Periodic fixed points of Bäcklund transformations and the Korteweg-de-Vries equation

John Weiss
Institute for Pure and Applied Physical Sciences,
University of California, San Diego, La Jolla, CA 92O93, U.S.A.

Abstract:

A new method for studying integrable systems based on the ``periodic fixed points" of Bäcklund transformations (BT's) is presented. Normally the BT maps an ``old" solution into a ``new" solution and requires a known ``seed" solution to get started. Besides this limitation, it can also be difficult to qualitatively classify the result of applying the BT several times to a known solution. By studying the periodic fixed points of the BT (regarded as a nonlinear map in a function space), integrable systems of equations of finite degree (equal to the order of the fixed point) and a method for the systematic classification of the solutions of the original system are obtained.