Bargmann type finite-dimensional reductions of the Lax integrable supersymmetric Boussinesq hierarchy and their integrability

For the supersymmetric Boussinesq hierarchy, related with the Lax type flows on the space dual to the Lie algebra of superintegro-differential operators of one anticommuting variable for some non-self-adjoint superdifferential operator, the method of the Bargmann type finite-dimensional reductions i...

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Datum:	2015
1. Verfasser:	Hentosh, O.E.
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spelling	irk-123456789-1772272021-02-13T01:25:55Z Bargmann type finite-dimensional reductions of the Lax integrable supersymmetric Boussinesq hierarchy and their integrability Hentosh, O.E. For the supersymmetric Boussinesq hierarchy, related with the Lax type flows on the space dual to the Lie algebra of superintegro-differential operators of one anticommuting variable for some non-self-adjoint superdifferential operator, the method of the Bargmann type finite-dimensional reductions is developed. We prove existence of an even exact supersymplectic structure on the corresponding invariant finite-dimensional supersubspace of the supersymmetric Boussinesq hierarchy as well as the Lax – Liouville integrability of commuting vector fields, generated by the hierarchy and reduced to this supersubspace. Для суперсиметричної iєрархiї Буссiнеска, пов’язаної з потоками типу Лакса на спряженому просторi до алгебри Лi суперiнтегро-диференцiальних операторiв однiєї антикомутативної змiнної для несамоспряженого супердиференцiального оператора, розвинено метод скiнченновимiрних редукцiй типу Баргмана. Доведено iснування парної точної суперсимплектичної структури на вiдповiдному iнварiантному скiнченновимiрному суперпiдпросторi суперсиметричної iєрархiї Буссiнеска та iнтегровнiсть за Лаксом – Лiувiллем редукованих на цей суперпiдпростiр комутуючих векторних полiв, породжених iєрархiєю. 2015 Article Bargmann type finite-dimensional reductions of the Lax integrable supersymmetric Boussinesq hierarchy and their integrability / O.E. Hentosh // Нелінійні коливання. — 2015. — Т. 18, № 4. — С. 454-474 — Бібліогр.: 30 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177227 517.9 en Нелінійні коливання Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	For the supersymmetric Boussinesq hierarchy, related with the Lax type flows on the space dual to the Lie algebra of superintegro-differential operators of one anticommuting variable for some non-self-adjoint superdifferential operator, the method of the Bargmann type finite-dimensional reductions is developed. We prove existence of an even exact supersymplectic structure on the corresponding invariant finite-dimensional supersubspace of the supersymmetric Boussinesq hierarchy as well as the Lax – Liouville integrability of commuting vector fields, generated by the hierarchy and reduced to this supersubspace.
format	Article
author	Hentosh, O.E.
spellingShingle	Hentosh, O.E. Bargmann type finite-dimensional reductions of the Lax integrable supersymmetric Boussinesq hierarchy and their integrability Нелінійні коливання
author_facet	Hentosh, O.E.
author_sort	Hentosh, O.E.
title	Bargmann type finite-dimensional reductions of the Lax integrable supersymmetric Boussinesq hierarchy and their integrability
title_short	Bargmann type finite-dimensional reductions of the Lax integrable supersymmetric Boussinesq hierarchy and their integrability
title_full	Bargmann type finite-dimensional reductions of the Lax integrable supersymmetric Boussinesq hierarchy and their integrability
title_fullStr	Bargmann type finite-dimensional reductions of the Lax integrable supersymmetric Boussinesq hierarchy and their integrability
title_full_unstemmed	Bargmann type finite-dimensional reductions of the Lax integrable supersymmetric Boussinesq hierarchy and their integrability
title_sort	bargmann type finite-dimensional reductions of the lax integrable supersymmetric boussinesq hierarchy and their integrability
publisher	Інститут математики НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/177227
citation_txt	Bargmann type finite-dimensional reductions of the Lax integrable supersymmetric Boussinesq hierarchy and their integrability / O.E. Hentosh // Нелінійні коливання. — 2015. — Т. 18, № 4. — С. 454-474 — Бібліогр.: 30 назв. — англ.
series	Нелінійні коливання
work_keys_str_mv	AT hentoshoe bargmanntypefinitedimensionalreductionsofthelaxintegrablesupersymmetricboussinesqhierarchyandtheirintegrability
first_indexed	2025-07-15T15:15:52Z
last_indexed	2025-07-15T15:15:52Z
