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Nonlinearity and Topology

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Emerging Frontiers in Nonlinear Science

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

Abstract

The interplay of nonlinearity and topology results in many novel and emergent properties across a number of physical systems such as chiral magnets, nematic liquid crystals, Bose–Einstein condensates, photonics, high energy physics, etc. It also results in a wide variety of topological defects such as solitons, vortices, skyrmions, merons, hopfions, monopoles to name just a few. The interaction among and collision of these nontrivial defects itself constitute topics of significant interest. Curvature and underlying geometry also affect the shape, interaction and behavior of these defects. Such properties can be studied using techniques such as, e.g. the Bogomolnyi decomposition. Some applications of this interplay, e.g. in nonreciprocal photonics as well as topological materials such as Dirac and Weyl semimetals, are also elucidated.

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Acknowledgements

A.S. and P.G.K. acknowledge the support of the U.S. Department of Energy. Specifically, LANL is operated by Triad National Security, L.L.C. for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. 892333218NCA000001. J.C.M. thanks financial support from MAT2016-79866-R project (AEI/FEDER, UE). This material is based upon work supported by the National Science Foundation under Grant No. DMS-1809074 (P.G.K.). Finally, P.G.K. gratefully acknowledges the support of the Leverhulme Trust towards a visiting fellowship at the University of Oxford and the kind hospitality of the Mathematical Institute of the University of Oxford.

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Authors and Affiliations

  1. Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Avadh Saxena

  2. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003-4515, USA

    Panayotis G. Kevrekidis

  3. Departamento de Fisica Aplicada I, Grupo de Fisica No Lineal, Universidad de Sevilla, Escuela Politécnica Superior, C/ Virgen de África 7, 41011, Sevilla, Spain

    Jesús Cuevas-Maraver

  4. Instituto de Matemáticas de la Universidad de Sevilla (IMUS), Edificio Celestino Mutis, Avda. Reina Mercedes s/n, 41012, Sevilla, Spain

    Jesús Cuevas-Maraver

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  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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Saxena, A., Kevrekidis, P.G., Cuevas-Maraver, J. (2020). Nonlinearity and Topology. 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_2

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