The discrete nonlinear Schrödinger type hierarchy, its finite dimensional reduction analysis and numerical integrability scheme

We investigate discretizations of the integrable nonlinear Schrodinger dynamical system, well known as the ¨ Ablowitz – Ladik equation, the related symplectic structures and its finite dimensional invariant reductions. An effective scheme of invariant reducing the corresponding infinite system of or...

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Hauptverfasser:	Prykarpatsky, A.K., Cieśliński, J.
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spelling	irk-123456789-1773042021-02-15T01:26:50Z The discrete nonlinear Schrödinger type hierarchy, its finite dimensional reduction analysis and numerical integrability scheme Prykarpatsky, A.K. Cieśliński, J. We investigate discretizations of the integrable nonlinear Schrodinger dynamical system, well known as the ¨ Ablowitz – Ladik equation, the related symplectic structures and its finite dimensional invariant reductions. An effective scheme of invariant reducing the corresponding infinite system of ordinary differential equations to an equivalent finite system of ordinary differential equations with respect to the evolution parameter is developed. A finite set of recurrent algebraic regular relations, allowing to generate solutions of the discrete nonlinear Schrodinger dynamical system, is constructed, the related functional spaces of ¨ solutions is discussed. Finally, the Fourier transform approach to studying the solution set of the discrete nonlinear Schrodinger dynamical system and its functional-analytical aspects is analyzed. Дослiджуються дискретизацiї iнтегровної нелiнiйної динамiчної системи Шрьодiнгера, вiдомої як рiвняння Абловiца – Ладiка, вiдповiднi симплектичнi структури та її скiнченновимiрнi iнварiантнi редукцiї. Побудовано ефективний алгоритм iнварiантної редукцiї вiдповiдної нескiнченної системи звичайних диференцiальних рiвнянь до еквiвалентної скiнченної системи звичайних диференцiальних рiвнянь вiдносно параметра еволюцiї. Побудовано скiнченну множину рекурентних алгебраїчних регулярних спiввiдношень, що дозволило побудувати розв’язки дискретної нелiнiйної динамiчної системи Шрьодiнгера, та розглянуто вiдповiднi функцiональнi простори розв’язкiв. Проведено аналiз пiдходу перетворення Фур’є до вивчення множини розв’язкiв дискретної нелiнiйної динамiчної системи Шрьодiнгера та її функцiонально-аналiтичних аспектiв. 2017 Article The discrete nonlinear Schrödinger type hierarchy, its finite dimensional reduction analysis and numerical integrability scheme / A.K. Prykarpatsky, J. Cieśliński // Нелінійні коливання. — 2017. — Т. 20, № 2. — С. 228-266 — Бібліогр.: 61 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177304 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	We investigate discretizations of the integrable nonlinear Schrodinger dynamical system, well known as the ¨ Ablowitz – Ladik equation, the related symplectic structures and its finite dimensional invariant reductions. An effective scheme of invariant reducing the corresponding infinite system of ordinary differential equations to an equivalent finite system of ordinary differential equations with respect to the evolution parameter is developed. A finite set of recurrent algebraic regular relations, allowing to generate solutions of the discrete nonlinear Schrodinger dynamical system, is constructed, the related functional spaces of ¨ solutions is discussed. Finally, the Fourier transform approach to studying the solution set of the discrete nonlinear Schrodinger dynamical system and its functional-analytical aspects is analyzed.
format	Article
author	Prykarpatsky, A.K. Cieśliński, J.
spellingShingle	Prykarpatsky, A.K. Cieśliński, J. The discrete nonlinear Schrödinger type hierarchy, its finite dimensional reduction analysis and numerical integrability scheme Нелінійні коливання
author_facet	Prykarpatsky, A.K. Cieśliński, J.
author_sort	Prykarpatsky, A.K.
