## Abstract

The first description of the soliton as a physical phenomenon was given by J. Scott-Russell in 1843 [1]. Much later, in 1895, Korteweg and de Vries [2] derived the nonlinear equation which describes the propagation of long water waves in a canal and which admits the soliton solution described by Russell. This is the celebrated
Thus, the background for the quantitative description of solitons was laid nearly a century ago. However, the modern history of solitons is still quite recent. In 1965, Zabusky and Kruskal showed by computer simulation that solitons of the KdV equation (1.1.1) emerge following interaction without change of shape. Indeed their speed is likewise unaltered [3]. An attempt to understand these unexpected and astonishing experimental facts led, two years later, to the discovery of the inverse scattering transform method for the analysis of nonlinear equations such as the KdV equation which possess solitonic behavior.

**Korteweg-de Vries (KdV) equation**$$\frac{{\partial u}}{{\partial t}} + \frac{{{\partial ^3}u}}{{\partial {x^3}}} + 6u\frac{{\partial u}}{{\partial x}} = 0$$

(1.1.1)

## Keywords

Spectral Problem Nonlinear Integrable Equation Recursion Operator Commutativity Condition Inverse Scatter Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1992