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Historical Overview of the \(\phi ^4\) Model

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A Dynamical Perspective on the ɸ4 Model

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 26))

Abstract

We provide an historical overview the \(\phi ^4\) model, placing it in the broad context of general KG theories and reviewing its applications to physics. While we will mention briefly other variants of the KG theory, we will focus chiefly on the history of the one-space, one-time dimensional [(1+1)D] degenerate minimum \(\phi ^4\) theory that is the central topic of this book. We will also compare this theory to other nonlinear Klein–Gordon equations, in particular, to the celebrated sine-Gordon theory. We review in some detail the history of the interrelated dynamical problems of kink-antikink scattering, contrasting \(\phi ^4\) and other non-integrable models with sine-Gordon and the search for a possible “breather” solution to \(\phi ^4\) in the continuum limit. Our discussion is intended to set the stage for more detailed expositions in the later chapters.

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Notes

  1. 1.

    This claim is made in a footnote on p. 259 in [44].

  2. 2.

    This is arguably the first introduction of a collective coordinate approach in studies of the \(\phi ^4\) theory. It was followed closely by the more detailed work of Sugiyama [60]. See the section of “Collective Coordinates” below and for a thorough discussion Chaps. 3 and 4 of this book.

  3. 3.

    For purposes of our later discussion and links to other chapters in this volume, we note that the location of the kink \(x_0\) and the amplitude of the shape mode can be taken as “collective coordinates” and used to reduce the PDE to coupled ODEs [60, 63, 64] (see also Chap. 3) or coupled maps [68, 69] (see also Chap. 4).

  4. 4.

    The binding energy is amount by which the kinks are trapped in the their common potential well. See [61] for details.

  5. 5.

    There was however a subtlety in this case. Reference [56] noted that there were “quasi-resonances” in the K\(\bar{\text {K}}\) related to excitation in the continuum spectrum around the kinks, and a later study by Quintero and Kevrekidis [71] showed how these “localized” phonons could in fact produce these “quasi-resonances”. Interested readers should refer to these two articles for details.

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Acknowledgements

It is a pleasure to thank my collaborators in the work I have described here —Michel Peyrard, Jonathan Schonfeld, Pasquale Sodano, and Chuck Wingate— and to recall the great times we shared in writing the original papers. I thank Michel Peyrard for regenerating some of the old figures. I am also grateful to Panos Kevrekidis and Jesús Cuevas-Maraver for helpful comments and assistance on the manuscript.

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  1. Department of Physics, Boston University, 590 Commonwealth Ave, Boston, MA, USA

    David K. Campbell

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Correspondence to David K. Campbell .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, USA

    Panayotis G. Kevrekidis

  2. Grupo de Física No Lineal, Universidad de Sevilla, Departamento de Física Aplicada I, Escuela Politécnica Superior, Sevilla, Spain

    Jesús Cuevas-Maraver

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Campbell, D.K. (2019). Historical Overview of the \(\phi ^4\) Model. In: Kevrekidis, P., Cuevas-Maraver, J. (eds) A Dynamical Perspective on the ɸ4 Model. Nonlinear Systems and Complexity, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-030-11839-6_1

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