title	The discrete nonlinear Schrödinger type hierarchy, its finite dimensional reduction analysis and numerical integrability scheme
title_short	The discrete nonlinear Schrödinger type hierarchy, its finite dimensional reduction analysis and numerical integrability scheme
title_full	The discrete nonlinear Schrödinger type hierarchy, its finite dimensional reduction analysis and numerical integrability scheme
title_fullStr	The discrete nonlinear Schrödinger type hierarchy, its finite dimensional reduction analysis and numerical integrability scheme
title_full_unstemmed	The discrete nonlinear Schrödinger type hierarchy, its finite dimensional reduction analysis and numerical integrability scheme
title_sort	discrete nonlinear schrödinger type hierarchy, its finite dimensional reduction analysis and numerical integrability scheme
publisher	Інститут математики НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/177304
citation_txt	The discrete nonlinear Schrödinger type hierarchy, its finite dimensional reduction analysis and numerical integrability scheme / A.K. Prykarpatsky, J. Cieśliński // Нелінійні коливання. — 2017. — Т. 20, № 2. — С. 228-266 — Бібліогр.: 61 назв. — англ.
series	Нелінійні коливання
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fulltext	UDC 517.9 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY, ITS FINITE DIMENSIONAL REDUCTION ANALYSIS AND NUMERICAL INTEGRABILITY SCHEME ДИСКРЕТНА НЕЛIНIЙНА IЄРАРХIЯ ТИПУ ШРЬОДIНГЕРА, АНАЛIЗ ЇЇ СКIНЧЕННОВИМIРНОЇ РЕДУКЦIЇ ТА ЧИСЕЛЬНА СХЕМА IНТЕГРУВАННЯ A. K. Prykarpatski AGH Univ. Sci. and Technology Krakow, Poland and Ivan Franko State Ped. Univ. Drohobych, Ukraine e-mail: pryk.anat@cybergal.com J. L. Cieśliński Univ. Białystok Lipowa Str., 41, Białystok, 15-424, Poland e-mail: janek@alpha.uwb.edu.pl We investigate discretizations of the integrable nonlinear Schrödinger dynamical system, well known as the Ablowitz – Ladik equation, the related symplectic structures and its finite dimensional invariant reducti- ons. An effective scheme of invariant reducing the corresponding infinite system of ordinary differential equations to an equivalent finite system of ordinary differential equations with respect to the evolution parameter is developed. A finite set of recurrent algebraic regular relations, allowing to generate solutions of the discrete nonlinear Schrödinger dynamical system, is constructed, the related functional spaces of solutions is discussed. Finally, the Fourier transform approach to studying the solution set of the discrete nonlinear Schrödinger dynamical system and its functional-analytical aspects is analyzed. Дослiджуються дискретизацiї iнтегровної нелiнiйної динамiчної системи Шрьодiнгера, вiдомої як рiвняння Абловiца – Ладiка, вiдповiднi симплектичнi структури та її скiнченновимiрнi iн- варiантнi редукцiї. Побудовано ефективний алгоритм iнварiантної редукцiї вiдповiдної нескiн- ченної системи звичайних диференцiальних рiвнянь до еквiвалентної скiнченної системи звичай- них диференцiальних рiвнянь вiдносно параметра еволюцiї. Побудовано скiнченну множину реку- рентних алгебраїчних регулярних спiввiдношень, що дозволило побудувати розв’язки дискрет- ної нелiнiйної динамiчної системи Шрьодiнгера, та розглянуто вiдповiднi функцiональнi прос- тори розв’язкiв. Проведено аналiз пiдходу перетворення Фур’є до вивчення множини розв’язкiв дискретної нелiнiйної динамiчної системи Шрьодiнгера та її функцiонально-аналiтичних аспек- тiв. 1. Introduction. As is well known, soliton equations with constitute an wide class of integrable dynamical