Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces

We obtain via Bäcklund transformation the Hamiltonian representation for a Lax type nonlinear dynamical system hierarchy on a dual space to the Lie algebra of super-integral-differential operators of one anticommuting variable, extended by evolutions of the corresponding spectral problem eigenfuncti...

Повний опис

Збережено в:

Бібліографічні деталі
Дата:	2006
Автор:	Hentosh, O.Ye.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2006
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/146454
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces / O.Ye. Hentosh // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 22 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-146454
record_format	dspace
spelling	irk-123456789-1464542019-02-10T01:24:00Z Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces Hentosh, O.Ye. We obtain via Bäcklund transformation the Hamiltonian representation for a Lax type nonlinear dynamical system hierarchy on a dual space to the Lie algebra of super-integral-differential operators of one anticommuting variable, extended by evolutions of the corresponding spectral problem eigenfunctions and adjoint eigenfunctions, as well as for the hierarchies of their additional symmetries. The relation of these hierarchies with the integrable by Lax (2\|1+1)-dimensional supersymmetric Davey-Stewartson system is investigated. 2006 Article Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces / O.Ye. Hentosh // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 22 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 35Q53; 35Q58; 37K10; 37K30; 37K35; 58A50 http://dspace.nbuv.gov.ua/handle/123456789/146454 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We obtain via Bäcklund transformation the Hamiltonian representation for a Lax type nonlinear dynamical system hierarchy on a dual space to the Lie algebra of super-integral-differential operators of one anticommuting variable, extended by evolutions of the corresponding spectral problem eigenfunctions and adjoint eigenfunctions, as well as for the hierarchies of their additional symmetries. The relation of these hierarchies with the integrable by Lax (2\|1+1)-dimensional supersymmetric Davey-Stewartson system is investigated.
format	Article
author	Hentosh, O.Ye.
spellingShingle	Hentosh, O.Ye. Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Hentosh, O.Ye.
author_sort	Hentosh, O.Ye.
title	Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces
title_short	Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces
title_full	Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces
title_fullStr	Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces
title_full_unstemmed	Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces
title_sort	lax integrable supersymmetric hierarchies on extended phase spaces
publisher	Інститут математики НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/146454
citation_txt	Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces / O.Ye. Hentosh // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 22 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT hentoshoye laxintegrablesupersymmetrichierarchiesonextendedphasespaces
first_indexed	2025-07-11T00:00:47Z
last_indexed	2025-07-11T00:00:47Z
_version_	1837306538804379648
