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The Inverse Scattering Transform‐Fourier Analysis for Nonlinear Problems
Author(s) -
Ablowitz Mark J.,
Kaup David J.,
Newell Alan C.,
Segur Harvey
Publication year - 1974
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1974534249
Subject(s) - inverse scattering transform , fourier transform , inverse scattering problem , mathematics , mathematical analysis , dispersion relation , scattering , nonlinear system , scattering theory , inverse , inverse problem , operator (biology) , partial differential equation , simple (philosophy) , similarity (geometry) , dispersion (optics) , fourier analysis , physics , computer science , quantum mechanics , geometry , philosophy , repressor , artificial intelligence , image (mathematics) , chemistry , biochemistry , epistemology , transcription factor , gene
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.