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Nonlinearity and Discreteness: Solitons in Lattices

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Emerging Frontiers in Nonlinear Science

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 32))

Abstract

An overview is given of basic models combining discreteness in their linear parts (i.e., the models are built as dynamical lattices) and nonlinearity acting at sites of the lattices or between the sites. The considered systems include the Toda and Frenkel–Kontorova lattices (along with their dissipative versions), as well as equations of the discrete nonlinear Schrödinger and Ablowitz–Ladik types, and their combination in the form of the Salerno model. The interplay of discreteness and nonlinearity gives rise to a variety of states, most important ones being discrete solitons. Basic results for 1D and 2D discrete solitons are collected in the review, including 2D solitons with embedded vorticity, and some results concerning mobility of discrete solitons. Main experimental findings are overviewed too. Models of the semi-discrete type, and basic results for solitons supported by them, are also considered, in a brief form. Perspectives for the development of topics covered in the review are discussed throughout the text.

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Notes

  1. 1.

    “Bad” implies that (4.11) gives rise to an unstable dispersion relation.

  2. 2.

    For velocities given by (4.19) with odd integer \(1+2N\) replaced by an even one, 2N, with \(N=1,2,...,\) the transfer of energy from the drive to the moving soliton averages to zero.

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Acknowledgements

I appreciate the invitation of the Editors of the volume Emerging Frontiers in Nonlinear Science to submit this Chapter. One of the Editors, Prof. J. Cuevas-Maraver, has provided a great deal of help in the course of preparing the manuscript. I would like to thank colleagues in collaboration with whom I have been working, in the course of a long time, on various topics related to the review: G.E. Astrakharchik, P. Beličev, A.R. Bishop, R. Blit, L.L. Bonilla, R. Carretero-González, Zhaopin Chen, Zhigang Chen, C. Chong, M. Cirillo, J. Cuevas-Maraver, J. D’Ambroise, F. K. Diakonos, S. V. Dmitriev, N. Dror, O. Dutta, A. Eckhard, L. M. Floría, D. J. Frantzeskakis, S. Fu, G. Gligorić, J. Gómez-Gardeñes, N. Grønbech-Jensen, L. Hadžievski, D. Herring, J. Hietarinta, P. Hauke, N.V. Hung, Y.V. Kartashov, T. Kapitula, D.J. Kaup, P.G. Kevrekidis, V.V. Konotop, T. Kuusela, M. Lewenstein, Y. Li, A. Maluckov, R.S. Osgood, N.C. Panoiu, I.E. Papachalarampus, J. Petrović, M.A. Porter, K.Ø. Rasmussen, H. Sakaguchi, L. Torner, M. Trippenbach, V. Ustinov, R.A. Van Gorder, M.I. Weinstein, and K. Zegadlo.

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Authors and Affiliations

  1. Department of Physical Electronics, Faculty of Engineering, School of Electrical Engineering, and Center for Light-Matter Interaction, Tel Aviv University, 69978, Tel Aviv, Israel

    Boris A. Malomed

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  1. Boris A. Malomed
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Correspondence to Boris A. Malomed .

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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 Fisica No Lineal, Departamento de Fisica Aplicada I, Escuela Politécnica Superior, Universidad de Sevilla, Seville, Spain

    Jesús Cuevas-Maraver

  3. Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA

    Avadh Saxena

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Malomed, B.A. (2020). Nonlinearity and Discreteness: Solitons in Lattices. In: Kevrekidis, P., Cuevas-Maraver, J., Saxena, A. (eds) Emerging Frontiers in Nonlinear Science. Nonlinear Systems and Complexity, vol 32. Springer, Cham. https://doi.org/10.1007/978-3-030-44992-6_4

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