Abstract
Splitting the algebra Psd of pseudodifferential operators into the Lie subalgebra of all differential operators without a constant term and the Lie subalgebra of all integral operators leads to an integrable hierarchy called the strict KP hierarchy. We consider two Psd modules, a linearization of the strict KP hierarchy and its dual, which play an essential role in constructing solutions geometrically. We characterize special vectors, called wave functions, in these modules; these vectors lead to solutions. We describe a relation between the KP hierarchy and its strict version and present an infinite-dimensional manifold from which these special vectors can be obtained. We show how a solution of the strict KP hierarchy can be constructed for any subspace W in the Segal–Wilson Grassmannian of a Hilbert space and any line ℓ in W. Moreover, we describe the dual wave function geometrically and present a group of commuting flows that leave the found solutions invariant.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. F. Helminck, A. G. Helminck, and E. A. Panasenko, “Integrable deformations in the algebra of pseudodifferential operators from a Lie algebraic perspective,” Theor. Math. Phys., 174, 134–153 (2013).
G. F. Helminck, A. G. Helminck, and E. A. Panasenko, “Cauchy problems related to integrable deformations of pseudo differential operators,” J. Geom. Phys., 85, 196–205 (2014).
G. F. Helminck, E. A. Panasenko, and S. V. Polenkova, “Bilinear equations for the strict KP hierarchy,” Theor. Math. Phys., 185, 1803–1815 (2015).
G. Segal and G. Wilson, “Loop groups and equations of KdV type,” Publ. Math. Inst. Hautes Études Sci., 61, 5–65 (1985).
L. A. Dickey, Soliton Equations and Hamiltonian Systems (Adv. Series Math. Phys., Vol. 26), World Scientific, Singapore (2003).
E. Date, M. Kashiwara, M. Jimbo, and T. Miwa, “Transformation groups for soliton equations,” in: Nonlinear Integrable Systems: Classical Theory and Quantum Theory (Kyoto, Japan, 13–16 May 1981, M. Jimbo and T. Miwa, eds.), World Scientific, Singapore (1983), pp. 39–119.
V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin (1991).
W. Oevel and W. Schief, “Darboux theorems and the KP hierarchy,” in: Applications of Analytic and Geometric Methods in Differential Equations (NATO ASI Ser. C: Math. Phys. Sci., Vol. 413, P. A. Clarkson, ed.), Springer, Dordrecht (1993), pp. 193–206.
A. Pressley and G. Segal, Loop Groups, Oxford Univ. Press, Oxford (1988).
N. H. Kuiper, “On the general linear group in Hilbert space,” Math. Centrum Amsterdam Afd. Zuivere Wisk., 1964, ZW-011 (1964).
M. F. Atiyah, K-theory, Benjamin, New York (1967).
G. D. Birkhoff, “Singular points of ordinary differential equations,” Trans. AMS, 10, 436–470 (1909).
G. D. Birkhoff, “Equivalent singular points of ordinary linear differential equations,” Math. Ann., 74, 134–139 (1913).
A. Grothendieck, “Sur la classification des fibrés holomorphes sur la sphère de Riemann,” Amer. J. Math., 79, 121–138 (1957).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 198, No. 1, pp. 54–78, January, 2019.
Rights and permissions
About this article
Cite this article
Helminck, G.F., Panasenko, E.A. Geometric Solutions of the Strict KP Hierarchy. Theor Math Phys 198, 48–68 (2019). https://doi.org/10.1134/S0040577919010045
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577919010045