dorsal/arxiv
View SchemaFactorization and the Dressing Method for the Gel'fand-Dikii Hierarch
| Authors | D. H. Sattinger, J. S. Szmigielski |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9801011 |
| URL | https://arxiv.org/abs/solv-int/9801011 |
| DOI | 10.1016/0167-2789(93)90247-X |
| Journal | Physica D, vol 64, (1993), 1-34 |
Abstract
The isospectral flows of an $n^{th}$ order linear scalar differential operator $L$ 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 $n>3$ 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.
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"abstract": "The isospectral flows of an $n^{th}$ order linear scalar differential\noperator $L$ under the hypothesis that it possess a Baker-Akhiezer function\nwere originally investigated by Segal and Wilson from the point of view of\ninfinite dimensional Grassmanians, and the reduction of the KP hierarchy to the\nGel\u0027fand-Dikii hierarchy. The associated first order systems and their formal\nasymptotic solutions have a rich Lie algebraic structure which was investigated\nby Drinfeld and Sokolov. We investigate the matrix Riemann-Hilbert\nfactorizations for these systems, and show that different factorizations lead\nrespectively to the potential, modified, and ordinary Gel\u0027fand-Dikii flows. Lie\nalgebra decompositions (the Adler-Kostant-Symes method) are obtained for the\nmodified and potential flows. For $n\u003e3$ the appropriate factorization for the\nGel\u0027fand-Dikii flows is not a group factorization, as would be expected; yet a\nmodification of the dressing method still works.\n A direct proof, based on a Fredholm determinant associated with the\nfactorization problem, is given that the potentials are meromorphic in $x$ and\nin the time variables. Potentials with Baker-Akhiezer functions include the\nmultisoliton and rational solutions, as well as potentials in the scattering\nclass with compactly supported scattering data. The latter are dense in the\nscattering class.",
"arxiv_id": "solv-int/9801011",
"authors": [
"D. H. Sattinger",
"J. S. Szmigielski"
],
"categories": [
"solv-int",
"nlin.SI"
],
"doi": "10.1016/0167-2789(93)90247-X",
"journal_ref": "Physica D, vol 64, (1993), 1-34",
"title": "Factorization and the Dressing Method for the Gel\u0027fand-Dikii Hierarch",
"url": "https://arxiv.org/abs/solv-int/9801011"
},
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