On spinor representation of the infinite dimensional grassmanian

Abstract

Fractional powers of the AKNS spectral operator L 3 = σ 3 ∂ x + qσ + − σ − are constructed as an expansion in terms of pseudodifferential operators. We found the existence of a quadruplet of operators L j , ( j = 0, 1, 2, 3), that share the same status as square roots of a common operator, L = L j 2, ( j = 0, 1, 2, 3). Completely integrable systems can then be represented as Lax pairs between L j and the differential operator part of the operator B n = L 0 L n−1 j ( L 0 and L j commute) on the infinite dimensional Grassmann manifold. The representation of the hierarchies of integrable equations can be obtained from an extension of the analysis to a “super algebra.”