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The Inverse Scattering Transform

  • Chapter
Solitons

Part of the book series: Topics in Current Physics ((TCPHY,volume 17))

Abstract

A detailed description of the inverse scattering transform associated with the generalized Zakharov-Shabat and Schrödinger eigenvalue problems is given. The close analogy with the ideas of the Fourier transform is emphasized and the general expansions for the unknown functions in terms of the squared eigenfunctions and their derivatives are developed for both eigenvalue problems. The results for the Schrödinger equation are new.1 The partial differential equations which are solvable by the inverse scattering transforms associated with these eigenvalue problems are identified, almost by inspection, and classified according to the nature of the dispersion relation. Of particular interest are those classes which are integrable but which do not possess conserved quantities, and also those equations which are integrable but for which the spectrum is not invariant. Several examples, the coherent pulse propagation problem, the nonlinear Schrödinger equation and the sine-Gordon equation, are used to illustrate some of the important points. Finally, a singular perturbation theory for examining the effects of perturbations over long times is given. Last-minute revisions and additions have been made to Sects.6.12 and 13 to reflect some recent developments.

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  1. A. C. Newell
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Editors and Affiliations

  1. Department of Mathematics, University of Manchester, Institute of Science and Technology, M60 1QD, Manchester, UK

    Robin K. Bullough  & Philip J. Caudrey  & 

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Newell, A.C. (1980). The Inverse Scattering Transform. In: Bullough, R.K., Caudrey, P.J. (eds) Solitons. Topics in Current Physics, vol 17. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-81448-8_6

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  • DOI: https://doi.org/10.1007/978-3-642-81448-8_6

  • Publisher Name: Springer, Berlin, Heidelberg

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