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

A new method for studying integrable systems based on the ‘‘periodic fixed points’’ of Backlund 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.