Factorization and the Dressing Method for the Gel'fand-Dikii Hierarch
arXiv:solv-int/9801011 · doi:10.1016/0167-2789(93)90247-X
Abstract
The isospectral flows of an order linear scalar differential operator under the hypothesis that it possess a Baker-Akhiezer function were originally investigated by Segal and Wilson from the point of view of infinite dimensional Grassmanians, and the reduction of the KP hierarchy to the Gel'fand-Dikii hierarchy. The associated first order systems and their formal asymptotic solutions have a rich Lie algebraic structure which was investigated by Drinfeld and Sokolov. We investigate the matrix Riemann-Hilbert factorizations for these systems, and show that different factorizations lead respectively to the potential, modified, and ordinary Gel'fand-Dikii flows. Lie algebra decompositions (the Adler-Kostant-Symes method) are obtained for the modified and potential flows. For the appropriate factorization for the Gel'fand-Dikii flows is not a group factorization, as would be expected; yet a modification of the dressing method still works. A direct proof, based on a Fredholm determinant associated with the factorization problem, is given that the potentials are meromorphic in and in the time variables. Potentials with Baker-Akhiezer functions include the multisoliton and rational solutions, as well as potentials in the scattering class with compactly supported scattering data. The latter are dense in the scattering class.