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fulltext	УДК 517.9 BARGMANN TYPE FINITE-DIMENSIONAL REDUCTIONS OF THE LAX INTEGRABLE SUPERSYMMETRIC BOUSSINESQ HIERARCHY AND THEIR INTEGRABILITY СКIНЧЕННОВИМIРНI РЕДУКЦIЇ ТИПУ БАРГМАНА IНТЕГРОВНОЇ ЗА ЛАКСОМ СУПЕРСИМЕТРИЧНОЇ IЄРАРХIЇ БУССIНЕСКА ТА ЇХ IНТЕГРОВНIСТЬ O. Ye. Hentosh Pidstryhach Inst. Appl. Probl. Mech. and Math. Nat. Acad. Sci. Ukraine Naukova Str., 3B, Lviv, 79060, Ukraine e-mail: ohen@ua.fm For the supersymmetric Boussinesq hierarchy, related with the Lax type flows on the space dual to the Lie algebra of superintegro-differential operators of one anticommuting variable for some non-self-adjoint superdifferential operator, the method of the Bargmann type finite-dimensional reductions is developed. We prove existence of an even exact supersymplectic structure on the corresponding invariant finite-dimen- sional supersubspace of the supersymmetric Boussinesq hierarchy as well as the Lax – Liouville integrabi- lity of commuting vector fields, generated by the hierarchy and reduced to this supersubspace. Для суперсиметричної iєрархiї Буссiнеска, пов’язаної з потоками типу Лакса на спряженому просторi до алгебри Лi суперiнтегро-диференцiальних операторiв однiєї антикомутативної змiнної для несамоспряженого супердиференцiального оператора, розвинено метод скiнченно- вимiрних редукцiй типу Баргмана. Доведено iснування парної точної суперсимплектичної струк- тури на вiдповiдному iнварiантному скiнченновимiрному суперпiдпросторi суперсиметричної iєрархiї Буссiнеска та iнтегровнiсть за Лаксом – Лiувiллем редукованих на цей суперпiдпростiр комутуючих векторних полiв, породжених iєрархiєю. 1. Introduction. In the framework of the different Lie-algebraic approaches, a wide class of supersymmetric nonlinear dynamical systems, possessing matrix Lax type representations [1 – 4] and infinite sequences of local conservation laws, has been constructed in [5 – 10] and many others. For such nonlinear dynamical systems, defined on suitable functional manifolds, a me- thod of reducing the system to the invariant subspaces, generated by critical points of the related conservation laws has been developed in [4, 11 – 14]. In particular, in [4, 12, 13] it has been shown that the exact symplectic structure on the corresponding invariant space can be obtained by means of the Gelfand – Dikii relationship [15, 16] for the differential of the Lagrangian functi- onal on a suitably extended phase space [11], and the corresponding Hamiltonian functions of the reduced vector fields generated by the systems have been constructed. In [13, 17, 18] the reduction method has been further developed for superconformal nonli- near dynamical systems as well as for supersymmetric ones defined on supermanifolds of one commuting and one anticommuting independent variables. In particular, in the paper [18] this method has been used for investigating Neumann type invariant reductions [19] of the Laberge – Mathieu supersymmetric hierarchy, related to the Lax type flows on the space dual to the Lie algebra of superintegro-differential operators of two anticommuting variables for some self- adjoint superdifferential operator. c© O. Ye. Hentosh, 2015 454 ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 BARGMANN TYPE FINITE-DIMENSIONAL REDUCTIONS . . . 455 In this article, the reduction method is applied to the supersymmetric Boussinesq hierarchy [6] associated with a non-self-adjoint superdifferential operator depending on one anticommu- ting variable. The second section contains a preliminary description of Lie-algebraic and differential- geometric properties, being important for a better understanding of the used techniques and the obtained results. In the third section, we establish existence of an even exact supersymplectic structure on the invariant supersubspace determined by the Bargmann type constraints [14] by means of the superanalog of the Gelfand – Dikii relationship [18, 20] and the Hamiltonian functions for the reduced commuting vector fields, generated by the hierarchy. In the fourth section making use of the differential-geometric properties of the supertrace gradient for the monodromy supermatrix of the related periodic matrix linear spectral problem, we obtain the Lax representations for these reduced vector fields. The algorithm for redu- cing of monodromy supermatrix upon the invariant supersubspace is described. A complete set of functionally independent conservation laws, being involutive with respect to the Poisson bracket related with the obtained even supersymplectic structure, is also found. It ensures the complete Liouville integrability [21] of the reduced vector fields. 