systems, which possess a lot very interesting mathematical properties, and describe diverse important physical phenomena. In particular, they are usually used to describe interacti- ons between different solitary waves having physical applications. For example, the nonlinear Schrödinger equation describes the soliton propagation in a medium with both resonant and nonresonant nonlinearities [7, 46, 48] and the nonlinear interaction of high-frequency electro- c© A. K. Prykarpatski, J. L. Cieśliński, 2017 228 ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 229 static wave with ion acoustic waves in plasma [18]. Due to the important role played by the soliton equations in many fields of physics such as hydrodynamics, solid-state physics, plasma physics, etc. they have received much attention in the literature. Especially there are of great interest their completely integrable descrete approximations of soliton equations, which have a great deal of applications in numerical analysis, computing simulations and related investigati- ons. Our work is devoted to a thorough investigation of the three-point discretization ( dun/dt dūn/dt ) = Kn[u, ū] := ( i(un+1 − 2un + un−1)− iūnun(un+1 + un−1) −i(2ūn − un+1 − un−1) + iūnun(ūn+1 + ūn−1) ) (1.1) for n ∈ Z of the integrable nonlinear Schrödinger dynamical system K̃[ψ,ψ∗] :=  d dt ψ = iψxx − 2iαψψψ∗, d dt ψ∗ = −iψ∗xx + 2iαψ∗ψψ∗ (1.2) on a functional manifold M̃ ⊂ L2(R;C2), studying their related symplectic structures and finite-dimensional invariant reductions. The set of equations (1.1) is well known as the Ablo- witz – Ladik (AL-DNLS) equation, whose Lax type integrability was first proven by Ablowitz and Ladik [1] and having many diverse applications [5, 21, 24, 30, 33, 35, 60] in physical and bilological sciences. Defining (1.1) as a smooth completely integrable Hamiltonian dynami- cal system on a discrete functional manifold M2 ⊂ l2(Z;C2) with respect to the evolution parameter t ∈ R, we developed an effective scheme of invariant reducing the correspon- ding infinite system of ordinary differential equations to a suitably determined finite system of ordinary differential equations, being a completely integrable finite-dimensional canoni- cal Hamiltonian flow. The dynamical system (1.1) appears also to be a bi-Hamiltonian flow on the discrete functional manifold M2 with respect to special noncanonical Poisson brackets (see, e.g., [9, 11, 22]). A constructed finite set of recurrent algebraic regular relations, allowing to generate solutions of the discrete nonlinear Schrödinger dynamical system, is analyzed in detail, the related functional spaces of solutions are also discussed. Based on the symplectic gradient-holonomic approach, devised before in [11, 54, 55] for the smooth nonlinear dynmami- cal systems on functional manifolds, we also investigate the differential-geometric and symplec- tic stuctures of the related hidden symmetries, responsible for the complete integrability of the Ablowitz – Ladik dynamical system (1.1). Finally, there is analyzed the Fourier transform appro- ach to constructing the solution set of the discrete nonlinear Schrödinger dynamical system and its functional-analytical aspects. 2. Discrete dynamical systems integrability and reduction analysis. 