fulltext	Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 001, 11 pages Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces Oksana Ye. HENTOSH Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, 3B Naukova Str., Lviv, 79060 Ukraine E-mail: dept25@iapmm.lviv.ua Received October 27, 2005, in final form December 21, 2005; Published online January 04, 2006 Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper001/ Abstract. We obtain via Bäcklund transformation the Hamiltonian representation for a Lax type nonlinear dynamical system hierarchy on a dual space to the Lie algebra of super-integral-differential operators of one anticommuting variable, extended by evolutions of the corresponding spectral problem eigenfunctions and adjoint eigenfunctions, as well as for the hierarchies of their additional symmetries. The relation of these hierarchies with the integrable by Lax (2\|1 + 1)-dimensional supersymmetric Davey–Stewartson system is investigated. Key words: Lax type flows; “ghost” symmetries; the Davey–Stewartson system 2000 Mathematics Subject Classification: 35Q53; 35Q58; 37K10; 37K30; 37K35; 58A50 1 Introduction Since the paper of M. Adler [1] there was an understanding that Lax forms for a wide class of integrable nonlinear dynamical system hierarchies on functional manifolds [2, 3, 4, 5] and their supersymmetric analogs [6, 7] could be considered as Hamiltonian flows on dual spaces to the Lie algebra of integro-differential operators. Those flows are generated by the R-deformed canonical Lie–Poisson bracket and Casimir functionals as Hamiltonian functions (see [1, 8, 9]). For a concrete integro-differential operator every Hamiltonian flow of such a type can be written as a compatibility condition for the corresponding isospectral problem in the case of an arbitrary eigenfunction and the suitable evolution of this function. Thus, the existence problem of a Hamiltonian representation for the Lax type hierarchy, extended by the evolutions of a finite set of eigenfunctions and appropriate adjoint eigenfunctions, arises. In [10, 11, 12] it was solved for the Lie algebra of integro-differential operators by use of the Casimir functionals’ invariant property under some Lie–Bäcklund transformation. Analogously we obtain in this paper the Hamiltonian representation of the extended Lax type system hierarchy for the Lie algebra of super-integro-differential operators of one anticommuting variable. The hierarchies of additional or “ghost” symmetries [13] for the extended Lax type system are also proved to be Hamiltonian. It is established that every additional symmetry hierarchy is gene- rated by the tensor product of the R-deformed canonical Lie–Poisson bracket with the Poisson bracket on a finite-dimensional superspace, possessing an odd supersymplectic structure [14, 15], and all natural powers of one eigenvalue from the mentioned above finite set as Hamiltonian functions. The additional symmetry hierarchy is used for introducing one more commuting va- riable into (1\|1 + 1)-dimensional supersymmetric nonlinear dynamical systems with preserving their integrability by Lax. By means of this approach a (2\|1 + 1)-dimensional supersymmetric analog of the Davey–Stewartson system [5, 16, 17] and its triple linearization of a Lax type are found. mailto:dept25@iapmm.lviv.ua http://www.emis.de/journals/SIGMA/2006/Paper001/ 2 O.Ye. Hentosh 2 The general algebraic scheme Let G be a Lie algebra of scalar super-integral-differential operators [6] of one anticommuting variable θ (θ2 = 0): a := ∂m + ∑ j<2m ajD j θ, m ∈ N, where the symbol ∂ := ∂/∂x designates differentiation with respect to the independent variable x ∈ R/2πZ ' S1, aj := aj(x, θ) = a0 j (x) + θa1 j (x), j ∈ Z, are smooth superfield functions (superfunctions), and the superderivative Dθ := ∂/∂θ + θ∂/∂x, for which D2 θ = ∂, satisfies the following relation for any smooth superfield functions u and v: Dθ(uv) = (Dθu)v + (−1)p(u)u(Dθv), where p(u) is a parity of an arbitrary superfunction u, which is equal to 0 for u, being even, and one for u, being odd. The usual Lie commutator on G is defined as [a, b] := a ◦ b− b ◦ a for all a, b ∈ G, where “◦” is an associative