2. The Lax integrability of the supersymmetric Boussinesq hierarchy. The supersymmetric Boussinesq hierarchy [6] can be represented in the form of the Lax type flows dl dtj = [ (l(3j+1)/3)+, l ] , dl dt̃j = [ (l(3j+2)/3)+, l ] , (1) where l = ∂3 + φDθ∂ + a∂ − χDθ − b ∈ G∗, w = (a, b, φ, χ)> ∈ M2\|2 ⊂ C∞(S1\|1;R2\|2), (x, θ) ∈ S1\|1, S1\|1 ' S × Λ1, S ' R/2πZ, Λ1 is a subalgebra of anticommuting elements of the Grassmann algebra Λ := Λ0 ⊕ Λ1 over the field R ⊂ Λ0, R2\|2 := Λ2 0 × Λ2 1, ∂ := ∂ ∂x , Dθ := ∂ ∂θ + θ ∂ ∂x is a superderivative and tj , t̃j ∈ R, j ∈ Z+ are evolution parameters. Here the lower index "+" denotes the pure differential part of a superintegro-differential operator from the space G∗ ' G being the dual space to the Lie algebra G of superintegro-differential operators A := ∂q + ∑ p<2q−1 apD p θ ∈ G, p ∈ Z, q ∈ N, with coefficients ap := ap(x, θ) = a0p(x) + θa1p(x), ap ∈ C∞(S1\|1; Λ0) if p = 2r and ap := := ap(x, θ) = a1p(x) + θa0p(x), ap ∈ C∞(S1\|1; Λ1) if p = 2r − 1, r < q, q ∈ N, subject to the scalar product (A,B) := 2π∫ 0 dx ∫ dθ resAB, A,B ∈ G, where the symbol "res" designates the coefficient at the operator D−1θ . The evolution with respect to the parameter t̃1 is given by the supersymmetric nonlinear dynamical system [6] such ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 456 O. YE. HENTOSH as da dt̃1 = 1 3 (4φχ+ 2φφx − 6bx − 3axx), db dt̃1 = 1 3 (2(Dθa)χ+ 2φ(Dθb) + 2aax − 2(Dθax)φ+ 3bxx + 2axxx), dφ dt̃1 = −2χx − φxx, dχ dt̃1 = 1 3 (2φ(Dθχ)− 2(Dθφ)χ+ 2(aφ)x + 2φ(Dθφx) + 3χxx + 2φxxx), which entails the Boussinesq system [22] at a = b = 0, φ = θu, χ = θv and (u, v)> ∈ ∈ C∞(S;R2). The evolution equations (1) can be considered as a compatibility condition for the spectral relationship ly = λy, (2) where λ ∈ Λ0 ⊃ C is a spectral parameter, being invariant with respect to the evolution flows (1), y ∈ L2(S1\|1;C1\|0), and the evolution equations dy dtj = (l(3j+1)/3)+y, and dy dt̃j = (l(3j+2)/3)+y. The corresponding adjoint spectral relationship and the adjoint evolutions take the form l∗z = λz, dz dtj = −(l(3j+1)/3)∗+z, (3) dz dt̃j = −(l(3j+2)/3)∗+z, where l∗ = −∂3 −Dθ∂φ − ∂a −Dθχ − b is the operator adjoint to l and defined by means of the integral relationship 2π∫ 0 dx ∫ dθz(ly) = 2π∫ 0 dx ∫ dθ(l∗z)y, for all y ∈ L2(S1\|1;C1\|0) and z ∈ L2(S1\|1;C0\|1). ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 BARGMANN TYPE FINITE-DIMENSIONAL REDUCTIONS . . . 457 The mentioned above spectral problems can be suitably rewritten in the equivalent matrix forms [6], DθY = AY, (4) DθZ = −A>sZ, (5) where A ∈ C∞(S1\|1; gl (3\|3)), A := A[w;λ], Y := Y (x, θ;λ) ∈ W := L2(S1\|1;C3\|3), Z := := Z(x, θ;λ) ∈ W, Y = (y0, y2, y4, y1, y3, y5) >, y0 := y, Z = (z0, z2, z4, z1, z3, z5) >, z5 := z and A :=  0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 −1 0 0 0 0 0 0 −1 0 0 0 b+ λ a 0 χ φ 0  . Here the upper index ">s" denotes the supermatrix supertransposition acting by the rule W>s = ( W11 W12 W21 W22 )>s = ( W>11 W>21 −W>12 W>22 ) for any supermatrix W ∈ gl(m\|n). The associated evolutions are written as dY dtj = BjY = (λjS)+Y, (6) dY dt̃j = B̃jY = (λjS2)+Y, (7) and dZ dtj = −IB>s j IZ, (8) dZ dt̃j = −IB̃>s j IZ, (9) where I := diag (1, 1, 1,−1,−1,−1), Bj := Bj [w;λ]Y = (λjS)+Y, B̃j := B̃j [w;λ]Y = = (λjS2)+Y, S ' ∑ j∈Z+ Šj−1λ −j+1 is an asymptotical expansion of the monodromy super- matrix S(x, θ;λ) := Y (x, x+ 2π, θ;λ) for the periodic matrix spectral problem (4) as \|λ\| → ∞, Y (x̆, x, θ;λ) is a fundamental solution of the linear equation (4), that is, Y (x, x, θ;λ) = 1, x̆ ∈ S, 1 ∈ gl (6) is the unit (6× 6)-matrix (see [18]), Š−1 =  0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0  , ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 458 O. YE. HENTOSH the lower index "+" denotes the polynomial part of the corresponding Laurent series. The hierarchy (1) possesses two sequences of the Casimir type conservation laws γj := 2π∫ 0 dx ∫ dθγj [w] = 3 3j + 1 2π∫ 0 dx ∫ dθres l(3j+1)/3, γ̃j := 2π∫ 0 dx ∫ dθ γ̃j [w] = 3 3j + 2 2π∫ 0 dx ∫ dθres l(3j+2)/3, (10) where some four of them are γ0 = − 2π∫ 0 dx ∫ dθφ, γ̃0 = 2π∫ 0 dx ∫ dθχ, γ1 = 1 3 2π∫ 0 dx ∫ dθ(aχ+ bφ+ φ(Dθχ)), γ̃1 = 1 27 2π∫ 0 dx ∫ dθ(−18bχ+ 9bxφ+ 3a2φ+ 9aχx + 6aφxx+ + 3aφ(Dθφ) + φ(Dθφ)2 + 9φ(Dθχx)− 9χ(Dθχ) + 3φ(Dθφxx), etc. These conservation laws are connected to each other by means of the Magri [23] relationships, Mϕ(x, θ; λ̄) = λ̄Lϕ(x, θ; λ̄), (11) M ϕ̃(x, θ; λ̄) = λ̄L ϕ̃(x, θ; λ̄), (12) where ϕ(x, θ; λ̄) = grad strS(x, θ;λ), ϕ̃(x, θ; λ̄) = grad strS2(x, θ;λ), ϕ(x, θ; λ̄) ' ∑ j∈Z+ ϕj λ̄ −j , ϕj = grad γj [w], ϕ̃(x, θ; λ̄) ' ∑ j∈Z+ ϕ̃j λ̄ −j , ϕ̃j = grad γ̃j [w], and L : T (M2\|2) → T ∗(M2\|2) and M : T (M2\|2) → T ∗(M2\|2), are a pair of compatible linear Poisson operators [23], constructed before in [6]. Here the symbol "grad" denotes, as usually, the left gradient of the corresponding functional. The operators L and M generate a bi-Hamiltonian representation for the hierarchy (1) in the form dw dtj = −L grad γj+1 = −M grad γj , dw dt̃j = −L grad γ̃j+1 = −M grad γ̃j , (13) ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 BARGMANN TYPE FINITE-DIMENSIONAL REDUCTIONS . . . 