2.1. Preliminary notions and definitions. We consider an infinite-dimensional discrete manifold Mm ⊂ l2(Z;Cm) for some integer m ∈ Z+ and a general nonlinear dynamical system of the form dw dt = K[w], (2.1) where w ∈ Mm and K : Mm → T (Mm) is a Fréchet smooth nonlinear local mapping of Mm into its tangent space T (Mm) and t ∈ R is the evolution parameter. As an example of the ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 230 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI dynamical system (2.1) at m = 2 on a discrete manifold M2 ⊂ l2(Z;C2), we will analyze the well-known [1, 44] AL-DNLS integrable system (1.1)( dun/dt dūn/dt ) = Kn[u, ū] := ( i(un+1 − 2un + un−1)− iūnun(un+1 + un−1) −i(2ūn − un+1 − un−1) + iūnun(ūn+1 + ūn−1) ) , (2.2) where the overbar denotes the complex conjugate on a functional manifold M2 with w = = (u, v)ᵀ ∈ M2, and which, as was before mentioned, has many interesting applications [20, 21, 35] in a wide range of modern physics and biology problems. To analyze the integrability properties of the differential-difference dynamical system (2.1), we shall develop a gradient-holonomic scheme related to those devised in [11, 32, 43, 54] for nonlinear dynamical systems defined on spatially one-dimensional functional manifolds and extended in [51] to include discrete manifolds. Denote by (·, ·) the standard bilinear form (or pairing) on the space T ∗(Mm) × T (Mm) naturally induced by the inner product in the Hilbert space l2(Z;Cm). We define D(Mm) to be the space of smooth functionals on Mm, so for any γ ∈ D(Mm) one can define the gradient grad γ[w] ∈ T ∗(Mm) as grad γ[u, ū] := γ′,∗[w] · 1, (2.3) where the prime denotes the Fréchet derivative and “∗” represents the conjugation with respect to the standard bracket on T (Mm)× T ∗(Mm). Definition 2.1. A linear smooth operator ϑ : T ∗(Mm) → T (Mm) is called Poissonian on the manifold Mm, if the bilinear bracket {·, ·} ϑ := (grad (·), ϑgrad (·)) satisfies [3, 6, 9, 25, 54] the Jacobi identity on the space D(Mm) of all smooth functionals on Mm. This means, in particular, that the bracket (2.3) satisfies the standard Jacobi identity on D(Mm). Definition 2.2. A linear smooth operator ϑ : T ∗(Mm) → T (Mm) is called Nötherian [9, 25, 54] with respect to the nonlinear dynamical system (2.1) if LKϑ = ϑ′K − ϑK ′,∗ −K ′ϑ = 0 (2.4) holds identically on the manifold Mm, where LK is the Lie-derivative along the vector field K : Mm → T (Mm). If the mapping ϑ : T ∗(Mm) → T (Mm) is invertible with inverse mapping ϑ−1 := Ω : T (Mm) → T ∗(Mm), it is called symplectic. It then follows easily from (2.4) that LKΩ = Ω′K + ΩK ′ +K ′,∗Ω = 0 (2.5) hold identically on Mm. Having now assumed that the manifold Mm ⊂ l2(Z;C2) is endowed with a smooth Poissonian structure ϑ : T ∗(Mm) → T (Mm), one can define the Hamiltonian system dw dt := −ϑgradH[w], (2.6) ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 231 corresponding to a Hamiltonian function H ∈ D(Mm). It follows directly from the definition (2.6) that the dynamical system dw dt = K[w] := −ϑ gradH[w] satisfies the Nötherian conditions (2.4). We are studying the integrability [6, 9, 11, 50] of the discrete dynamical system (2.1). Accordingly we need to construct invariants with respect to it functions, called conservation laws, which are mutually commuting with respect to the Poisson bracket (2.3). The following Lax criterion [11, 39, 54] proves to be very useful. Lemma 2.1. Any smooth solution ϕ ∈ T ∗(Mm) to the Lax equation LKϕ = dϕ dt +K ′,∗ϕ = 0, (2.7) satisfying the symmetry condition ϕ′ = ϕ′.