Contemporary Mathematics

Volume 301, 2002

The Legacy of the 1ST

David

J.

Kaup

ABSTRACT.

We provide a brief review of some of the major research results

arising from the method of the Inverse Scattering Transform.

1.

Introduction

I will give a brief review of several items in the Legacy of the Inverse Scattering

Transform. In no way is this to be a complete review, since the Legacy has become

so vast. However, I will treat those items with which I am most familiar, and try

to detail their significance and importance.

There is no doubt that the most important contribution was the famous clas-

sical Gardner, Greene, Kruskal and Miura (GGKM) work

[1]

of 1967 on the KdV

equation. This was the starting point. They had found a very strange and new

method for solving the initial value problem of a nonlinear evolution equation, the

KdV. At that time, and even for several years later, this strange new method was

considered to be only a novelty, since it would only work for that one equation, the

KdV. Shortly thereafter, as a prelude to what was to follow, Peter Lax

[2]

showed

that if given an appropriate linear operator,

L,

dependent on a potential, u(x), then

one could always construct an infinite sequence of evolution operators,

B,

each of

which would satisfy

{1.1) BL-LB

=

8tL.

This sequence of evolution operators could be generated by simply increasing the

order of the spatial differentials contained in B. Then from (1.1) one would obtain

additional nonlinear evolution equations, each of the form

(1.2)

OtU

=

K(u)

where K was some (nonlinear) operator. All these additional higher order evolution

equations would be solvable by this same technique. This collection is now known

as the KdV hierarchy.

1991 Mathematics Subject Classification. Primary 01A65; Secondary 35Q51.

Key words and phrases. Solitons, Inverse Scattering Transform.

The author was supported in part by NSF Grant #0129714. The author thanks an anonymous

referee for his comments, and also H. Steudel for his comments.

©

2002 American Mathematical Society

http://dx.doi.org/10.1090/conm/301/05156