459 where L =  0 2φ+ 3Dθ∂ 0 −3∂ 2φ+ 3Dθ∂ 2χ+ (Dθa) + φx −3∂ φDθ 0 −3∂ 0 0 −3∂ −(Dθφ) + φDθ 0 0  and M has a more cumbersome form (see [6]). The vector fields d dtj1 and d dtj2 , j1, j2 ∈ Z+, commute with each other, i.e.,[ d dtj1 , d dtj2 ] = 0, [ d dtj1 , d dt̃j2 ] = 0, [ d dt̃j1 , d dt̃j2 ] = 0. (14) The existence of conservation laws (10) and matrix Lax type linearizations (4), (6), (7) and (5), (8), (9) allow us to reduce the hierarchy (1) to its invariant supersubspaces, M 2\|2 N = {w ∈ M2\|2 : gradLN [w] = 0}, generated by the Lagrangian functionals LN := 2π∫ 0 dx ∫ dθLN [w] = P∑ k1=0 ak1γk1 + Q∑ k2=0 bk2 γ̃k2 + N∑ i=1 ciλi, where ak1 , bk2 , ci ∈ Λ0 ⊃ C are some coefficients and λi ∈ Λ0 ⊃ C, i = 1, N, are different eigenvalues of the periodic spectral problem (4) for arbitrarily chosen orders P, Q, N ∈ Z+. 3. The supersymplectic structure on some invariant supersubspace. Below we shall study the reductions of the vector fields d dtj and d dt̃j , j ∈ Z+, on the invariant supersubspace determined by the Bargmann type constraints [14] M 2\|2 N = { w ∈ M2\|2 : gradLN [w] = 0 } , LN := 2π∫ 0 dx ∫ dθLN [w] = −3γ1 + N∑ i=1 ciλi, (15) where λi ∈ Λ0 ⊃ C, are some different eigenvalues of the periodic spectral problem (2), being considered as smooth by Frechet functionals on M2\|2, i.e., λi ∈ D(M2\|2), with the corresponding eigenvectors Yi = (y0i, y2i, y4i, y1i, y3i, y5i) > ∈ W and adjoint eigenvectors Zi = (z0i, z2i, z4i, z1i, z3i, z5i) > ∈ W, ci ∈ Λ0 ⊃ C, i = 1, N. First, we shall analyze the differential-geometric structure of the invariant supersubspace M 2\|2 N ⊂ M2\|2. To describe this supersubspace, evidently we shall construct the gradients of the eigenvalues λi ∈ D(M2\|2), i = 1, N, making use the relationships 2π∫ 0 dx ∫ dθ〈DθYi, Z̄i〉 = 2π∫ 0 dx ∫ dθ〈A[w, λi]Yi, Z̄i〉, i = 1, N, (16) ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 460 O. YE. HENTOSH where the brackets 〈 , 〉 denotes the standard scalar product on C6N \|6N and Z̄i = (z̄0i, z̄2i, z̄4i, z̄1i, z̄3i, z̄5i) > is complex conjugate to the vector Zi, which follows from the spectral problem (4). These gradients are written as gradλi = − 1 µi (ȳ2iz̄5i, ȳ0iz̄5i, ȳ3iz̄5i, ȳ1iz̄5i) > , where Ȳi = (ȳ0i, ȳ2i, ȳ4i, ȳ1i, ȳ3i, ȳ5i) > is complex conjugate to the vector Yi, µi := 2π∫ 0 ∫ dθ y0iz5i, i = 1, N, are normalizing multipliers, being invariant with respect to the vector fields d dtj and d dt̃j for all j ∈ Z+. In case of µi = −ci, i = 1, N, the condition (15) takes the form of the Bargmann type constraints, M 2\|2 N ⋂ Hc = { w ∈ M2\|2 : a = − N∑ i=1 y0iz4i, b = − N∑ i=1 y2iz4i, φ = N∑ i=1 y0iz5i, χ= N∑ i=1 y2iz5i } , (17) where Hc := {(w,Y,Z)> ∈ M̂2\|2 : µi = −ci, ci ∈ Λ0, i = 1, N} are common level surfaces of the invariant functionals µi, i = 1, N, in the phase space M̂2\|2 := M2\|2×W 2N of the hierarchies of coupled dynamical systems (13), (6), (7) and (8), (9) with the parameters λi, i = 1, N, and Y := (Y1, Y2, . . . , YN )>, Z := (Z1, Z2, . . . , ZN )>. From the relationships (17) it follows that the solutions to the supersymmetric Boussinesq hierarchy on the invariant supersubspace (17) are expressed by means of the coordinates of the eigenvectors Yi and Zi, i = 1, N. The exact supersymplectic structure on the invariant supersubspace M2\|2 N ⊂ M2\|2 can be obtained by means of the analog [18, 20] of the Gelfand – Dikii relationship on the functional supermanifold M2\|2 similarly as it was done in the paper [16] for subspaces of critical points of local conservation laws. To make use this relationship we need the evident forms of the Frechet smooth functionals λi, i = 1, N, on Hc. From the equalities (16) we have λ′i = 2π∫ 0 dx ∫ dθ ( − 5∑ s=0 (Dθysi)zsi + y1iz0i + y3iz2i + y5iz4i− − y2iz1i − y4iz3i + by0iz5i + ay2iz5i + χy1iz5i + φy3iz5i ) , (18) ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 BARGMANN TYPE FINITE-DIMENSIONAL REDUCTIONS . . . 