Nonlinear Equations

Abstract

In 1967 Gardner, Green, Kruskal, and Miura [6] discovered a number of profound connections between the Korteweg-deVries (KdV) equation

$${v_t}-6v{v_x}+{v_{xxx}}=0$$

((4.1.1))

which describes the motion of waves in shallow water, and the spectral properties of the family

$$L = -\frac{{{d^2}}}{{d{x^2}}}+v\left({x,t}\right)\quad\left({-\infty<x<\infty}\right)$$

((4.1.2))

of Sturm-Liouville operators generated by a solution v(x, t) of equation (4.1.1). These connections allowed them to find the solution to the Cauchy problem

$$v\left({x,0} \right)={v_{0\;}},\;\mathop{\lim}\limits_{x\to\pm\infty}v\left({x,t}\right)=0$$

((1))

for the KdV equation using the inverse problem of scattering theory (this is presently known as the “inverse scattering method” (ISM); translator’s note). The fundamental idea of their method was developed further in the work of Lax [12], in which the notion of an operator L-A (or Lax) pair was introduced.