On the structure of symplectic operators and hereditary symmetries

In the last fifteen years, there has been a remarlmble development in the exact analysis of certain nonlinear evolution equations, like tbe Korteweg-de Vries equation. I t is weH known that among the surprising features of these so-called exactly solvable equations is the possession of infinitely many symmetries and conservation laws, of N-soliton solutions and Bäcklund transformations. It has turned out that considering an operator which maps symmetries (1) onto symmetries of a given equation yields a useful approach to all these features. 'l'his operator is called a strong synunetry (2 ) ( or recursion operator (3 ) ). It is particularly useful because its transpose generates conserved covariants from given ones and because its eigenfunctions are also symmetries (which actually characterize the N-soliton solutions). One way of finding strong symmetries is to use the fact that any translation-invariant operator fP(u), possessing the property defined by v below is a strong symmetry for the hierarchy of equations u, = ( fP(u) )" Uz, n = 0, 1, 2, ... . These operators are called hereditary symmetries. The strong symmetries of all well-known exactly solvable equations are hereditary (2 ). Recently there has also been progress in understanding the Hamiltonian structure of these evolution equations (4). An evolution equations is said to be a Hamiltonian system if it can be written in the form u 1 = O(u)f(u), where O(u) is impletic (which is, roughly speaking, the same as saying that 0-1(u) is sympletic) and where f(u) is the gradient of a suitable potential. For these systems tbe operator-valued function O(u) is of particular interest because it is a N oether operator, i.e. it maps conserved covariants onto symmetries. Our paper is related to .1\Iagri's work wbo considered bi-Hamiltonion systems u, = = 81(u)j1(u) = 82(u)/2(u) and who sbowed that these equations have fP(u) =81(u) 8;(u) as strong symmetries.