Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-18T17:56:58.433Z Has data issue: false hasContentIssue false
  • English
  • Français

Darboux Transformations for the KP Hierarchy in the Segal-Wilson Setting

Published online by Cambridge University Press:  20 November 2018

G. F. Helminck  and
J. W. van de Leur
G. F. Helminck
Affiliation:
Faculty of Applied Mathematics University of Twente P.O.Box 217 7500 AE Enschede The Netherlands, email: helminck@math.utwente.nl
J. W. van de Leur
Affiliation:
Faculty of Mathematics University of Utrecht P.O.Box 80010 3508 TA Utrecht The Netherlands, email: vdleur@math.uu.nl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper it is shown that inclusions inside the Segal-Wilson Grassmannian give rise to Darboux transformations between the solutions of the $\text{KP}$ hierarchy corresponding to these planes. We present a closed form of the operators that procure the transformation and express them in the related geometric data. Further the associated transformation on the level of $\tau $-functions is given.

Keywords

22E6522E7035Q5335Q5858B25KP hierarchyDarboux transformationGrassmann manifold
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 2 , 01 April 2001 , pp. 278 - 309
Copyright
Copyright © Canadian Mathematical Society 2001

References

[1] Adler, M. and van Moerbeke, P., Birkhoff Strata, Bäcklund Transformations and Regularization of Isospectral Operators. Adv. in Math. 108 (1994), 140–204.Google Scholar
[2] Bakalov, B., Horozov, E. and Yakimov, M., Bäcklund-Darboux transformations in Sato's Grassmannian. Serdica Math. J. 22 (1996), 571–586.Google Scholar
[3] Crum, M. M., Associated Sturm-Liouville systems. Quart. J. Math. 6 (1955), 121–127.Google Scholar
[4] Date, E., Jimbo, M., Kashiwara, M. and Miwa, T., Transformation groups for soliton equations. In: Nonlinear integral systems—classical theory and quantum theory (eds. M. Jimbo and T. Miwa), World Scientific, 1983, 39–120.Google Scholar
[5] Helminck, G. F. and Post, G. F., Geometrical interpretation of the bilinear equations for the KP hierarchy. Letters Math. Phys. 16 (1988), 359–364.Google Scholar
[6] Helminck, G. F. and van de Leur, J. W., An analytic description of the vector constrained KP hierarchy. Comm. Math Phys. 193 (1998), 627–641 (solv-int 9706004).Google Scholar
[7] Helminck, G. F. and van de Leur, J. W., Constrained and Rational Reductions of the KP hierarchy. Preprint, to appear in “Supersymmetry and IntegrableModels”, Springer Lecture Notes in Physics 502, 1998, 167–182.Google Scholar
[8] Kadomtsev, B. B. and Petviashvilii, V. I., On the stability of solitary waves in weakly dispersing media. Sov. Phys. Doklady 15, 539–541.Google Scholar
[9] Matveev, V. B. and Salle, M. A., Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics, Springer, Berlin, 1991.Google Scholar
[10] Oevel, W., Darboux theorems and Wronskian formulas for integrable systems (I. Constrained KP flows). Physica A 195(1993), 533–576.Google Scholar
[11] Oevel, W. and Schief, W., Darboux theorems and the KP hierarchy. In: Applications of analytic and geometric Methods in Differential Equations (Proceedings of the NATO Advanced Research Workshop, Exeter, 14–17 July 1992, UK) (ed. P. A. Clarkson), Kluwer Publ., 1993, 193–206.Google Scholar
[12] Oevel, W. and Strampp, W., Wronskian solutions of the constrained Kadomtsev-Petviashvili hierarchy. Journ. Math. Phys. 37 (1996), 6213–6219.Google Scholar
[13] Pressley, A. and Segal, G., Loop groups. Oxford Mathematical Monographs, Oxford University Press, Oxford, 1988.Google Scholar
[14] Segal, G. and Wilson, G., Loop groups and equations of KdV type. Inst. Hautes Études Sci. Publ. Math. 63 (1985), 1–64.Google Scholar