On the Poisson structure for the Korteweg-de Vries equation. (English. Russian original) Zbl 0701.35132

Arkad’ev, V. A.; Pogrebkov, A. K.; Polivanov, M. K.

Sov. Phys., Dokl. 33, No. 1, 24-26 (1988); translation from Dokl. Akad. Nauk SSSR 298, No. 2, 324-328 (1988).

The paper deals with the equation \(u_ t-6uu_ x+u_{xxx}=0\) on the whole axis \(x\in R\) in the case where u is rapidly decreasing as \(| x| \to \infty\). More precisely, u is a real function, continuously differentiable with respect to t, and for each t u(t,\(\cdot)\in S\), the Schwarz space. Various forms of the Poisson brackets are discussed: - the Gardner bracket, - its antisymmetrized version, - a bracket with special boundary terms. Follow-up results are announced.

Reviewer: Z.Kamont

MSC:

35Q53

KdV equations (Korteweg-de Vries equations)

Keywords:

rapidly decreasing; Poisson brackets; Gardner bracket