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The averaging of local field-theoretic Poisson brackets. (English. Russian original) Zbl 0962.37035

Russ. Math. Surv. 52, No. 2, 409-411 (1997); translation from Usp. Mat. Nauk 52, No. 2, 177-178 (1997).
Let the evolution system \[ \psi_t^i= Q^i(\psi, \psi_x,\dots), \qquad i=1,\dots, n, \tag{1} \] be Hamiltonian with respect to a local field-theoretic Poisson bracket of the form \[ \{\psi^i(x), \psi^i(y)\}= \sum_{k\geq 0} B_k^{ij} (\psi, \psi_x,\dots) \delta^k(x-y) \] (finitely many terms are present in the sum) with Hamiltonian \[ H= \int {\mathcal P}_H (\psi,\psi_x,\dots) dx. \] Let (1) have an \(N\)-parameter family of \(m\)-phase solutions \[ \varphi^i(x,t)= \Phi^i(k(U)x+ \omega(U)t+ \theta_0,U), \] where \(U= (U^1,\dots, U^N)\) are the parameters of the family, \(\Phi^i(\theta,U)\) are \(2\pi\)-periodic functions of \(\theta= (\theta^1,\dots, \theta^m)\), and \(\theta_0\) is the initial phase. In this paper the author describes a procedure for averaging local field-theoretic Hamiltonian structures defining a Poisson bracket.

MSC:

37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
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