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Inverse scattering transform, KdV, and solitons. (English) Zbl 1062.37087

Ball, Joseph A. (ed.) et al., Current trends in operator theory and its applications. Proceedings of the international workshop on operator theory and its applications (IWOTA), Virginia Tech, Blacksburg, VA, USA, August 6–9, 2002. Basel: Birkhäuser (ISBN 3-7643-7067-X/hbk). Operator Theory: Advances and Applications 149, 1-22 (2004).
Summary: In this review paper, the Korteweg-de Vries equation (KdV) is considered, and it is derived by using the Lax method and the AKNS method. An outline of the inverse scattering problem and of its solution is presented for the associated Schrödinger equation on the line. The inverse scattering transform is described in order to solve the initial-value problem for the KdV, and the time evolution of the corresponding scattering data is obtained. Soliton solutions to the KdV are derived in several ways.
For the entire collection see [Zbl 1050.47002].

MSC:

37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
81U40 Inverse scattering problems in quantum theory
34A55 Inverse problems involving ordinary differential equations
34L25 Scattering theory, inverse scattering involving ordinary differential operators
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