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On the Hamiltonian formalism for Korteweg-de Vries type hierarchies. (English. Russian original) Zbl 0790.58022
Funct. Anal. Appl. 26, No. 2, 140-142 (1992); translation from Funkts. Anal. Prilozh. 26, No. 2, 79-82 (1992).
For most of the known hierarchies of integrable nonlinear equations, the phase space admits a pair of Poisson structures that satisfies the following Adler-Kostant theorem [M. A. Semenov-Tyan-Shanskij, Funct. Anal. Appl. 17, 259-272 (1983); translation from Funkts. Anal. Prilozh. 17, No. 4, 17-33 (1983; Zbl 0535.58031)]:
Suppose that for \(\xi\) and \(\eta\) from the cotangent bundle of the phase space there is a solution \(R\) of the modified classical Yang-Baxter equation \[ [R\xi,R\eta] - R([R\xi, \eta]+ [\xi, R\eta)] + \alpha[\xi,\eta] = 0.\tag{1} \] Let \(H_ 1\) be a Hamiltonian operator, and let \(\{I_ n\}\) be the ring of annihilators of the Poisson bracket determined by \(H_ 1\). Then the \(I_ n\) are in involution with respect to the Poisson bracket associated with the Hamiltonian operator \(H_ 2 = H_ 1R + R^*H_ 1\).
In this paper we study the Hamiltonian structure on the phase space that is the dual space of the algebra of Laurent series with coefficients in the algebra of vector fields on the circle \(S^ 1\). A hierarchy of integrable systems constructed according to the Adler-Kostant scheme is shown to exist in this phase space. Under additional conditions, this hierarchy reduces to Korteweg-de Vries (KdV) type hierarchies in the form introduced by S. I. Al’ber in J. Lond. Math. Soc., II. Ser. 19, 467-480 (1979; Zbl 0413.35064).

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
17B65 Infinite-dimensional Lie (super)algebras
35Q53 KdV equations (Korteweg-de Vries equations)
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