product of super-integro-differential operators. On the Lie algebra G there exists the ad-invariant nondegerated symmetric bilinear form: (a, b) := ∫ 2π 0 resDθ (a ◦ b) dx, (1) where resDθ -operation for all a ∈ G is given by the expression: resDθ a := a−1. By means of the scalar product (1) the Lie algebra G is transformed into a metrizable one. As a consequence, its dual linear space of scalar super-integro-differential operators G∗ is identified with the Lie algebra G, that is G∗ ' G. The linear subspaces G+ ⊂ G and G− ⊂ G G+ := a := ∂m + 2m−1∑ j=0 ajD j θ : j = 0, 2m− 1  , G− := { b := ∞∑ l>0 blD −l θ : l ∈ N } , (2) where aj and bl are smooth superfunctions, forms Lie subalgebras in G and G = G+ ⊕ G−. Because of the splitting of G into the direct sum (2) of its Lie subalgebras one can construct a Lie–Poisson structure [1, 8, 9] on G∗, using the special linear endomorphism R of G: R := (P+ − P−)/2, P±G := G±, P±G∓ = 0. For any smooth by Frechet functionals γ, µ ∈ D(G∗) the Lie–Poisson bracket on G∗ is given by the expression: {γ, µ}R (l) = (l, [∇γ(l),∇µ(l)]R) , (3) Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces 3 where l ∈ G∗ and for all a, b ∈ G the R-deformed commutator has the form: [a, b]R := [Ra, b] + [a,Rb]. (4) The linear space G with the commutator (4) also becomes a Lie algebra. The gradient ∇γ(l) ∈ G of some functional γ ∈ D(G∗) at the point l ∈ G∗ with respect to the scalar product (1) is defined as δγ(l) := (∇γ(l), δl) , where the linear space isomorphism G ' G∗ is taken into account. Every Casimir functional γ ∈ I(G∗), being invariant with respect to Ad∗-action of the corre- sponding Lie group G, obeys the following condition at the point l ∈ G∗: [l,∇γ(l)] = 0. (5) The relationship (5) is satisfied by the hierarchy of functionals γn ∈ I(G∗), n ∈ Z+, taking the forms: γn(l) = 1 n+ 1 (l1/m, ln/m). (6) The Lie–Poisson bracket (3) generates the hierarchy of Hamiltonian dynamical systems on G∗: dl/dtn := [R∇γn(l), l] = [(∇γn(l))+, l], (7) with the Casimir functionals (6) as Hamiltonian functions. The latter equation is equivalent to the usual commutator Lax type representation. It is easy to verify that for every n ∈ Z+ the relationship (7) is a compatibility condition for such linear integral-differential equations: lf = λf, (8) and df/dtn = (∇γn(l))+f, (9) where λ ∈ C is a spectral parameter, f ∈W 1\|0 := L∞(S1×Λ1; C1\|0) if f is an even superfunction and f ∈W 0\|1 := L∞(S1×Λ1; C0\|1) if f is an odd one. Here Λ := Λ0⊕Λ1 is a Grassmann algebra over C, Λ0 ⊃ R. The associated with (9) dynamical system for the adjoint superfunction f∗ takes the form: df∗/dtn = −(∇γn(l))∗+f ∗, (10) where (f, f∗)T ∈W 1\|1 := L∞(S1×Λ1; C1\|1) or (f∗, f)T ∈W 1\|1 and superfunction f∗ is a solution of the adjoint spectral problem: l∗f∗ = λf∗. The objects of further investigations are some algebraic properties of equation (7) together with 2N ∈ N copies of equation (9): dfi/dtn = (∇γn(l))+fi, dΦi/dtn = (∇γn(l))+Φi, (11) for even fi ∈W 1\|0 and odd Φi ∈W 0\|1 eigenfunctions of the spectral problem (8), corresponding to the eigenvalues λi, i = 1, N , and the same number of copies of equation (10): df∗i /dtn = −(∇γn(l))∗+f ∗ i , dΦ∗i /dtn = −(∇γn(l))∗+Φ∗i , (12) for corresponding odd f∗i ∈ W 0\|1 and even Φ∗i ∈ W 1\|0 adjoint eigenfunctions, as a coupled evolution system on the space G∗ ⊕W 2N \|2N . 