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Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations (Paperback, Softcover reprint of hardcover 1st ed. 2000)
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Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations (Paperback, Softcover reprint of hardcover 1st ed. 2000)
Series: Mathematics and Its Applications, 507
Expected to ship within 10 - 15 working days
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To our wives, Masha and Marian Interest in the so-called completely
integrable systems with infinite num ber of degrees of freedom was
aroused immediately after publication of the famous series of
papers by Gardner, Greene, Kruskal, Miura, and Zabusky [75, 77, 96,
18, 66, 19J (see also [76]) on striking properties of the
Korteweg-de Vries (KdV) equation. It soon became clear that systems
of such a kind possess a number of characteristic properties, such
as infinite series of symmetries and/or conservation laws, inverse
scattering problem formulation, L - A pair representation,
existence of prolongation structures, etc. And though no
satisfactory definition of complete integrability was yet invented,
a need of testing a particular system for these properties
appeared. Probably one of the most efficient tests of this kind was
first proposed by Lenard [19]' who constructed a recursion operator
for symmetries of the KdV equation. It was a strange operator, in a
sense: being formally integro-differential, its action on the first
classical symmetry (x-translation) was well-defined and produced
the entire series of higher KdV equations; but applied to the
scaling symmetry, it gave expressions containing terms of the type
J u dx which had no adequate interpretation in the framework of the
existing theories. It is not surprising that P. Olver wrote "The de
duction of the form of the recursion operator (if it exists)
requires a certain amount of inspired guesswork. . . " [80, p.
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