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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.

##### MSC:
 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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##### References:
 [1] Beals, R., The inverse problem for ordinary differential operators on the line, Am. J. math., 107, 281-366, (1985) · Zbl 0579.34008 [2] Beals, R.; Deift, P.; Tomei, C., Direct and inverse scattering on the line, (1989), Amer. Math. Soc Providence, RI [3] Beals, R.; Sattinger, D.H., Action-angle variables for the Gel’fand-Dikii flows, Z. angew. mat. phys., 43, (1992) · Zbl 0754.35134 [4] R. Beals and D.H. Sattinger, Integrable systems and isomonodromy deformations, Physica D, to appear. · Zbl 0795.35077 [5] J. Dorfmeister, Banach manifolds of solutions to nonlinear partual differential equations, and relations with finite dimensional manifolds, in: Differential Geometry, eds. R.E. Greene and S.-T. Yau (AMS, Providence, RI). · Zbl 0798.58010 [6] Deift, P.; Zhou, X., Inverse scattering for nth order differential operators, Commun. pure appl. math., 54, 485-533, (1991) · Zbl 0734.34073 [7] Drinfel’d, V.G.; Sokolov, V.V., Lie algebras and equations of Korteweg-de Vries type, J. sov. math., 30, 1975-2036, (1985) · Zbl 0578.58040 [8] Gel’fand, I.M.; Dikii, L.A., Fractional powers of operators and Hamiltonian systems, Funct. anal. appl., 10, 259-273, (1976) · Zbl 0356.35072 [9] Gohberg, I.C.; Krein, M.G., Systems of integral equations on a half line with kernels depending on the difference of arguments, Uspehi mat. nauk, American math. soc. transl., 14, 217-287, (1960) · Zbl 0098.07501 [10] Kac, V., Infinite dimensional Lie algebras, (1985), Cambridge Univ. Press Cambridge [11] Kuperschmidt, B.A.; Wilson, G., Modifying Lax equations and the second Hamiltonian structure, Invent. math., 62, 403-436, (1981) · Zbl 0464.35024 [12] Malgrange, B., La classification des connexions irregulieres a une variable, (1982), Université de Grenoble France [13] Miwa, T., Painlevé property of monodromy preserving equations and the analyticity of the τ function, Publ. R.I.M.S. Kyoto, 17, 703-721, (1981) · Zbl 0605.34005 [14] Pressley, A.; Segal, G., Loop groups, (1986), Oxford Univ. Press Oxford, UK · Zbl 0618.22011 [15] Schilling, R., A loop algebra decomposition for Korteweg-de Vries equations, () [16] Segal, G.; Wilson, G., Loop groups and equations of KdV type, Publications I.H.E.S., 61, (1985) · Zbl 0592.35112 [17] Simon, B., Trace ideals and their applications, Vol. 35, (1979), London Mathematical Society Lecture Note Series · Zbl 0423.47001 [18] Szmigielski, J., Infinite dimensional homogeneous manifolds with translational symmetry and nonlinear partial differential equations, () · Zbl 0644.58024 [19] Zakharov, V.E.; Shabat, A., Integration of nonlinear equations of mathematical physics by the method of the inverse scattering, Funkt. anal. pril., 13, 13-22, (1979), [Functional Analysis and its Applications]
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