461 where λ′i := λi\|M̂2\|2 ⋂Hc , i = 1, N, on the level surfaces Hc, c := (c1, c2, . . . , cN )> ∈ ΛN0 , in the phase space M̂2\|2. Therefore, taking into account the evident dependence (18) of λ′i ∈ D(M2\|2), i = 1, N, on the functions (w,Y,Z)> ∈ M̂2\|2 on the level surfacesHc, c ∈ ΛN0 ,we can apply an analog of the Gelfand – Dikii relationship to the Lagrangian functional L̂N := ∫ 2π 0 dx ∫ dθL̂N [w,Y,Z] ∈ ∈ D(M̂2\|2) such as L̂N = −3γ1 + N∑ i=1 λ′i + N∑ i=1 siµi, where si ∈ Λ0 ⊃ C, i = 1, N, are Lagrangian multipliers. Owing to the well known Lax theorem [1, 4], the condition grad L̂N [w,Y,Z] = 0 determi- nes an invariant supersubspace M̃2\|2 N ⊂ M̂2\|2 of the hierarchy of the coupled dynamical systems (13), (6), (7) and (8), (9) with the parameters λi, i = 1, N, such as M̃ 2\|2 N = { (w,Y,Z)> ∈ M2\|2 : a = − N∑ i=1 y0iz4i, b = − N∑ i=1 y2iz4i, φ = N∑ i=1 y0iz5i, χ = N∑ i=1 y2iz5i, DθYi = A[w; si]Yi, DθZi = −A>s [w; si]Zi, i = 1, N } . Thus, the supersubspace M2\|2 N ⋂ Hc ⊂ M2\|2 is diffeomorphic to the supersubspace M̃2\|2 N ⊂ ⊂ M̂2\|2 if si = λi, i = 1, N, for every c ∈ ΛN0 . By means of the analog of Gelfand – Dikii differential relationship [18, 20] for the Lagrangi- an functional L̂N ∈ D(M̂2\|2), dL̂N [w,Y,Z] = 〈 (dw, dY, dZ)>, grad L̂N [w,Y,Z] 〉 +Dθα (1), (19) where (φ, χ,Y,Z)> are coordinates on a suitably truncated functional supermanifold M̂2\|2 N ⊂ ⊂ M̂2\|2, "d" is a symbol of the exterior differentiation in the Grassmann algebra of differential forms on C(6N+2)\|(6N+2) and the brackets 〈 , 〉 denotes the standard scalar product on C(6N+2)\|(6N+2), we can construct the even exact two-form ω̂(2) = dα(1), ω̂(2) = N∑ i=1 5∑ s=0 ysi ∧ zsi + dφ ∧ dχ, (20) where "∧" is a symbol of the exterior product on the Grassmann algebra of differential forms on C(6N+2)\|(6N+2). The reduced two-form ω(2) := ω̂(2) ∣∣ M̃ 2\|2 N defines a supersymplectic structure on the supersubspace M2\|2 N ⋂ Hc ' M̃ 2\|2 N ⊂ M̂ 2\|2 N , which is smoothly embedded in the superspace M̂ 2\|2 N owing to the relationships (17). ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 462 O. YE. HENTOSH The expression (19) ensures the invariance of the reduced two-form ω(2) with respect to the superdifferentiation Dθ, that is, Dθω (2) = 0. Since D2 θ = d dx , the two-form ω(2) is also invariant with respect to the vector field d dx on M̂2\|2 N . Taking into account that the supersubspace M 2\|2 N ⋂ Hc ⊂ M2\|2 is diffeomorphic to the finite-dimensional supersubmanifold MF ⊂ R6N \|(6N+2), determined by the constraints F1 := φ− N∑ i=1 y0iz5i = 0, F2 := χ− N∑ i=1 y2iz5i, (21) in the superspace R6N \|(6N+2), we can obtain a supersymplectic structure on M 2\|2 N ⋂ Hc as a natural Dirac type reduction of the two-form ω̂(2) on MF . The two-form ω̂(2) generates the Poisson bracket on the superspace R6N \|(6N+2), {F,G}ω̂(2) = N∑ i=1 ∑ s=0,2,4 ( ∂F ∂zsi ∂G ∂ysi − ∂F ∂ysi ∂G ∂zsi ) − − N∑ i=1 ∑ s=1,3,5 ( ∂rF ∂ysi ∂lG ∂zsi + ∂rF ∂zsi ∂lG ∂ysi ) + ∂rF ∂φ ∂lG ∂χ + ∂rF ∂χ ∂lG ∂φ , (22) where ∂l ∂ζ and ∂r ∂ζ denote operators of the left and the right derivatives with respect to the anticommuting variable ζ ∈ Λ1, for arbitrary smooth functions F ∈ C∞(R6N \|(6N+2);R1\|0) or C∞(R6N \|(6N+2); R0\|1) and G ∈ C∞(R6N \|(6N+2);R1\|0) or C∞(R6N \|(6N+2);R0\|1). Since the matrix of constraints {Fκ1 , Fκ2}ω̂(2) , κ1, κ2 = 1, 2, is nondegenerate, the standard Dirac type reduction procedure [4, 25] entails the following Poisson bracket: {F,G} ω (2) F = {F,G}ω̂(2) − {F, F1}ω̂(2){F2, G}ω̂(2) − {F, F2}ω̂(2){F1, G}ω̂(2) = {F,G}ω̂(2)− − ( N∑ i1=1 ( − ∂F ∂z0i1 z5i1 − ∂rF ∂y5i1 y0i1 ) + ∂rF ∂χ ) × × ( N∑ i2=1 ( z5i2 ∂G ∂z2i2 − y2i2 ∂lG ∂y5i2 ) + ∂lG ∂φ ) − − ( N∑ i1=1 ( − ∂F ∂z2i1 z5i1 − ∂rF ∂y5i1 y2i1 ) + ∂rF ∂φ ) × × ( N∑ i2=1 ( z5i2 ∂G ∂z0i2 − y0i2 ∂lG ∂y5i2 ) + ∂lG ∂χ ) , (23) ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 BARGMANN TYPE FINITE-DIMENSIONAL REDUCTIONS . . . 463 related with the supersymplectic structure ω(2) F := ω(2) on MF ' M 2\|2 N . From the equalities dLN dtj = 0, dLN dt̃j = 0, j ∈ Z+, it follows that there exist functions ĥ(tj), ĥ(t̃j) ∈ D(M̂2\|2), obeying the relationships〈( dw dtj , dY dx , dZ dx )> , grad L̂N [w,Y,Z] 〉 = Dθĥ (tj), 〈( dw dt̃j , dY dx , dZ dx )> , grad L̂N [w,Y,Z] 〉 = Dθĥ (t̃j), (24) where the brackets 〈 , 〉 denotes the standard scalar product on C(6N+2)\|(6N+2). The functions ĥ(tj) and ĥ(t̃j) on M̃2\|2 N satisfy the following equalities: i d dtj ω̂(2) = −dĥ(tj), i d dt̃j ω̂(2) = −dĥ(t̃j), j ∈ Z+, (25) where i d dtj , i d dt̃j are inner differentiations with respect to the vector fields d dtj : M̃ 2\|2 N → → T (M̃ 2\|2 N ) and d dt̃j : M̃ 2\|2 N → T (M̃ 2\|2 N ), j ∈ Z+, in the Grassmann algebra of differential forms on C(6N+2)\|(6N+2). To state the first equality in (25) we need to calculate the expressions i d dtj 〈 (dw, dY, dZ)> , grad L̂N [w,Y,Z] 〉 = Dθĥ (tj). Then we have di d dtj 〈 (dw, dY, dZ)> , grad L̂N [w,Y,Z] 〉 = −Dθdĥ (tj). From (19) we easily obtain that the identities d 〈 (dw, dY, dZ)> , grad L̂N [w,Y,Z] 〉 = Dθdα (1) and i d dtj d 〈 (dw, dY, dZ)> , grad L̂N [w,Y,Z] 〉 = i d dtj Dθω̂ (2) (26) hold owing to the relations dDθ = −Dθd, i d dtj Dθ = −Dθi d dtj . ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 464 O. YE. HENTOSH Since the Lie derivative with respect to the vector field d dtj : M̃ 2\|2 N → T (M̃ 2\|2 N ) can be repre- sented as d dtj = di d dtj + i d dtj d the resulting relationship d dtj 〈 (dw, dY, dZ)> , grad L̂N [w,Y,Z] 〉 = −Dθ ( dĥ(tj) + i d dtj ω̂(2) ) holds on M̃2\|2.The latter proves the first equality in (25), which takes place on M̃2\|2 N .The second equality in (25) can be proved analogously. The obtained relationships (25) lead to important equalities, d dtj ω(2) = 0, d dt̃j ω(2) = 0, on M2\|2 N ⋂ Hc ' M̃ 2\|2 N . Therefore, the functions h(tj) := ĥ(tj) ∣∣∣ M 2\|2 N ⋂ Hc , h(t̃j) := ĥ(t̃j) ∣∣∣ M 2\|2 N ⋂ Hc are Hamiltonians subject to the vector fields d dtj and d dt̃j on M2\|2 N ⋂ Hc ' MF for all j ∈ Z+ if si = λi, i = 1, N, that is, i d dtj ω (2) F = −dh(tj), i d dt̃j ω (2) F = −dh(t̃j), j ∈ Z+. For example, the Hamiltonian function of the vector field d dx := d dt0 on the supersubspace M 2\|2 N ⋂ Hc ⊂ M2\|2 equals, by definition, h(x) := ĥ(x) ∣∣∣ M 2\|2 N ⋂ Hc , where ĥ(x) = N∑ i=1 ( λi(y0iz4i + y1iz5i)− y2iz0i − y4iz2i − y3iz1i − y5iz3i+ + φ(y4iz5i − y2iz3i)− χ(y2iz5i − y0iz3i)+ + ( N∑ k=1 y0kz5k ) (y3iz4i + y2iz3i) + ( N∑ k=1 y2kz5k ) (y1iz4i + y0iz3i)+ + ( N∑ k=1 y1kz5k ) y3iz5i − ( N∑ k=1 y2kz4k ) y0iz4i ) , h(x) = N∑ i=1 ( λi(y0iz4i + y1iz5i)− y2iz0i − y4iz2i − y3iz1i − y5iz3i+ + ( N∑ k=1 y0kz5k ) (y3iz4i + y4iz5i) + ( N∑ k=1 y2kz5k ) y1iz4i+ ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 BARGMANN TYPE FINITE-DIMENSIONAL REDUCTIONS . . . 465 + ( N∑ k=1 y1kz5k ) y3iz5i − ( N∑ k=1 y2kz4k ) y0iz4i ) . Analogously we can find Hamiltonians of the other vector fields d dtj and d dt̃j , j ∈ Z+, on the supersubspace M2\|2 N ⋂ Hc ⊂ M2\|2. Therefore, the following theorem holds. Theorem 1. The vector fields d dtj and d dt̃j , j ∈ Z+, generated by the Boussinesq hierarchy (1), allow invariant reductions on the finite-dimensional supersubspaces M2\|2 N ⋂ Hc ⊂ M2\|2 for each N ∈ N, which are diffeomorphic to the finite-dimensional supermanifold MF , smoothly embedded into the superspace R6N \|(6N+2) and endowed with the even, reduced via the Dirac scheme, Poisson bracket (23). On these supersubspaces the vector fields d dtj and d dt̃j , j ∈ Z+, generated by the equations (6), (7) and (8), (9) under λ = λi, i = 1, N, are Hamiltonian with respect to the Poisson bracket (23). The corresponding Hamiltonians h(tj), h(t̃j) ∈C∞(R6N \|(6N+2); R1\|0) are reductions onM2\|2 N ⋂ Hc ⊂ M2\|2 of suitably constructed functions ĥ(tj), ĥ(t̃j) ∈ D(M̂2\|2) satisfying the equalities (24). The relationships (17) describe all periodic and quasiperiodic soluti- ons to the Boussinesq hierarchy (1) on the supersubspace M2\|2 N ⋂ Hc. 4. The Lax – Liouville integrability of the reduced commuting vector fields. To state the Liouville integrability of the Hamiltonian vector fields d dtj and d dt̃j , j ∈ Z+, on M̃2\|2 N for allN ∈ ∈ N we need to construct for these flows the related matrix Lax type representations, depending on the spectral parameter λ ∈ C, making use of the reduction procedure for the monodromy supermatrix of the periodic spectral problem (4). We can formulate the following theorem. Theorem 2. On the finite-dimensional supersymplectic superspace M2\|2 N ⋂ Hc, c ∈ ΛN0 , there exist matrix Lax representations dSN dtj = [Bj,N , SN ], (27) dSN dt̃j = [B̃j,N , SN ], (28) for the Hamiltonian vector fields d dtj and d dt̃j , j ∈ Z+, where Bj,N := Bj,N (Y,Z;λ) = B[w;λ]\| M 2\|2 N ⋂ Hc , and B̃j,N := B̃j,N (Y,Z;λ) = B̃[w;λ] ∣∣∣ M 2\|2 N ⋂ Hc are projections of the corresponding supermatrices on M2\|2 N ⋂ Hc and the reduced monodromy ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 466 O. YE. HENTOSH supermatrix SN := SN (Y,Z;λ) = S(x, θ;λ)‖ M 2\|2 N ⋂ Hc equals SN := N∑ i=1 Si λ− λi + S0 + S−1λ = = N∑ i=1 1 λ− λi  y0iz0i y0iz2i y0iz4i −y0iz1i −y0iz3i −y0iz5i y2iz0i y2iz2i y2iz4i −y2iz1i −y2iz3i −y2iz5i y4iz0i y4iz2i y4iz4i −y4iz1i −y4iz3i −y4iz5i y1iz0i y1iz2i y1iz4i −y1iz1i −y1iz3i −y1iz5i y3iz0i y3iz2i y3iz4i −y3iz1i −y3iz3i −y3iz5i y0iz0i y5iz2i y5iz4i −y5iz1i −y5iz3i −y5iz5i + +  0 3 0 0 0 0 a 0 3 φ 0 0 −b+ N∑ `=1 y0`z2` −2a 0 −χ− N∑ `=1 y0`z3` −2φ 0 φ 0 0 0 3 0 χ+ (Dθa) φ 0 a+ (Dθφ) 0 3 −2(Dθb)− (Dθax)+ −χ− (Dθa)+ −2φ −b− (Dθχ)− −2a− 2(Dθφ) 0 + N∑ `=1 y4`z5` + N∑ `=1 y1`z4` − N∑ `=1 y1`z3`  + +  0 0 0 0 0 0 0 0 0 0 0 0 −3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −3 0 0 λ, (29) with the functions a, b, φ, χ given by the expressions (17) and the differential relationships Dθa = − N∑ i=1 y1iz4i − N∑ i=1 y0iz3i, Dθb = − N∑ i=1 y3iz4` − N∑ i=1 y2iz3i, Dθφ = N∑ i=1 y1iz5i − a, Dθχ = N∑ i=1 y3iz5i − b, Dθax = N∑ i=1 y3iz4` + N∑ i=1 y2iz3i − N∑ i=1 y1iz2i − N∑ i=1 y0iz1i + aφ, Proof. Making use of the spectral problem (4) we can express the gradient ϕ(x, θ; λ̄) of the ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 BARGMANN TYPE FINITE-DIMENSIONAL REDUCTIONS . . . 