Integrable Systems

9The Korteweg–de Vries Equation

The Korteweg–de Vries (KdV) equation, which is a nonlinear partial differential equation of the third order, is a universal mathematical model for the description of weakly nonlinear long wave propagation in dispersive media. It is a most remarkable nonlinear partial differential equation in 1 + 1 dimensions whose solutions can be exactly specified; it has a soliton-like solution or a solitary wave of sech² form. Various physical systems of dispersive waves admit solutions in the form of generalized solitary waves. The study of this equation is the archetype of an integrable system and is one of the most fundamental equations of soliton phenomena and a topic of active mathematical research. Our purpose here is to give a motivated and brief overview of this interesting subject. One of the objectives of this chapter is to study the KdV equation and the inverse scattering method (based on Schrödinger and Gelfand–Levitan equations) used to solve it.

9.1. Historical aspects and introduction

Korteweg and de Vries established a nonlinear partial differential equation describing the gravitational wave propagating in a shallow channel (Korteweg and de Vries 1895) and possessing remarkable mathematical properties:

[9.1]

where u(x, t) is the amplitude of the wave at the point x and the time t. The equation thus bearing their name (abbreviated KdV) presents ...