4 O.Ye. Hentosh 3 Tensor product of Poisson structures and its Bäcklund transformation To compactify the description below one shall use the following designation of the left gradient vector: ∇γ(l̃, f̃i, Φ̃∗i , f̃ ∗ i , Φ̃i) := ( δγ δl̃ , δγ δf̃i , δγ δΦ̃∗i , δγ δf̃∗i , δγ δΦ̃i )T , where i = 1, N , at a point (l̃, f̃i, Φ̃∗i , f̃ ∗ i , Φ̃i)T ∈ G∗ ⊕ W 2N \|2N for any smooth functional γ ∈ D(G∗ ⊕W 2N \|2N ). On the spaces G∗ and WN ⊕W ∗N there exist a Lie–Poisson structure [1, 8, 9] δγ/δl̃ : Θ̃→ [ l̃, ( δγ δl̃ ) + ] − [ l̃, δγ δl̃ ] + , (13) where Θ̃ : G → G∗, at a point l̃ ∈ G∗ and the canonical Poisson structure [14, 15]( δγ δf̃i , δγ δΦ̃∗i , δγ δf̃∗i , δγ δΦ̃i )T : J̃→ ( − δγ δf̃∗i , δγ δΦ̃i , δγ δf̃i , − δγ δΦ̃∗i , )T , (14) J̃ : T ∗(W 2N \|2N ) → T (W 2N \|2N ), corresponding to the odd symplectic form ω(2) = N∑ i=1 (f̃i ∧ f̃∗i − Φ̃i∧Φ̃∗i ), at a point (f̃i, Φ̃∗i , f̃ ∗ i , Φ̃)T ∈W 2N \|2N . It should be noted that the Poisson structure (13) generates equation (7) for any Casimir functional γ ∈ I(G∗). Thus, on the extended phase space G∗ ⊕W 2N \|2N one can obtain a Poisson structure as the tensor product L̃ := Θ̃⊗ J̃ of (13) and (14). Consider the following Bäcklund transformation: (l̃, f̃i, Φ̃∗i , f̃ ∗ i , Φ̃i)T : B7→ (l(l̃, f̃i, Φ̃∗i , f̃ ∗ i , Φ̃i), fi = f̃i,Φ∗i = Φ̃∗i , f ∗ i = f̃∗i ,Φi = Φ̃i)T , (15) generating on G∗ ⊕ W 2N \|2N a Poisson structure L with respect to variables (l, fi,Φ∗i , f ∗ i ,Φi), i = 1, N , of the coupled evolution equations (7), (11) and (12). The main condition for the mapping (15) is coincidence of the dynamical system (dl/dt, dfi/dt, dΦ∗i /dt, df ∗ i /dt, dΦi/dt)T := −L∇γn(l, fi,Φ∗i , f ∗ i ,Φi) (16) with equations (7), (11) and (12) in the case of γn ∈ I(G∗), n ∈ Z+, i.e. when the functional γn is taken to be not dependent of variables (fi,Φ∗i , f ∗ i ,Φi)T ∈ W 2N \|2N . To satisfy that condition, one should find a variation of some Casimir functional γn ∈ I(G∗), n ∈ Z+, at δl̃ = 0, taking into account the evolutions (11), (12) and the Bäcklund transformation (15): δγn(l̃, f̃i, Φ̃∗i , f̃ ∗ i , Φ̃i) ∣∣∣ δl̃=0 = N∑ i=1 ( 〈δf̃i, δγn δf̃i 〉+ 〈δΦ̃∗i , δγn δΦ̃∗i 〉+ 〈δf̃∗i , δγn δf̃∗i 〉+ 〈δΦ̃i, δγn δΦ̃i 〉 ) = N∑ i=1 ( 〈δf̃i,− df̃∗i dtn 〉+ 〈δΦ̃∗i , dΦ̃i dtn 〉+ 〈δf̃∗i , df̃i/dtn〉+ 〈δΦ̃i,− dΦ̃∗i dtn 〉 ) = N∑ i=1 ( 〈δfi, (∇γn(l))∗+f ∗ i 〉+ 〈δΦ∗i , (∇γn(l))+Φi〉+ 〈δf∗i , (∇γn(l))+fi〉 Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces 5 + 〈δΦi, (∇γn(l))∗+Φ∗i 〉 ) = N∑ i=1 ( 〈(∇γn(l))+δfi, f ∗ i 〉+ 〈(∇γn(l))+fi, δf ∗ i 〉 + 〈(∇γn(l))+(δΦi),Φ∗i 〉+ 〈(∇γn(l))+Φi, δΦ∗i 〉 ) = N∑ i=1 ( (∇γn(l), δ(fiD −1 θ f∗i )) + (∇γn(l), δ(ΦiD −1 θ Φ∗i )) ) = ( ∇γn(l), δ N∑ i=1 (fiD −1 θ f∗i + ΦiD −1 θ Φ∗i ) ) := (∇γn(l), δl), (17) where γn ∈ I(G∗), n ∈ Z+, at the point l ∈ G∗ and the brackets 〈·, ·〉 designate paring of the spaces W 1\|0 and W 0\|1. As a result of the expression (17) one obtains the relationships: δl\|δl̃=0 = δ N∑ i=1 (fiD −1 θ f∗i + ΦiD −1 θ Φ∗i ). Having assumed the linear dependence of l from l̃ ∈ G∗ one gets right away that l = l̃ + N∑ i=1 (fiD −1 θ f∗i + ΦiD −1 θ Φ∗i ). (18) Thus, the Bäcklund transformation (15) can be written as (l̃, f̃i, Φ̃∗i , f̃ ∗ i , Φ̃i)T B7→ ( l = l̃ + N∑ i=1 (fiD −1 θ f∗i + ΦiD −1 θ Φ∗i ), fi,Φ∗i , f ∗ i ,Φi )T . (19) The expression (19) generalizes the result obtained in the papers [10, 11, 12] for the Lie algebra of integral-differential operators. The existence of the Bäcklund transformation (19) makes it possible to formulate the following theorem. Theorem 1. The dynamical system on G∗ ⊕W 2N \|2N , being Hamiltonian with respect to the Poisson structure L̃ : T ∗(G∗ ⊕ W 2N \|2N ) → T (G∗ ⊕ W 2N \|2N ), in the form of the following evolution equations: dl̃ dtn = [( δγn δl̃ ) + , l̃ ] − [ δγn δl̃ , l̃ ] + , df̃i dtn = δγn δf̃∗i , dΦ̃∗i dtn = −δγn δΦ̃i , df̃∗i dtn = −δγn δf̃i , dΦ̃i dtn = δγn δΦ̃∗i , where i = 1, N and γn ∈ I(G∗), n ∈ Z+, is a Casimir functional at the point l ∈ G∗, con- nected with l̃ ∈ G∗ by (18), is equivalent to the system (9), (13) and (14) via the Bäcklund transformation (19). By means of simple calculations via the formula: L = B ′L̃B′∗, 6 O.Ye. Hentosh where B ′ : T (G∗⊕W 2N \|2N ) → T (G∗⊕W 2N \|2N ) is a Frechet derivative of (19), one brings about the