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 research was supported in part by the Ministry
of Education and Science of the Russian Federation (Grant
No. 3.8515.2017/8.9 in the framework of the base part of the Government
order) and the Russian Foundation for Basic Research (Grant
No. 17-01-00553).
Citation:
G. F. Helminck, E. A. Panasenko, “Geometric solutions of the strict KP hierarchy”, TMF, 198:1 (2019), 54–78; Theoret. and Math. Phys., 198:1 (2019), 48–68
This publication is cited in the following 12 articles:
G. F. Helminck, J. A. Weenink, “LU Factorizations for ℕ × ℕ-Matrices and Solutions of the k[S]-Hierarchy and Its Strict Version”, Geometry, 2:2 (2025), 4
Aloysius G. Helminck, Gerardus F. Helminck, “A construction of solutions of an integrable deformation of a commutative Lie algebra of skew hermitian Z×Z-matrices”, Indagationes Mathematicae, 2024
G. F. Helminck, V. A. Poberezhny, S. V. Polenkova, “Darboux transformations for the discrete versions of the KP and strict KP hierarchies”, Theoret. and Math. Phys., 221:3 (2024), 2031–2048
G. F. Helminck, V. A. Poberezhny, S. V. Polenkova, “Connecting KP and Strict KP with Their Discrete Versions”, Lobachevskii J Math, 45:10 (2024), 4644
G. F. Helminck, E. A. Panasenko, “Darboux transformations for the strict KP hierarchy”, Theoret. and Math. Phys., 206:3 (2021), 296–314
G. F. Helminck, J. A. Weenink, “Homogeneous spaces yielding solutions of the $k[S]$-hierarchy and its strict version”, Vestnik rossiiskikh universitetov. Matematika, 26:135 (2021), 315–336
G. F. Helminck, V. A. Poberezhny, S. V. Polenkova, “Extensions of the discrete KP hierarchy and its strict version”, Theoret. and Math. Phys., 204:3 (2020), 1140–1153
G. F. Helminck, E. A. Panasenko, “Reductions of the strict KP hierarchy”, Theoret. and Math. Phys., 205:2 (2020), 1411–1425
G. F. Khelmink, E. A. Panasenko, “Svoistva algebry psevdodifferentsialnykh operatorov, svyazannye s integriruemymi ierarkhiyami”, Vestnik rossiiskikh universitetov. Matematika, 25:130 (2020), 183–195
G. F. Helminck, E. A. Panasenko, “Scaling invariance of the strict KP hierarchy”, Vestnik rossiiskikh universitetov. Matematika, 25:131 (2020), 331–340
G. F. Helminck, E. A. Panasenko, “Expressions in Fredholm determinants for solutions of the strict KP hierarchy”, Theoret. and Math. Phys., 199:2 (2019), 637–651
Helminck G.F., Poberezhny V.A., Polenkova S.V., “A Geometric Construction of Solutions of the Strict Dkp(Lambda(0)) Hierarchy”, J. Geom. Phys., 131 (2018), 189–203
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