences |
Norman J. Zabusky |
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Abstract
As the lead-off speaker I would like to reflect on the previous School of Nonlinear Mathematics and Physics [Zabusky, 1968] that Martin Kruskal and I organized 14 years ago. Several members of the present faculty or their teachers were there. W. Heisenberg, well-known for his pioneering contributions to quantum mechanics, opened the School with a talk “Nonlinear Problems in Physics” [Heisenberg, 1967] that I take the liberty of distilling into four aphorisms:
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Introduction to Nonlinear Waves |
Alwyn C. Scott |
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Abstract
A funny thing happened to solitary wave research over the past decade: it became respectable. No longer is it possible for all soliton buffs to meet in a small room; nor can one now read the important papers in a few weeks. The early, innocent days are gone, and (as Fig. 1 shows) soliton research output has entered a period of exponential growth with a doubling time of about 18 months. The solitary wave concept has emerged as a widely accepted paradigm for exploring and modeling the dynamics of the real world.
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Remarks on Nonlinear Evolution Equations and the Inverse Scattering Transform |
Mark J. Ablowitz |
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Abstract
In recent years there has been considerable attention devoted to a new and rapidly developing area of mathematical physics, namely the Inverse Scattering Transform (I.S.T. for short). This method has allowed us to solve certain physically interesting nonlinear evolution equations. By now there are a number of review articles [for example, see Scott et al, 1973; Miura, 1976; Ablowitz, 1978] on this subject as well as some new books [for example, see Zakharov et al, 1980; Ablowitz and Segur, to appear], all of which contain numerous references.
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The Linearity of Nonlinear Soliton Equations and the Three Wave Resonance Interaction |
D. J. Kaup |
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Abstract
The purpose of a school such as this is to provide a background of information for those interested in a given particular subject. The basic fundamentals of solitons have already been well expanded on by the previous talks, and in general, the most that I could add without excessive duplication is simply additional references [Kaup, 1977; Kaup and Newell, 1978a, b; Kaup et al, 1979], which describe my viewpoint on these basics. However, there is one basic point that still has not been emphasized here, and which I have always considered to be striking and important. That is the fact that although these systems are indeed nonlinear, their behavior so closely mocks or imitates linear systems, that one is frequently ahead if he simply forgets that it is nonlinear, and looks upon the system as being essentially linear. For example, Professor Newell [Kaup and Newell, 1978b] has demonstrated a striking representation of the general solution for q in terms of the squared eigenstates.
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Contour Dynamics: A Boundary Integral Evolutionary Method for Inviscid Incompressible Flows |
Norman J. Zabusky |
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Abstract
The method of contour dynamics, a generalization of the water- bag model, is ideally suited to study the evolution of incompressible and nondissipative fluids in two dimensions. For example, the method is applicable to the Euler equations, in homogeneous and stratified media and the equations for a “deformable” dielectric (or an ionospheric plasma cloud). In essence the “sources” of motion are singular points and/or piecewise-constant regions of “density” whose boundaries are advected with the local flow velocity. This velocity is derived from a stream-function that is obtained by solving an elliptic equation. In all cases to date the inviscid flows are n area-preserving mappings of the regions.
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Numerical Computation of Nonlinear Waves |
Bengt Fornberg |
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Abstract
Equations that can be described as wave equations arise in a large number of physical situations. Most often, they take a form which includes partial derivatives. In contrast to the case of ordinary differential equations, it tends to be true that each equation involving partial derivatives requires a special solution technique. There are of course some very general approaches, like finite differences, finite elements, spectral decomposition etc. but there are likely to be lots of details that differ from case to case. Program packets which more or less automatically handle a wide range of different problems have so far had little or no impact on wave calculations and I do not think any major changes in that respect are in sight.
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Bifurcations, Fluctuations and Dissipative Structures |
G. Nicolis |
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Abstract
The purpose of the present lectures is to discuss the emergence of structures from dissipative processes in macroscopic systems. This class of phenomena, generally studied by means of nonequilibrium thermodynamics, is quite different from solitons and other structures appearing in nonlinear Hamiltonian systems, which were covered by many lecturers at this school.
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Chemical Oscillations |
Louis N. Howard |
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Abstract
The general point of view of my lectures will be that of the applied mathematician interested in problem-solving. Some general techniques, such as bifurcation methods and various aspects of singular perturbations, will be discussed, but mostly in the context of specific examples related to chemical oscillations and waves. The more general background can be found in the paper by Professor Nicolis [1973].
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Models in Neurobiology |
John Rinzel |
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Abstract
The most familiar mode of neural communication is electrical and synaptic signaling by individual nerve cells. Here we shall consider a few stereotypical phenomena of such signaling. A primary observable is the potential V (deviation from rest) across the cell membrane which responds to applied stimulating current I. and to changes in membrane permeability to the various ion species. These permeability changes also usually depend on V and, for excitable membrane, result in the generation of the nerve impulse. We will describe and present results for models (some quantitative but others more qualitative) of excitable membrane behavior.
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Nonlinear Waves in Neuronal Cortical Structures |
Robert M. Miura |
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Abstract
This quote, from Aris’ [1978] interesting book on mathematical modelling, gives a basic description of a mathematical model. As Aris points out, mathematical modelling largely remains an art and it is difficult to communicate modelling skills to the uninitiated.
