Geodesic
A geometric construction of solutions of the strict $\mathbf h$-hierarchy
Teoretičeskaâ i matematičeskaâ fizika, Tome 200 (2019) no. 1, pp. 72-95 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $\mathbf{h}$ be a complex commutative subalgebra of the $n{\times}n$ matrices $M_n(\mathbb{C})$. In the algebra MPsd of matrix pseudodifferential operators in the derivation $\partial$, we previously considered deformations of $\mathbf{h}[\partial]$ and of its Lie subalgebra $\mathbf{h}[\partial]_{>0}$ consisting of elements without a constant term. It turned out that the different evolution equations for the generators of these two deformed Lie algebras are compatible sets of Lax equations and determine the corresponding $\mathbf{h}$-hierarchy and its strict version. Here, with each hierarchy, we associate an $MPsd$-module representing perturbations of a vector related to the trivial solution of each hierarchy. In each module, we describe so-called matrix wave functions, which lead directly to solutions of their Lax equations. We next present a connection between the matrix wave functions of the $\mathbf{h}$-hierarchy and those of its strict version; this connection is used to construct solutions of the latter. The geometric data used to construct the wave functions of the strict $\mathbf{h}$-hierarchy are a plane in the Grassmanian $Gr(H)$, a set of $n$ linearly independent vectors $\{w_i\}$ in $W$, and suitable invertible maps $\delta\colon S^1\to\mathbf{h}$, where $S^1$ is the unit circle in $\mathbb{C}^*$. In particular, we show that the action of a corresponding flow group can be lifted from $W$ to the other data and that this lift leaves the constructed solutions of the strict $\mathbf{h}$-hierarchy invariant. For $n>1$, it can happen that we have different solutions of the strict $\mathbf{h}$-hierarchy for fixed $W$ and $\{w_i\}$. We show that they are related by conjugation with invertible matrix differential operators.
Keywords: matrix pseudodifferential operator, strict $\mathbf{h}$-hierarchy, linearization, matrix wave function.
Mots-clés : Lax equation 
@article{TMF_2019_200_1_a3,
     author = {G. F. Helminck},
     title = {A~geometric construction of solutions of the~strict $\mathbf h$-hierarchy},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {72--95},
     year = {2019},
     volume = {200},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2019_200_1_a3/}
}
TY  - JOUR
AU  - G. F. Helminck
TI  - A geometric construction of solutions of the strict $\mathbf h$-hierarchy
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2019
SP  - 72
EP  - 95
VL  - 200
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2019_200_1_a3/
LA  - ru
ID  - TMF_2019_200_1_a3
ER  - 
%0 Journal Article
%A G. F. Helminck
%T A geometric construction of solutions of the strict $\mathbf h$-hierarchy
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2019
%P 72-95
%V 200
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2019_200_1_a3/
%G ru
%F TMF_2019_200_1_a3
G. F. Helminck. A geometric construction of solutions of the strict $\mathbf h$-hierarchy. Teoretičeskaâ i matematičeskaâ fizika, Tome 200 (2019) no. 1, pp. 72-95. http://geodesic.mathdoc.fr/item/TMF_2019_200_1_a3/

[1] G. F. Helminck, “Integrable deformations in the matrix pseudo differential operators”, J. Geom. Phys., 113 (2017), 104–116 | DOI | MR

[2] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, “Operator approach to the Kadomtsev–Petviashvili equation – Transformation Groups for Soliton Equations III –”, J. Phys. Soc. Japan, 50:11 (1981), 3806–3812 | DOI | MR

[3] G. Wilson, “Commuting flows and conservation laws for Lax equations”, Math. Proc. Cambridge Philos. Soc., 86:1 (1979), 131–143 | DOI | MR

[4] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, “Transformation groups for soliton equations”, Nonlinear Integrable Systems – Classical Theory and Quantum Theory (Kyoto, Japan, May 13–16, 1981), eds. M. Jimbo, T. Miwa, World Sci., Singapore, 1983, 39–119 | MR

[5] G. Segal, G. Wilson, “Loop groups and equations of KdV type”, Publ. Math. IHES, 61 (1985), 5–65 | DOI | MR

[6] A. Pressli, G. Sigal, Gruppy petel, Mir, M., 1990 | MR

[7] M. J. Bergvelt, A. P. E. ten Kroode, “Differential-difference AKNS equations and homogeneous Heisenberg algebras”, J. Math. Phys., 28:2 (1987), 302–306 | DOI | MR

[8] G. F. Helminck, J. W. van de Leur, “Darboux transformations for the KP-hierarchy in the Segal–Wilson setting”, Canad. J. Math., 53:2 (2001), 278–309 | DOI | MR