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Integrability and Nonlinear Waves

  • Mark J. AblowitzEmail author
Chapter
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Part of the Nonlinear Systems and Complexity book series (NSCH, volume 32)

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.

Notes

Acknowledgements

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

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of ColoradoBoulderUSA

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