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Factorization and the dressing method for the Gel’fand-Dikii hierarchy. (English) Zbl 0770.34060
Summary: The isospectral flows of an \(n\)th order linear scalar differential operator \(L\) under the hypothesis that it possesses a Baker-Akhiezer function were originally investigated by Segal and Wilson from the point of view of infinite dimensional Grassmannians, and the reduction of the KP hierarchy to the Gel’fand-Dikii hierarchy. The associated first order system 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 \(n>3\) 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 \(x\) 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.

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
Full Text: DOI
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