467 supertrace of the corresponding monodromy supermatrix S :=  S11 S12 S13 S14 S15 S16 S21 S22 S23 S24 S25 S26 S31 S32 S33 S34 S35 S36 S41 S42 S43 S44 S45 S46 S51 S52 S53 S54 S55 S56 S61 S62 S63 S64 S65 S66  (30) by means of its elements ϕ(x, θ; λ̄) =  str (IS̄IAa) str (IS̄IAb) str (S̄Aφ) str (S̄Aχ)  = −  S̄26 S̄16 S̄56 S̄46  , where S̄ is a supermatrix with the elements being complex conjugate to the corresponding ones of S. Using the equality ϕ(x, θ; λ̄i) = ( d dλ strS(x, θ;λ) ∣∣∣∣ λ=λi ) gradλi, we can obtain, in particular, the Magri type relationship [23] M gradλi = λ̄i L gradλi, for all i = 1, N as well as expansions of the elements S16, S26, S46 and S56 of the reduced on M 2\|2 N monodromy supermatrix SN at their poles. Another elements of the supermatrix SN can be extracted from the supergeneralization of Novikov – Lax [2, 3, 18] monodromy supermatrix equation DθS = AS − (ISI)A. (31) From this equation we directly obtain the elements S13, S23, S43, S15, S45 and S12. To find the other monodromy supermatrix elements we can use the relationship strS := S11 + S22 + S33 − (S44 + S55 + S66) ≡ C(λ), Dθ C(λ) = 0, where C(λ) is some Laurent series with constant even coefficients, which follows from the equation (31). Since Dθ(S14 + S25 + S36) = (S44 + S55 + S66 − S11 + S22 + S33)− χS16 − φS26, this property of the monodromy supermatrix supertrace allows us to obtain S25 from the equ- ality Dθ(3S25 + aS16 + φS46 +Dθ(S45 − S12) +Dθ(S23 − S56)) = −χS16 − φS26 − C(λ). ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 468 O. YE. HENTOSH It is evident that the function C(λ) can be chosen arbitrarily since the supermatrix YC(λ)Y −1 with Y := Y (x, θ;λ) is some even fundamental supermatrix for the spectral problem (4) and C(λ) is some even supermatrix with elements in the forms of Laurent series with constant coeffecients and satisfies the equation (31). If C(λ) = N∑ i=1 σi λ− λi , σi = 5∑ s=0 ysizsi, (32) the elements S25, S14 and S36 look like S25 = N∑ i=1 y2iz3i λ− λi + C1(λ), S14 = N∑ i=1 y0iz1i λ− λi + C1(λ), S36 = N∑ i=1 y4iz5i λ− λi + C1(λ), DθC1(λ) = 0, where C1(λ) is some Laurent series with odd coefficients. Then the elements S42, S53, S44 − S11, S55 − S22, S66 − S33, S44 − S22, S55 − S33, S24 and S35 can be found successfully. From the evident expressions for the differences S44 − S11, S55 − S22, S66 − S33, S44 − S22 and S55−S33 we further obtain the diagonal elements of the reduced monodromy supermatrix SN in the forms S11 = N∑ i=1 y0iz0i λ− λi + P, S22 = N∑ i=1 y2iz2i λ− λi + P, S33 = N∑ i=1 y4iz4i λ− λi + P, S44 = − N∑ i=1 y1iz1i λ− λi + P, S55 = − N∑ i=1 y3iz3i λ− λi + P, S66 = − N∑ i=1 y5iz5i λ− λi + P, where P = P(Y,Z;λ) is some still undefined function on M2\|2 N . From the relationships for the superderivatives of the diagonal elements of SN we look for the elements S41, S52 and S63, depending on (Dθ P). Moreover, from the similar relationships for the elements in the first column of SN we obtain expressions for the elements S21, S51, S31, S61 and the differential relationship −DθPxx = −aPx + χ(DθP)− φ(DθPx). (33) Thereby, the elements S54, S32, S65, S34, S62 and S64 can also be found. ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 BARGMANN TYPE FINITE-DIMENSIONAL REDUCTIONS . . . 469 Having calculated now the elements S65, S62 and S64 successfully in other way we obtain the expressions Px = 2 3 φC1(λ), (Dθφ)C1(λ) = 0, 3Pxx = −(2χ+ (Dθa))C1(λ) + φxC1(λ)− φ(Dθ P). From these equations it follows that DθPx = 0, −2 3 aφC1(λ) + χ(Dθ P) = 0, −(2χ+ (Dθa))C1(λ)− φ(Dθ P) = 0. allowing for the vanishing solution for P and C1(λ). The latter entails the reduced monodromy supermatrix SN expression (29). In addition, the relationships (27), (28) follow from the compatibility conditions of the equations (4) and (6), dA dtj − (DθBj) = (IBjI)A−ABj , j ∈ Z+, as well as from those for the equations (5) and (7) dA dt̃j − (DθB̃j) = (IB̃jI)A−AB̃j , j ∈ Z+. This finishes the proof. Owing to the equations (27), (28) the functionals 1 α strSαN , α ∈ N, are invariant with respect to the vector fields d dtj and d dt̃j , j ∈ Z+. Then the coefficients in the expansions of these functionals at their poles appear to be conservation laws of the reduced on M2\|2 N ⋂ Hc vector fields from the hierarchy (1). Among them the coefficients σi, σ̂i, σ̆i ∈ C∞(R6N \|(6N+2);R1\|0), i = 1, N, are given by the expansions of the invariant functionals strSN , 1 2 strS2 N and 1 3 strS3 N as follows: strSN = N∑ i=1 σi λ− λi , 1 2 strS2 N = 1 2 N∑ i=1 σ2i (λ− λi)2 + N∑ i=1 σ̂i λ− λi , (34) ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 470 O. YE. HENTOSH σ̂i = N∑ k=1,k 