following form of the Poisson structure L on G∗ ⊕W 2N \|2N 3 (l, fi,Φ∗i , f ∗ i ,Φi)T : ∇γ(l, fi,Φ∗i , f ∗ i ,Φi) L→  [ l̃, ( δγ δl̃ ) + ] − [ l̃, δγ δl̃ ] + + N∑ i=1 ( fiD −1 θ δγ δfi − − δγ δf∗i D−1 θ f∗i + ΦiD −1 θ δγ δΦi − δγ δΦ∗i D−1 θ Φ∗i ) − δγ δf∗i − ( δγ δl ) + fi δγ δΦi + ( δγ δl )∗ + Φ∗i δγ δf∗i + ( δγ δl )∗ + f∗i − δγ δΦ∗i − ( δγ δl ) + Φi  , (20) where γ ∈ D(G∗ ⊕ W 2N \|2N ) is an arbitrary smooth functional and i = 1, N , that makes it possible to formulate the theorem. Theorem 2. For every n ∈ Z+ the coupled dynamical system (7), (11) and (12) is Hamiltonian with respect to the Poisson structure L in the form (20) and the functional γn ∈ I(G∗). Using the expression (18) one can construct the hierarchy of Hamiltonian evolution equa- tions, describing commutative flows, generated by involutive with respect to the Lie–Poisson bracket (3) Casimir invariants γn ∈ I(G∗), n ∈ Z+, on the extended space G∗ ⊕ W 2N \|2N at a fixed element l̃ ∈ G∗. For every n ∈ Z+ the equation of such a type is equivalent to the system (7), (11) and (12). 4 Hierarchies of additional symmetries The evolution type hierarchy (7), (11) and (12) possesses another set of invariants, which includes all natural powers of the eigenvalues λi, i = 1, N . They can be considered as smooth by Frechet functionals on the extended space G∗ ⊕W 2N \|2N due to the representation: λs k = 〈lsfk, f ∗ k 〉+ 〈lsΦk,Φ∗k〉, (21) where s ∈ N, taking place for all k = 1, N under the normalizing condition: 〈fk, f ∗ k 〉+ 〈Φk,Φ∗k〉 = 1. In the case of l := l+ + N∑ i=1 (fiD −1 θ f∗i + ΦiD −1 θ Φ∗i ) (22) the formula (21) leads to the following variation of the functionals λs k ∈ D(G∗ ⊕ W 2N \|2N ), k = 1, N : δλs k = 〈(δls)fk, f ∗ k 〉+ 〈(δls)Φk,Φ∗k〉 + 〈ls(δfk), f∗k 〉+ 〈lsfk, δf ∗ k 〉+ 〈ls(δΦk),Φ∗k〉+ 〈lsΦk, δΦ∗k〉 Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces 7 = (δl+,M s k) + N∑ i=1 ( 〈δfi, (−M s k + δi kl s)∗f∗i 〉+ 〈δf∗i , (−Mn k + δi kl s)fi〉 + 〈δΦi, (−M s k + δi kl s)∗Φ∗i 〉+ 〈δΦ∗i , (−Mn k + δi kl s)Φi〉 ) , where δi k is a Kronecker symbol and the operator M s k , s ∈ N, is determined as M s k := s−1∑ p=0 ( (lpfk)D−1 θ (l∗s−1−pf∗k ) + (lpΦk)D−1 θ (l∗s−1−pΦ∗k) ) = λs−1 k M1 k . Thus, one obtains the exact forms of gradients for the functionals λs k ∈ D(Ĝ∗ ⊕ W 2N \|2N ), k = 1, N : ∇λs k(l+, fi,Φ∗i , f ∗ i ,Φi) =  M s k (−M s k + δi kl s)∗f∗i (−Mn k + δi kl s)Φi (−Mn k + δi kl s)fi (−M s k + δi kl s)∗Φ∗i  , (23) where i = 1, N . By means of the expression (23) the tensor product L̃ of Poisson structures (13) and (14) generates the hierarchy of coupled evolution equations on G∗ ⊕W 2N \|2N : dl+/dτs,k = −[M s k , l̂+]+, (24) dfi/dτs,k = (−M s k + δi kl s)fi, df∗i /dτs,k = (M s k − δi kl s)∗f∗i , (25) dΦi/dτs,k = (−M s k + δi kl s)Φi, dΦ∗i /dτs,k = (M s k − δi kl s)∗Φ∗i , (26) where i = 1, N , for every k = 1, N . Because of the Bäcklund transformation (19) the equa- tion (24) is equivalent to the commutator relationship: dl/dτs,k = −[M s k , l] = −λs−1 k [M1 k , l] = λs−1 k dl/dτ1,k, (27) and the following theorem takes place: Theorem 3. For every k = 1, N and s ∈ N the coupled dynamical system (24), (25) and (26) is Hamiltonian one with respect to the Poisson structure L in the form (20) and the functional λn k ∈ D(G∗ ⊕W 2N \|2N ). The coupled dynamical systems (24), (25) and (26) represent flows on G∗ ⊕W 2N \|2N , com- muting one with each other. Theorem 4. For k = 1, N the coupled evolution equations (24), (25) and (26) form a set of additional symmetry hierarchies for the coupled dynamical system (7), (11) and (12). Proof. To prove the theorem it is sufficient to show that [d/dtn, d/dτ1,k] = 0, [d/dτ1,k, d/dτ1,q] = 0, (28) where k, q = 1, N and n ∈ N. The first equality in the formula (28) follows from the identities: d(∇γn(l))+/dτ1,k = [(∇γn(l))+,M1 