∗, with respect to bracket (·, ·), is related to the conservation law γ := 1∫ 0 dλ(ϕ[wλ], w). (2.8) Proof. The expression (2.8) follows easily from the well-known Volterra homology equali- ties γ = 1∫ 0 dγ[wλ] dλ dλ = 1∫ 0 dλ ( 1, γ′[wλ] · w ) = 1∫ 0 dλ ( γ′,∗[wλ] · 1, w ) = 0∫ 1 dλ(grad γ[wλ], w) and (grad γ[w])′ = (grad γ[w])′,∗, holding identically onMm.Whence, one finds that there exists a function γ ∈ D(Mm) such that LKγ = 0, grad γ[w] = ϕ[w] for any w ∈ Mm. Lemma 2.1 is proved. This result of Lax lemma is a direct consequence of the following generalized Nöther type result. Lemma 2.2. Let a smooth element ψ ∈ T ∗(Mm) satisfy the Nöther condition LKψ = dψ dt +K ′,∗ψ = gradLψ (2.9) for some smooth functional Lψ ∈ D(Mm). Then the Hamiltonian representation K = −ϑgradHϑ ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 232 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI holds, where ϑ := ψ′ − ψ′,∗ and the Hamiltonian function is Hϑ = (ψ,K)− Lψ. It is easy to see that Lemma 2.1 follows from Lemma 2.2, if the conditions ψ′ = ψ′,∗ and Lψ = 0 are imposed on (2.9). Assume now that equation (2.9) allows an additional (nonsymmetric) smooth solution φ ∈ ∈ T ∗(Mm): LKφ = dφ dt +K ′,∗φ = gradLφ. (2.10) This means that our system (2.1) is bi-Hamiltonian: −ϑgradHϑ = K = −ηgradHη, where, by definition, η := φ ′ − φ′∗, Hη = (φ,K)− Lφ. Definition 2.3. One says that two Poissonian structures ϑ, η : T ∗(Mm) → T (Mm) on Mm are compatible [9, 25, 42, 54], if for any λ, µ ∈ R the linear combination λϑ + µη : T ∗(Mm) → → T (Mm) will be also Poissonian on Mm. It is easy to see that this condition is satisfied if, for instance, there exist an inverse ϑ−1 : T (Mm) → T ∗(Mm) and the composite map η(ϑ−1η) : T ∗(Mm) → T (Mm) is also Poissonian on Mm. Concerning the complete integrability of the infinite-dimensional dynamical system (2.1) on the discrete manifold Mm it is, in general, necessary, but not sufficient [11, 50, 54], to prove the existence of an infinite hierarchy of mutually commuting conservation laws with respect to the Poissonian structure (2.3). Since in the case of Lax integrability of (2.1) there exist compatible Poissonian structures and related hierarchies of conservation laws, we shall focus our analysis by devising an integrabi- lity algorithm under the a priori assumption that the nonlinear dynamical system (2.1) on the manifoldMm is Lax integrable. This means that it possesses a Lax representation in the followi- ng general form: ∆fn := fn+1 = ln[w;λ]fn, (2.11) where f := {fn ∈ Cr : n ∈ Z} ⊂ l2(Z;Cr) for some integer r ∈ Z+ and the matrices ln[w;λ] ∈ ∈ EndCr, n ∈ Z, in (2.11) are local matrix-valued functionals on Mm, depending on the “spectral” parameter λ ∈ C, invariant with respect to our dynamical system (2.1). As the Lax representation (2.11) is “local” with respect to the discrete variable n ∈ Z, we shall assume for convenience that our manifold Mm := M (N) m ⊂ l∞(Z/NZ;Cm) is periodic with respect to the discrete index n ∈ ZN , that is for any n ∈ ZN := Z/NZ and λ ∈ C ln[w;λ] = ln+N [w;λ] (2.12) ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 233 for some integer N ∈ Z+. In this case the smooth functionals on M (N) m can be represented as γ := ∑ n∈ZN γn[w] for some local Fréchet smooth densities γn : M (N) m → C, n ∈ ZN . 2.2. The gradient-holonomic scheme. Consider the representation (2.11) and define its fun- damental solution Fm,n(λ) ∈ Aut (Cr), m, n ∈ ZN , satisfying the equation Fm+1,n(λ) = lm[w;λ]Fm,n(λ) and the condition Fm,n(λ)\|m=n = 1 for all λ ∈ C and n ∈ ZN . Then the matrix function Sn(λ) := Fn+N,n(λ) (2.13) is called the monodromy matrix for the linear equation (2.12) and satisfies for all n ∈ ZN the Novikov – Lax relationship Sn+1(λ)ln = lnSn(λ). (2.14) It easy to compute that Sn(λ) := ∏N−1 k=0 ln+k[u;λ] owing to the periodicity condition (2.12). Construct now the generating functional