×

On the Adler-Gel’fand-Dickey bracket. (English) Zbl 0737.35111

Hamiltonian systems, transformation groups and spectral transform methods, Proc. CRM Workshop, Montréal/Can. 1989, 77-85 (1990).
Summary: [For the entire collection see Zbl 0726.00014.]
The Adler-Gel’fand-Dickey (AGD) bracket is very interesting Poisson bracket first discovered by Adler in connexion with the theory of certain integrable systems that generalize the Korteweg-de Vries (KdV) equation. It is also known as the second Hamiltonian structure for scalar Lax equations, or (in the recent Russian literature) simply as the Gel’fand- Dickey bracket. A key property of the AGD bbracket is that it makes the composition of (ordinary) differential operators into a canonical transformation in the sense of Hamiltonian mechanics. The author sketches a way of understanding this result even ore thoroughly than was done in the earlier literature.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
17B66 Lie algebras of vector fields and related (super) algebras

Citations:

Zbl 0726.00014

Software:

AGD
PDFBibTeX XMLCite