The hierarchy of the Poisson brackets between the elements of the scattering matrix for the general differential spectral problem

Abstract

The infinite family of Poisson brackets (S(i1k1)( lambda 1), S(i2k2)( lambda 2))(n(1),n(2)) (n1, n2=0, 1,.) between the elements of the scattering matrix S( lambda ) is calculated for the differential spectral problem of an arbitrary order N with respect to the family of brackets of the form (F, H)(n(1),n(2))=1/2N integral - infinity + infinity dx tr ( delta F/ delta VT. (L++)n1I(L-)n2 delta H/ delta V)-(F to or from H) where I is Gelfand-Dikij Hamiltonian operator and L+, L- are recursion operators. It is shown that the highest Poisson brackets (S,S)(n(1),n(2)) (n1+n2>or=1) in general contain new t-type terms in addition to the classical r-matrices terms. The classical r matrices depend only on n1+n2. The t terms depend separately on n1 and n2. The t-type terms are absent for the two brackets (S, S)(0,0) and (S, S)(0,1). It is demonstrated that both the classical r matrices and t terms are bounded only for special values of arg lambda 1 and arg lambda 2. The cases N=2 and N=3 are considered in detail. For N=2, 3 the quadratic algebras of the Poisson brackets (,)(n(1),n(2)) between scattering data are also calculated. The t-type terms do not contribute to these Poisson brackets and the algebras of the Poisson brackets of the scattering data are the same for different n1 and n2 with a fixed value of n1+n2.