Abstract
We consider the averaging of local field-theoretic Poisson brackets in the multi-dimensional case. As a result, we construct a local Poisson bracket for the regular Whitham system in the multidimensional situation. The procedure is based on the procedure of averaging of local conservation laws and follows the Dubrovin–Novikov scheme of the bracket averaging suggested in one-dimensional case. However, the features of the phase space of modulated parameters in higher dimensions lead to a different natural class of the averaged brackets in comparison with the one-dimensional situation. Here we suggest a direct procedure of construction of the bracket for the Whitham system for d > 1 and discuss the conditions of applicability of the corresponding scheme. At the end, we discuss canonical forms of the averaged Poisson bracket in the multidimensional case.
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