Abstract
A new regular method for (a) classification of integrable equations, (b) constructing an infinite set of differential substitutions, and (c) reducing every nonlocal Hamiltonian structure into canonical form is presented. An explanation of the origin of these differential substitutions is given via a relationship between the spectral problem and a Miura transformation. A relationship of these differential substitutions with a generating function of conservation law densities is found. Every Hamiltonian structure of the Korteweg-de Vries equation possesses a transformation to the canonical “ d dx ”- type by a combination of some differential substitutions and reciprocal transformations. Some well-known equations are embedded into a unified chain.
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