1 ]+, dM1 1 /dtn = [(∇γn(l̂))+,M1 1 ]−, the second one being a consequence of the relationship: dM1 k/dτ1,q − dM1 q /dτ1,k = [M1 k ,M 1 q ]. � 8 O.Ye. Hentosh When N ≥ 2, a new class of nontrivial Hamiltonian flows d/dTn,K := d/dtn + K∑ k=1 d/dτn,k, n ∈ N, K = 1, N − 1, in a Lax form on G∗ ⊕ W 2N \|2N can be constructed by use of the set of additional symmetry hierarchies for the Lie algebra of super-integro-differential operators. Acting on the functions fi, f∗i , Φi, Φ∗i , i = 1, N , these flows generate ((1+K)\|1+1)-dimensional supersymmetric nonlinear dynamical systems. For the first time the additional symmetries in the case of N = 2 were applied by E. Nissimov and S. Pacheva [13] to obtain a Lax integrable supersymmetric analog of the (2+1)-dimensional Davey–Stewartson system. If l := ∂ + f1D −1 θ f∗1 + f2D −1 θ f∗2 + Φ1D −1 θ Φ∗1 + Φ2D −1 θ Φ∗2 ∈ G∗, where (f1, f2,Φ∗1,Φ ∗ 2, f ∗ 1 , f ∗ 2 ,Φ1,Φ2)T ∈ W 4\|4, the flows ∂/∂τ := d/dτ1,1 and d/dT := d/dT2,1 = d/dt2 + d/dτ2,1 on G∗ ⊕W 4\|4, acting on the functions fi, f∗i , Φi, Φ∗i , i = 1, 2, by the following way: f1,τ = f1,x + u1f2 − α1Φ2, f2,τ = −ū1f1 + ᾱ2Φ1, f∗1,τ = f∗1,x + ū1f ∗ 2 − ᾱ1Φ∗2, f∗2,τ = −u1f ∗ 1 − α2Φ∗1, Φ1,τ = Φ1,x − α2f2 + u2Φ2, Φ2,τ = −ᾱ1f1 − ū2Φ1, Φ∗1,τ = Φ∗1,x − ᾱ2f ∗ 2 − ū2Φ∗2, Φ∗2,τ = α1f ∗ 1 − u2Φ∗1, (29) and f1,T = f1,xx + f1,ττ + w1Dθf1 + w0f1 + 2v1,τf1 − 2βτΦ1, f2,T = f2,xx + w1Dθf2 + w0f2 − ū1f1,τ + ᾱ2Φ1,τ + ū1,τf1 − ᾱ2,τΦ1,τ , f∗1,T = −f∗1,xx − f∗1,ττ −Dθ(w1f ∗ 1 )− w0f ∗ 1 − 2v1,τf ∗ 1 + 2β̄τΦ∗1, f∗2,T = −f∗2,xx −Dθ(w1f ∗ 2 )− w0f ∗ 2 + u1f ∗ 1,τ + α2Φ∗1,τ − u1,τf ∗ 1 + α2,τΦ∗1,τ , Φ1,T = Φ1,xx + Φ1,ττ + w1DθΦ1 + w0Φ1 + 2f1β̄τ + 2v2,τΦ1, Φ2,T = Φ2,xx + w1DθΦ2 + w0Φ2 − ᾱ1f1,τ − ū2Φ1,τ + ᾱ1,τf1 + ū2,τΦ1,τ , Φ∗1,T = −Φ∗1,xx − Φ1,ττ −Dθ(w1Φ∗1)− w0Φ∗1 − 2f∗1βτ − 2v2,τΦ∗1, Φ∗2,T = −Φ∗2,xx −Dθ(w1Φ∗2)− w0Φ∗2 − α1f ∗ 1,τ + u2Φ∗1,τ + α1,τf ∗ 1 − u2,τΦ∗1,τ , Dθu1 = f1f ∗ 2 , Dθu2 = Φ1Φ∗2, Dθū1 = f∗1 f2, Dθū2 = Φ∗1Φ2, Dθv1 = f1f ∗ 1 , Dθv2 = Φ1Φ∗1, Dθα1 = f1Φ∗2, Dθα2 = Φ1f ∗ 2 , Dθᾱ1 = f∗1 Φ2, Dθᾱ2 = Φ∗1f2, Dθβ = f1Φ∗1, Dθβ̄ = f∗1 Φ1, (30) where (∇γ2(l))+ := ∂2 +w1Dθ +w0, represent (2\|1 + 1)-dimensional supersymmetric nonlinear dynamical system. The system (29) and (4) possesses an infinite sequence of local conservation laws, which can be found by the formula (6), and a Lax representation, given by the spectral problem (8) and the evolution equations: fτ = −M1 1 f, (31) fT = ((∇γ2(l))+ −M2 1 )f, (32) for an arbitrary eigenfunction f ∈ W 1\|0 or f ∈ W 0\|1. The relationships and (32) lead to additional nonlinear constraints such as w0,τ = 2w1(f1f ∗ 1 − Φ1Φ∗1) + 2(f1(Dθf ∗ 1 ) + Φ1(DθΦ∗1))x, w1,τ = −2(f1f ∗ 1 )x + 2(Φ1Φ∗1)x. (33) Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces 9 When f1 := ψ, f∗1 := θψ∗, f2 = f∗2 = 0 and Φ1 = Φ∗1 = Φ2 = Φ∗2 = 0, the equations and are reduced to the Lax integrable (2 + 1)-dimensional Davey–Stewartson system [5, 16, 17]: ψ1,T = ψ1,xx + ψ1,ττ + 2(S − 2ψψ∗)ψ, ψ∗1,T = −ψ∗1,xx − ψ∗1,ττ − 2(S − 2ψψ∗)ψ∗, Sxτ = (∂/∂x+ ∂/∂τ)2ψψ∗, where 2S := w0 0 + 2v0 1,τ + 4ψψ∗, w0 := w0 0, v1,τ := v0 1,τ and ψ,ψ∗ ∈ L∞(S1; C). The Lax representation (10), (4) and (32) for the (2\|1 + 1)-dimensional supersymmetric nonlinear dynamical Davey–Stewartson system (29), (4) and (4) has equivalent matrix form: DθF =  0 0 0 0 0 1 f∗1 0 0 0 0 0 f∗2 0 0 0 0 0 Φ∗1 0 0 0 0 0 Φ∗2 0 0 0 0 0 λ −f1 −f2 −Φ1 −Φ2 0 F, dF dτ =  0 −f1 0 −Φ1 0 0 Dθf ∗ 1 −λ ū1 0 ᾱ1 −f∗1 0 −u1 0 α2 0 0 DθΦ∗1 0 ᾱ2 −λ ū2 Φ∗1 0 −α1 0 −u2 0 0 Φ1Φ∗1 − f1f ∗ 1 −Dθf1 0 −DθΦ1 0 0 F, dF dT = CF, where F = (F 0 := f, F 2, F 4, F 1, F 3, F 5)τ ∈W 3\|3, C := (Cmn) ∈ gl(3\|3), m,n = 1, 6, and C11 = λ2 + 1 2 w0 + f1Dθf ∗ 1 + Φ1DθΦ∗1, C12 = −(2λf1 + f1,x + f1,τ ), C13 = −(λf2 + f2,x) + ū1f1 − ᾱ2Φ1, C14 = −(2λΦ1 + Φ1,x + Φ1,τ ), C15 = −(λΦ2 + Φ2,x)− ū2Φ1 + ᾱ1f1, C16 = 1 2 w1 − f1f ∗ 1 + Φ1Φ∗1, C21 = −w1f ∗ 1 + 2Dθ(−f∗1,x + λf∗1 )− ū1Dθf ∗ 2 − ᾱ1DθΦ∗2, C22 = −λ2 − 2Dθ(f1f ∗ 1 )− u1ū1 + α1ᾱ1, C23 = −Dθ(f2f ∗ 1 ) + λū1 − ū1,τ , C24 = −2Dθ(Φ1f ∗ 1 ) + ū1α2 − u2ᾱ1, C25 = −Dθ(Φ2f ∗ 1 ) + λᾱ1 − ᾱ1,τ , C26 = 2(−λf∗1 + f∗1,x) + ū1f ∗ 2 − ᾱ1Φ∗2, C31 = −1 2 w1f ∗ 2 +Dθ(−f∗2,x + λf∗2 ) + u1Dθf ∗ 1 + α2DθΦ∗1, C32 = −Dθ(f1f ∗ 2 )− λu1 − u1,τ , C33 = −Dθ(f2f ∗ 2 ) + u1ū1 − α2ᾱ2, C34 = −Dθ(Φ1f ∗ 2 ) + λα2 + α2,τ , C35 = −Dθ(Φ2f ∗ 2 ) + u1ᾱ1 − ū2α2, C36 = (−λf∗2 + f∗2,x)− u1f ∗ 1 + α2Φ∗1, C41 = −w1Φ∗1 + 2Dθ(−Φ∗1,x + λΦ∗1) + ū2DθΦ∗2 − ᾱ2Dθf ∗ 2 , C42 = −2Dθ(f1Φ∗1)− u1ᾱ2 − ū2α1, C43 = −Dθ(f2Φ∗1) + λᾱ2 − ᾱ2,τ , C44 = −λ2 − 2Dθ(Φ1Φ∗1)− u2ū2 − α2ᾱ2, C45 = −Dθ(Φ2Φ∗1) + λū2 − ū2,τ , C46 = 2(−λΦ∗1 + Φ∗1,x) + ū2Φ∗2 + ᾱ2f ∗ 2 , C51 = −1 2 w1Φ∗2 +Dθ(−Φ∗2,x + λΦ∗2) + α1Dθf ∗ 1 + u2DθΦ∗1, 10 O.Ye. Hentosh C52 = −Dθ(f1Φ∗2)− λα1 − α1,τ , C53 = −Dθ(f2Φ∗2) + α1ū1 − ᾱ2u2, C54 = −Dθ(Φ1Φ∗2)− λu2 − u2,τ , C55 = −Dθ(Φ2Φ∗2) + α1ᾱ1 + u2ū2, C56 = (−λΦ∗2 + Φ∗2,x)− α1f ∗ 1 + u2Φ∗1, C61 = 1 2 Dθw0 + (Dθf1)Dθf ∗ 1 + (DθΦ1)DθΦ∗1 − (f1,τf ∗ 1 + f2,xf ∗ 2 − Φ1,τΦ∗1 − Φ2,xΦ∗2) + ū1f1f ∗ 2 − ᾱ1f1Φ∗2 + ᾱ2Φ1f ∗ 2 + ū2Φ1Φ∗2, C62 = −Dθ(2λf1 + f1,x + f1,τ ) + 1 2 w1f1 + f1(−f1f ∗ 1 + Φ1Φ∗1), C63 = −Dθ(λf2 + f2,x) + 1 2 w1f2 + ū1(Dθf1) + ᾱ2(DθΦ1) C64 = −Dθ(2λΦ1 + Φ1,x + Φ1,τ ) + 1 2 w1Φ1 − f1f ∗ 1 Φ1, C65 = −Dθ(λΦ2 + Φ2,x) + 1 2 w1Φ2 − ū2DθΦ1 − ᾱ1Dθf1, C66 = λ2 + 1 2 w0 + 1 2 Dθw1 − (Dθf1)f∗1 + (DθΦ1)Φ∗1. In fact, one has found a triple matrix linearization for a (2\|1+1)-dimensional dynamical system, that is important for the standard method of inverse scattering transformation [3] as well as for the reduction procedure [18, 19] upon invariant subspaces of associated spectral problem eigenvalues. The method of additional symmetries is effective for constructing a wide class of (2\|1 + 1)- dimensional supersymmetric nonlinear dynamical systems with a triple matrix linearization. 5 Conclusion By now several regular Lie-algebraic approaches existed to constructing Lax integrable (2 + 1)- dimensional nonlinear dynamical systems on functional manifolds, which were presented in [12, 20, 21, 22]. In this paper a new Lie-algebraic method is devised for introducing one more commuting variable into (1\|1+1)-dimensional dynamical systems with preserving their integra- bility by Lax. It involves use of additional symmetries [13] for a Hamiltonian flow hierarchy on extended dual space to some operator Lie algebra. Any integrable (2\|1+1)-dimensional supersymmetric nonlinear dynamical system obtained by means of the method possesses an infinite sequence of local conservation laws and a triple mat- rix linearization of a Lax type. These properties make it possible to apply the standard inverse scattering transformation [3] and the reduction procedure [18, 19] upon invariant subspaces. If N > 2 in the representation (22), the hierarchies of additional symmetries can be used for constructing Lax integrable ((1 + K)\|1 + 1)-dimensional supersymmetric systems, where K = 1, N − 1. Analyzing the structure of the Bäcklund type transformation (19) as a key point of the method, one can observe that it strongly depends on an ad-invariant scalar product chosen for an operator Lie algebra G and a Lie algebra decomposition like (2). Since there are other possibilities of choosing ad-invariant scalar products on G and such decompositions, they give rise naturally to other Bäcklund transformations. Acknowledgements The author thanks Professor A.K. Prykarpatsky for useful discussions and Organizers of Sixth International Conference “Symmetry in Nonlinear Mathematical Physics” (2005, Kyiv) for in- vitation to take part in the conference. The present paper is the written version of the talk delivered by the author at this conference. Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces 11 [1] Adler M., On a trace functional for formal pseudo-differential operators and the symplectic structures of a Korteweg–de Vries equation, Invent. Math., 1979, V.50, N 2, 219–248. [2] Lax P.D., Periodic solutions of the KdV equation, Comm. Pure Appl. Math., 1975, V.28, 141–188. [3] Novikov S.P. (Editor), Soliton theory: method of the inverse problem, Moscow, Nauka, 1980 (in Russian). [4] Prykarpatsky A.K., Mykytiuk I.V., Algebraic integrability of nonlinear dynamical systems on manifolds: classical and quantum aspects, Dordrecht–Boston–London, Kluwer Academic Publishers, 1998. [5] Blaszak M., Multi-Hamiltonian theory of dynamical systems, Verlag–Berlin–Heidelberg, Springer, 1998. [6] Manin Yu.I., Radul A.O., A supersymmetric extension of the Kadomtsev–Petviashvili hierarchy, Comm. Math. Phys., 1985, V.98, 65–77. [7] Oevel W., Popowicz Z., The bi-Hamiltonian structure of fully supersymmetri Korteweg–de Vries systems, Comm. Math. Phys., 1991, V.139, 441–460. [8] Semenov-Tian-Shansky M.A., What is theR-matrix, Funct. Anal. Appl., 1983, V.17, N 4, 17–33 (in Russian). [9] Oevel W., R-structures, Yang–Baxter equations and related involution theorems, J. Math. Phys., 1989, V.30, N 5, 1140–1149. [10] Oevel W., Strampp W., Constrained KP hierarchy and bi-Hamiltonian structures, Comm. Math. Phys., 1993, V.157, 51–81. [11] Prykarpatsky Ya.A., Structure of integrable Lax flows on nonlocal manifolds: dynamical systems with sources, Math. Methods and Phys.-Mech. Fields, 1997, V.40, N 4, 106–115 (in Ukrainian). [12] Prykarpatsky A.K., Hentosh O.Ye., The Lie-algebraic structure of (2 + 1)-dimensional Lax type integrable nonlinear dynamical systems, Ukrainian Math. J., 2004, V.56, N 7, 939–946. [13] Nissimov E., Pacheva S., Symmetries of supersymmetric integrable hierarchies of KP type, J. Math. Phys., 2002, V.43, N 5, 2547–2586; nlin.SI/0102010. [14] Berezin F.A., Introduction to algebra and analysis with anticommuting variables, Moscow Univ., 1983 (in Russian). [15] Shander V.N., Analogues of the Frobenius and Darboux theorems, Reports of Bulgarian Academy of Sciences, 1983, V.36, N 3, 309–311. [16] Ablowitz M.J., Segur H., Solitons and the inverse scattering transform, Philadelphia, SIAM, 1981. [17] Kulish P.P., Lipovsky V.D., On Hamiltonian interpretation of the inverse problem method for the Davey– Stewartson equation, LOMI Proceedings, Leningrad, Nauka, 1987, V.161, 54–71 (in Russian). [18] Prykarpatsky A.K., Hentosh O.E., Blackmore D.L., The finite-dimensional Moser type reductions of modified Boussinesq and super-Korteweg–de Vries Hamiltonian systems via the gradient-holonomic algorithm and the dual moment maps. I, J. Nonlinear Math. Phys., 1997, V.4, N 3–4, 455–469. [19] Prykarpatsky A.K., Blackmore D., Strampp W., Sydorenko Yu., Samuliak R., Some remarks on Lagrangian and Hamiltonian formalism, related to infinite-dimensional dynamical systems with symmetries, Condensed Matter Phys., 1995, N 6, 79–104. [20] Sato M., Soliton equations as dynamical systems on infinite Grassmann manifolds, RIMS Kokyuroku, Kyoto Univ., 1981, V.439, 30–40. [21] Reiman A.G., Semenov-Tian-Shansky M.A., The Hamiltonian structure of Kadomtsev–Petviashvili type equations, LOMI Proceedings, Leningrad, Nauka, 1987, V.164, 212–227 (in Russian). [22] Prykarpatsky A.K., Samoilenko V.Hr., Andrushkiw R.I., Mitropolsky Yu.O., Prytula M.M., Algebraic struc- ture of the gradient-holonomic algorithm for Lax integrable nonlinear systems. I, J. Math. Phys., 1994, V.35, N 4, 1763–1777. 1 Introduction 2 The general algebraic scheme 3 Tensor product of Poisson structures and its Bäcklund transformation 4 Hierarchies of additional symmetries 5 Conclusion

Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces

Репозитарії

Схожі ресурси