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Multi-Lump Structures in the Kadomtsev–Petviashvili Equation

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Advances in Dynamics, Patterns, Cognition

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 20))

Abstract

The lumps are specific types of solitary waves completely localised in the two-dimensional space. They represent exact solutions of the Kadomtsev–Petviashvili equation with the positive dispersion. A summary of up-to-date knowledge on lump solutions is presented. It is shown that they can form stationary bound states, and their interaction is very nontrivial.

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Notes

  1. 1.

    At that time I was involved into one of the joint researches with him [11].

  2. 2.

    MIR was actually the first to invent and realise jointly with S. V. Kiyashko and A. S. Pikovsky in 1980, the simplest example of a strange attractor [19], now known as the Chua attractor [10].

  3. 3.

    MIR wrote a section about this in the classical book by L.D. Landau and E.M. Lifshitz, Hydrodynamics [22].

  4. 4.

    A rigorous substantiation of the convergence of the Petviashvili method was published in [31].

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Acknowledgements

The author is thankful to N.B. Krivatkina for her help with editing the paper.

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Authors and Affiliations

  1. University of Southern Queensland, West Street, Toowoomba, QLD, 4350, Australia

    Yury Stepanyants

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  1. Yury Stepanyants
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Correspondence to Yury Stepanyants .

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Editors and Affiliations

  1. Pennsylvania State University, University Park, Pennsylvania, USA

    Igor S. Aranson

  2. Institute for Physics and Astronomy, University of Potsdam, Potsdam, Germany

    Arkady Pikovsky

  3. BioCircuits Institute, University of California San Diego, La Jolla, California, USA

    Nikolai F. Rulkov

  4. BioCircuits Institute, University of California San Diego, La Jolla, California, USA

    Lev S. Tsimring

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Stepanyants, Y. (2017). Multi-Lump Structures in the Kadomtsev–Petviashvili Equation. In: Aranson, I., Pikovsky, A., Rulkov, N., Tsimring, L. (eds) Advances in Dynamics, Patterns, Cognition. Nonlinear Systems and Complexity, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-319-53673-6_19

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  • DOI: https://doi.org/10.1007/978-3-319-53673-6_19

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