The Inverse Scattering Transform‐Fourier Analysis for Nonlinear Problems
1974/12/01 by Mark J. Ablowitz, David J. Kaup, D. J. Kaup +2 · 3,241 citations
Mathematics · Physics and Astronomy · #Advanced Fiber Laser Technologies #Applied mathematics #Computer science #Dispersion (optics) #Dispersion relation #Fourier analysis #Fourier transform #Geometry #Inverse #Inverse problem #Inverse scattering problem #Inverse scattering transform #Mathematical analysis #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Operator (biology) #Partial differential equation #Physics #Quantum mechanics #Scattering #Scattering theory #Similarity (geometry) #Simple (philosophy)
paper · doi:10.1002/sapm1974534249
published in Studies in Applied Mathematics 53(4), 249-315 (Wiley)
openalex publication_date 1974/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Abstract
A systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering. The form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro‐differential operator. A comprehensive presentation of the inverse scattering method is given and general features of the solution are discussed. The relationship of the scattering theory and Backlund transformations is brought out. In view of the role of the dispersion relation, the comparatively simple asymptotic states, and the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems.
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