The Inverse Scattering Transform‐Fourier Analysis for Nonlinear Problems

TLDR

The theory extends Fourier analysis to nonlinear problems by characterizing evolution equations through their dispersion relations and integro‑differential operators, noting simple asymptotic states and a close similarity to Fourier transforms. The authors develop a systematic method to identify evolution equations solvable by inverse scattering and provide a comprehensive presentation of the method and its general solution features. The framework links scattering theory with Backlund transformations, enabling the systematic identification of solvable evolution equations.

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.