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Bifurcations in Insect Morphogenesis I |
Stuart A. Kauffman |
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Abstract
The past decade has witnessed a renewal of deep interest in the problem of pattern formation in developmental biology. In large measure, the resurgence of enthusiasm coincides with Wolpert’s reformulation of this fundamental problem in terms of the concept of positional information [Wolpert, 1969, 1971]. Novel features of Wolpert’s theory have led both to proposals of alternative “coordinate systems” supplying positional information, and to a rich variety of experiments designed to test the alternatives. The present article provides a brief review of the major alternatives which leads to the formulation of a new class of models based on the unrecognized, but common capacity of biochemical reaction-diffusion systems to generate transverse (cross) gradients in growing asymmetrical tissue domains.
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Bifurcations in Insect Morphogenesis II |
Stuart A. Kauffman |
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Abstract
Two of the most fundamental problems in developmental biology are the manner in which cells in different regions of an embryo come to adopt different developmental programs, and the relation between such different programs. The purpose of this lecture is to indicate how bifurcations in reaction-diffusion systems may offer new insights into these problems. The discussion will center on the fruit fly, .
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Selection and Evolution in Molecular Systems |
Peter Schuster |
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Abstract
The capability of self-replication is one of the unique features of living beings. Outside biology we encounter this property only occasionally. Examples of such exceptions are autocatalytic reactions in chemistry which play an important role in the formation of chemical oscillations and waves as well as in vapor phase combustion. In this class of chemical reactions the numbers of small molecules, radicals or ions double or, in general, multiply. There are also examples of self-replication in the intellectual world like the highly sophisticated programs of learning computers or some complicated mathematical games. In this contribution we shall summarize a series of studies on molecular self-replication by means of chemical kinetics. This work has been initiated in 1971 by the comprehensive paper by Eigen [1971]. Later on, several publications continued work along these lines [Eigen and Winkler-Oswatitsch, 1975, 1980; Eigen, 1976; Eigen and Schuster, 1977, 1978 a, b; Schuster, 1980; Schuster and Sigmund, 1980]. Finally, we shall apply the concept of self-organization through self-replication to two vividly discussed problems in biological evolution, to the emergence of life from pre
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Escape from Domains of Attraction for Systems Perturbed by Noise |
Donald Ludwig |
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Abstract
Deterministic theories have been remarkably successful in interpreting and explaining the world, although more precise formulations will always involve random effects. An outstanding example is classical mechanics, whose utility is hardly impaired by the existence of quantum effects. Intuitively, we may think of the deterministic trajectories as being smeared out by the random effects. Even if the smearing is large, the qualitative behavior may still be captured by the deterministic result.
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Abstract
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Error Propagation in Translation and its Relevance to the Nucleation of Life |
Geoffrey W. Hoffmann |
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Abstract
Some of the enzymes produced by the translation of information stored in nucleic acid sequences participate as adaptors in the translation processes itself. The more errors there are in the adaptors doing the translation, the lower the quality (and therefore the lower the accuracy in action) of the adaptors that are produced. Conceivably the level of errors could escalate to give an “error catastrophe” [Orgel, 1963]. A mathematical model of this process has been formulated, based on a simple “all-or-none” classification of the amino acid residues of a typical adaptor residue [Hoffmann, 1974, 1975]. The model permits an estimate to be made of the threshold level of specificity, required by the components of a workable primitive translation apparatus. In the model, a certain number of the amino acid residues of an adaptor are presumed to be critical for the correct folding of the adaptor, (such that if an error is made there, all activity is lost), and a certain number are presumed to be critical for the specificity of adaptors (such that an error made in those sites results in an adaptor that makes correct each of the possible incorrect assignments with equal rates). The remaining a
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Pseudopotentials and Symmetries for Generalized Nonlinear Schrödinger Equations |
J. Harnad,P. Winternitz |
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Abstract
Applying the method of differential ideals and prolongations to equations of the type . where f(Z,Z*) is any smooth function, we arrive at eight types of equations admitting “pseudopotentials” (in the sense of Wahlquist and Estabrook [1976]). For five of these, the integrability conditions are equivalent to conservation laws. For the remaining three, including the usual cubically non-linear Schrodinger equation, there are non-trivial pseudo-potentials, which may be interpreted as defining Bácklund transformations or linear scattering equations. For the two cases other than the standard one, however, there is no parameter identifiable as an eigenvalue and no Lie symmetry (other than translation invariance) to generate such a parameter. For the standard case, we show that the real and imaginary parts of the parameter occuring in the Backlund transformation (or scattering equation) may be generated by composing a given transformation with the 2-parameter Lie symmetry group consisting of Galilean boosts and dilations.
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Superposition Principles for Nonlinear Differential Equations |
R. L. Anderson,J. Harnad,P. Winternitz |
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Abstract
Consider a system of n first order quasilinear ordinary differential equations ..
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On Some Nonlinear Schrödinger Equations |
H. Lange |
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Abstract
The equation which is studied in this note is the following nonlinear Schrodinger equation . where . and (with r = |x|) ..
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发表于 2025-3-21 18:03:38