6=i str (SiSk) λi − λk + λi str (S−1Si) + str (S0Si) = = N∑ k=1,k 6=i (∑5 s=0 ysizsk ) (∑5 r=0 yrkzri ) λi − λk − λigii + fii, and 1 3 strS3 N = 1 3 N∑ i=1 σ3i (λ− λi)3 + N∑ i=1 σiσ̂i (λ− λi)2 + N∑ i=1 σ̆i λ− λi , (35) σ̆i = N∑ k,`=1, k, 6̀=i, k 6=` str (SiSkS`) (λi − λk)(λi − λ`) + N∑ k=1, k 6=i str (SiSk) (σk − σi) (λi − λk)2 + + N∑ k=1, k 6=i λi str (S−1(SiSk + SkSi)) λi − λk + + N∑ k=1, k 6=i str (S0(SiSk + SkSi)) λi − λk + + λi str ((S−1S0 + S0S−1)Si) + str (S2 0Si) = = N∑ k,`=1, k, 6̀=i, k 6=` (∑5 s=0 ysizs` ) (∑5 κ=0 yκ`zκk ) (∑5 r=0 yrkzri ) (λi − λk)(λi − λ`) + + N∑ k=1, k 6=i (∑5 s=0 ysizsk ) (∑5 r=0 yrkzri ) ∑5 κ=0(yκizκi − yκkzκk) (λi − λk)2 + + N∑ k=1, k 6=i λigik (∑5 s=0 yskzsi ) + λigki (∑5 s=0 ysizsk ) λi − λk + + N∑ k=1, k 6=i fik (∑5 s=0 yskzsi ) + fki (∑5 s=0 ysizsk ) λi − λk − λipii + qii, ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 BARGMANN TYPE FINITE-DIMENSIONAL REDUCTIONS . . . 471 where gik = 3(y0iz4k + y1iz5k), fik = 3y2iz0k + ay0iz2k + 3y4iz2k + φy1iz2k + ( −b+ N∑ `=1 y0`z2` ) y0iz4k − 2ay2iz4k− − ( χ+ N∑ `=1 y0`z3` ) y1iz4k − 2φy3iz4k + φy0iz1k + 3y3iz1k+ + (χ+ (Dθa))y0iz3k + φy2iz3k + (a+ (Dθφ))y1iz3k+ + 3y5iz3k − ( 2(Dθb) + (Dθax)− N∑ `=1 y4`z5` ) y5iz0k− − ( χ+ (Dθa)− N∑ `=1 y1`z4` ) y2iz5k − 2φy4iz5k− − ( b+ (Dθχ) + N∑ `=1 y1`z3` ) y1iz5k − 2(a+ (Dθφ))y3iz5k, pik = 9(y0iz2k + y2iz4k + y1iz3k + y3iz5k)− 3φy0iz5k, qik = 3ay0iz0i + 9y4iz0i + 3φy1iz0i + 3 ( −b+ N∑ `=1 y0`z2` ) y0iz2k− − 3ay2iz2k − 3 ( χ+ N∑ `=1 y0`z3` ) y1iz2k − 3φy3iz2k+ + ( −2a2 + φ ( −χ− 2(Dθa) + N∑ `=1 y0`z3` )) y0iz4k+ + 3 ( −b+ N∑ `=1 y0`z2` ) y2iz4k − 6ay4iz4k − 2φ(2a+ (Dθφ))y1iz4k− − 3 ( χ+ N∑ `=1 y0`z3` ) y3iz4k − 6φy5iz4k + 3(χ+ (Dθa))y0iz1k+ + 6φy2iz1k + 3(a+ (Dθφ))y1iz1k + 9y5iz1k+ + ( φ(2a+ (Dθφ))− 6(Dθb)− 3(Dθax) + 3 N∑ `=1 y4`z5` ) y0iz3k+ ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 472 O. YE. HENTOSH + 3 ( N∑ `=1 y1`z4` ) y2iz3k − 3φy4iz3k− − 3 ( b+ (Dθχ) + N∑ `=1 y1`z3` ) y1iz3k − 3(a+ (Dθφ))y3iz3k+ + ( −3a(χ+ (Dθa)) + a N∑ `=1 y1`z4` − 2(Dθφ)(χ+ (Dθa)) + + φ ( b− (Dθχ)− N∑ `=1 (y1`z3` + 2y0`z2`) )) y0iz5k+ + ( 2φ(a− (Dθφ))− 6(Dθb)− 3(Dθax) + 3 N∑ `=1 y4`z5` ) y2iz5k− − 3 ( χ+ (Dθa)− N∑ `=1 y1`z4` ) y4iz5k+ + ( −2(a+ (Dθφ))2 + φ ( 3χ+ N∑ `=1 (y0`z3` − 2y1`z4`) )) y1iz5k− − 3 ( b+ (Dθχ) + N∑ `=1 y1`z3` ) y3iz5k − 6(a+ (Dθφ))y5iz5k, and are functionally independent on M 2\|2 N ⋂ Hc and involutive with respect to the Poisson bracket {., .}ω(2) on M 2\|2 N ⋂ Hc. Thus, owing to the superanalog [29] of Liouville integrabili- ty theorem the vector fields d dtj and d dt̃j , j ∈ Z+, are superintegrable flows on the finite- dimensional supersubspace M2\|2 N ⋂ Hc ⊂ M2\|2. 5. Conclusion. In the present paper the generalized invariant reduction technique, devi- sed before in [18], for investigating Lax type integrable supersymmetric nonlinear dynamical systems, has been used to study the Bargmann type reductions of the vector fields generated by the supersymmetric Boussinesq hierarhy related with a non-self-adjoint superdifferential operator of one anticommuting variable. It has been established that the corresponding invari- ant finite-dimensional supersubspace is diffeomorphic to some supersymplectic supermani- fold, smoothly embedded into superspace R6N \|(6N+2), N ∈ N, with an even supersymplectic structure. The invariant reduction procedure can be applied to a wide class of other Lax integrable supersymmetric nonlinear dynamical systems on the functional supermanifolds of one commu- ting and one anticommuting independent variables, associated with the linear matrix spectral relationships. The devised technique can also be effectively used for investigating reductions of (2\|1 + 1)- and (2\|2 + 1)-dimensional supersymmetric nonlinear dynamical systems with triple matrix Lax linearizations, described before in the papers [26, 27, 28], upon suitably determi- ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 4 BARGMANN TYPE FINITE-DIMENSIONAL REDUCTIONS . . . 473 ned invariant finite-dimensional supersubspaces. The latter is planned to be a subject of future studies. It is worth to mention here that the reduction method contributes to solving Lax integrable supersymmetric nonlinear dynamical systems on functional supermanifolds (of one commuting and one anticommuting independent variables) by means of the integration of the Liouville integrable systems on suitably determined finite-dimensional supermanifolds with even super- symplectic structures. 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Нелiнiйнi коливання, 2015, т . 18, N◦ 4

Bargmann type finite-dimensional reductions of the Lax integrable supersymmetric Boussinesq hierarchy and their integrability

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