Summary: SKETCHES OF KDV
To Herb Clemens, on his 60th birthday.
Abstract. This is a survey of some of the links between geometry and the
theory of Korteweg de Vries equations.
A few years ago, Herb asked me to tell him the story of the connections between
algebraic geometry and the KdV equation. I wrote for him a few handwritten
pages, mostly on the origin of the fascinating interaction between these seemingly
distant subjects. More recently, on the occasion of a Colloquium talk at the Courant
Institute, I thought again about the multiple facets of these subtle connections. The
present survey grew out of these two occurences. It reflects the particular path I
happened to take in learning this subject and it exposes the many things I do not
know about it. This is also an opportunity to thank the friends and collegues at
Courant for their kindness and their precious hospitality and Domenico Fiorenza
for very helpful suggestions.
1. Solitons and theta-function solutions of KdV
Few papers or books on this subject resist the temptation of quoting John Scott
Russel's beautiful prose describing his first encounter of a solitary wave at the
time when he was offering his engineering's work to the Union Canal Society of