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Scientific articles
Homogeneous spaces yielding solutions of the k[S]k[S]-hierarchy and its strict version
G. F. Helmincka, J. A. Weeninkb a Korteweg-de Vries Institute, University of Amsterdam
b Bernoulli Institute, University of Groningen
Abstract:
The k[S]k[S]-hierarchy and its strict version are two deformations of the commutative algebra k[S]k[S], k=R or C, in the N×N-matrices, where S is the matrix of the shift operator.
In this paper we show first of all that both deformations correspond to conjugating k[S] with elements from an appropriate group. The dressing matrix of the deformation is unique in the case of the k[S]-hierarchy and it is determined up to a multiple of the identity in the strict case. This uniqueness enables one to prove directly the equivalence of the Lax form of the k[S]-hierarchy with a set of Sato–Wilson equations. The analogue of the Sato–Wilson equations for the strict k[S]-hierarchy always implies the Lax equations of this hierarchy. Both systems are equivalent if the setting one works in, is Cauchy solvable in dimension one.
Finally we present
a Banach Lie group G(S2), two subgroups P+(H) and U+(H) of G(S2), with U+(H)⊂P+(H), such that one can construct from
the homogeneous spaces G(S2)/P+(H) resp. G(S2)/U+(H)
solutions of respectively the k[S]-hierarchy
and its strict version.
Keywords:
homogeneous spaces, integrable hierarchies, Lax equations, Sato-Wilson form, wave matrices.
Received: 17.06.2021
Citation:
G. F. Helminck, J. A. Weenink, “Homogeneous spaces yielding solutions of the k[S]-hierarchy and its strict version”, Russian Universities Reports. Mathematics, 26:135 (2021), 315–336
Linking options:
https://www.mathnet.ru/eng/vtamu234 https://www.mathnet.ru/eng/vtamu/v26/i135/p315
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Abstract page: | 124 | Full-text PDF : | 36 | References: | 31 |
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