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To our wives, Masha and Marian Curiosity inside the so-known as completely integrable strategies with infinite num ber of ranges of freedom was aroused immediately after publication of the nicely-recognized assortment of papers by Gardner, Greene, Kruskal, Miura, and Zabusky [75, 77, 96, 18, 66, 19J (see also ) on placing properties of the Korteweg-de Vries (KdV) equation. It shortly turned clear that methods of such a kind possess quite a lot of attribute properties, just like infinite assortment of symmetries and/or conservation authorized tips, inverse scattering disadvantage formulation, L – A pair illustration, existence of prolongation buildings, and so forth. And though no satisfactory definition of full integrability was however invented, a necessity of testing a selected system for these properties appeared. Perhaps one of many surroundings pleasant exams of this sort was first proposed by Lenard ' who constructed a recursion operator for symmetries of the KdV equation. It was a wierd operator, in a approach: being formally integro-differential, its movement on the first classical symmetry (x-translation) was correctly-outlined and produced all the gathering of higher KdV equations; nevertheless utilized to the scaling symmetry, it gave expressions containing phrases of the type J u dx which had no adequate interpretation inside the framework of the prevailing theories. It is not surprising that P. Olver wrote "The de duction of the kind of the recursion operator (if it exists) requires a certain amount of impressed guesswork. . . " [eighty, p.
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