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
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