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Integrability and Nonlinear Waves

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Emerging Frontiers in Nonlinear Science

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

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Abstract

Major developments in the study of nonlinear wave equations were the discovery of the soliton, its connection to the eigenvalue of a linear operator and solutions of the underlying equations by the inverse scattering transform. Inverse scattering transform provides the solution to the initial value problem of a class of nonlinear equations. In this article the background, key ideas and methodology of inverse scattering transform are discussed in connection with some well-known physically important nonlinear equations including the Korteweg–de Vries and nonlinear Schrödinger equations. More recently a new class of nonlocal nonlinear Schrödinger type equations have been found to be integrable; they arise from new symmetries in the associated scattering problem that had not been previously known. The solution of these novel systems is also discussed in this review. Other methods of solution of soliton/integrable equations and comments regarding the future are also included.

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Notes

  1. 1.

    By solved we mean linearized in terms of integral equations.

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Acknowledgements

MJA was partially supported by NSF under Grant No. DMS-1712793.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Colorado, Campus Box 526, Boulder, CO, 80309-0526, USA

    Mark J. Ablowitz

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  1. Mark J. Ablowitz
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Correspondence to Mark J. Ablowitz .

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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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Ablowitz, M.J. (2020). Integrability and Nonlinear Waves. 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_7

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