Geodesic
Kdv flow on generalized reflectionless potentials
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 4 (2008), pp. 490-528.

Voir la notice de l'article provenant de la source Math-Net.Ru

The purpose of this article is to construct KdV flow on a space of generalized reflectionless potentials by applying Sato's Grassmannian approach. The point is that the base space contains not only rapidly decreasing potentials but also oscillating ones such as periodic ones, which makes it possible for us to discuss the shift invariant probability measures on it. 
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S. Kotani. Kdv flow on generalized reflectionless potentials. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 4 (2008), pp. 490-528. http://geodesic.mathdoc.fr/item/JMAG_2008_4_a2/

[1] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, “Transformation Groups for Soliton Equations, VII”, Publ. RIMS, Kyoto Univ., 18 (1982), 1077–1110 | DOI | MR | Zbl

[2] P. L. Duren, Theory of $H^p$ spaces, Pure and Appl. Math., 38, Acad. Press, 1970 | MR

[3] F. Dyson, “Fredholm Determinants and Inverse Scattering Problems”, Comm. Math. Phys., 47 (1976), 171–183 | DOI | MR | Zbl

[4] F. Gesztesy, W. Karwowski, Z. Zhao, “Limit of Soliton Solutions”, Duke Math. J., 68 (1992), 101–150 | DOI | MR | Zbl

[5] F. Gesztesy, B. Simon, “A New Approach to Inverse Spectral Theory. II. General Real Potentials and the Connection to the Spectral Measure”, Ann. Math., 152 (2000), 593–643 | DOI | MR | Zbl

[6] N. Ikeda, S. Kusuoka, S. Manabe, “Lévy's Stochastic Area Formula for Gaussian Processes”, Comm. Pure and Appl. Math., XLVII (1994), 329–360 | DOI | MR | Zbl

[7] A. R. Its, V. B. Matveev, “Hill's Operator with Finitely Many Gaps”, Funct. Anal. Appl., 9 (1975), 65–66 | DOI | MR | Zbl

[8] K. Iwasaki, “On the Inverse Sturm–Liouville Problem with Spatial Symmetry”, Funkc. Ekvacioj, 31 (1988), 25–74 | MR | Zbl

[9] R. A. Johnson, “On the Sato–Segal–Wilson Solutions of the KdV Equation”, Pacific J. Math., 132 (1988), 343–355 | DOI | MR | Zbl

[10] S. Kotani, “Lyapounov Indices Determine Absolutely Continuous Spectra of Stationary One-Dimensional Schrödinger Operators”, Proc. Taniguchi Symp. SA. Katata, 1984, 225–248 | MR

[11] D. S. Lundina, “Compactness of the Set of Reflectionless Potentials”, Teor. Funkts., Funkts. Anal. i Prilozh., 44 (1985), 55–66 (Russian)

[12] V. A. Marchenko, Sturm–Liouville Operators and Applications, Birkhäuser, Boston, 1986 | MR | Zbl

[13] V. A. Marchenko, Nonlinear Equations and Operator Algebras, D. Reidel, Dordrechet, 1987 | MR

[14] V. A. Marchenko, “The Cauchy Problem for the KdV Equation with Nondecreasing Initial Data”, What is Integrability?, Springer Ser. in Nonlinear Dynamics, ed. V. E. Zakharov, 1990, 273–318 | MR

[15] H. P. McKean, “Geometry of KdV. V. Scattering from the Grassmannian Viewpoint”, Comm. Pure and Appl. Math., XLII (1989), 687–701 | DOI | MR | Zbl

[16] G. Segal, G. Wilson, “Loop Groups and Equations of KdV Type”, Inst. Haute Étude Sci. Publ. Math., 61 (1985), 5–65 | DOI | MR | Zbl