γ̄(λ) := trSn(λ), (2.15) where tr is the standard trace map, having the asymptotic expansion γ̄(λ) ∼ ∑ j∈Z+ γ̄jλ j0−j (2.16) as λ → ∞ for some fixed j0 ∈ Z+. Then, owing to the obvious condition Dnγ(λ) = 0 for all n ∈ ZN , where we have introduced the “discrete” derivative Dn := ∆− 1, we find that all functionals γ̄j ∈ D(M (N) m ), j ∈ Z+, are independent of the discrete index n ∈ ZN and are simultaneously conservation laws for the dynamical system (2.1). We now make an additional natural assumption, namely that the gradient vector ϕ̄(λ) := grad γ̄(λ)[w] = tr l′,∗n ( Sn(λ)l−1 n ) , (2.17) ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 234 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI solving the Lax determining equation (2.7), satisfies, owing to (2.14), for all λ ∈ C, z(λ)ϑ ϕ̄(λ) = ηϕ̄(λ), (2.18) where z : C → C is a meromorphic function, and ϑ and η : T ∗(M (N) m ) → T (M (N) m ) are compatib- le Poissonian operators on the manifold M (N) m that are Nötherian with respect to the dynamical system (2.1). Then it follows at once that the generating functional γ(λ) ∈ D(M (N) m ) satisfies the commutation relationships {γ̄(λ), γ̄(µ)}ϑ = 0 = {γ̄(λ), γ̄(µ)}η (2.19) for all λ, µ ∈ C. Consequently, if we define on M(N) a generating dynamical system dw dτ := −ϑgrad γ̄(λ)[w] as λ → ∞, it follows from (2.19) that the hierarchy of functionals defined by the coefficients in (2.16) comprise its conservation laws. With the importance of invariants and Poissonian structures related to the linear spectral problem (2.11) firmly in mind, we now describe its main Lie-algebraic properties and connecti- ons with the whole hierarchy of integrable differential-difference dynamical systems on the manifold Mm. More precisely, we sketch the Lie-algebraic aspects [22, 49, 57, 58] of the diffe- rential-difference dynamical systems associated with the Lax linear difference spectral problem (2.11). In this process we shall assume that ln := ln[w;λ] ∈ Gn := GL2(C)⊗ C(λ, λ−1) for n ∈ ZN := Z/NZ as λ → ∞. To describe the related Lax integrable dynamical systems, we first define first the matrix product-group GN := ⊗Nj=1Gj and its action GN ×M (N) G → M (N) G on the phase space M (N) G := {ln ∈ Gn : n ∈ ZN}, given as {gn ∈ Gn : n ∈ ZN} × {ln ∈ Gn : n ∈ ZN} = {gnlng−1 n+1 ∈ Gn : n ∈ ZN}. A functional γ ∈ D(M (N) G ) is invariant for this action iff the following discrete relationship: grad γ(ln)ln = ln+1grad γ(ln+1) (2.20) holds for all n ∈ ZN . We assume further that the matrix group GN is identified with its tangent spaces Tl(GN ), l ∈ GN , which is locally isomorphic to the Lie algebra G(N), where G(N) is the corresponding Lie algebra of the Lie group GN , which is isomorphic to the tangent space Te(GN ) at the group unity e ∈ GN . With any element l ∈ GN there are associated, respectively, the left ηl : G(N) → → Tl(G N ) and right ρl : G(N) → Tl(G N ) differentials of the left and right translations on the Lie group GN , and their adjoint mappings ρ∗l : T ∗l (GN ) → G(N),∗ and η∗l : T ∗l (GN ) → G(N),∗, where (ρ∗l grad γ(l), X) = (gradγ(l), Xl) = (l grad γ(l), X) := Tr (l grad γ(l)X), (η∗l grad γ(l), X) = (grad γ(l), lX) = (grad γ(l)l,X) := Tr (grad γ(l)lX) (2.21) ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 235 for any X ∈ G(N) and smooth functional γ ∈ D(GN ). Here Tr: GN → C is the trace operation on the group GN defined as TrA := res λ=∞ ∑ j∈ZN SpAj [u, ū;λ] for any A ∈ GN . By virtue of (2.20) and (2.21), we can define the set {Φn = grad γ(ln)ln ∈ G∗n := T ∗e (G), n ∈ ZN} belonging to the space G(N),∗ ' T ∗e (GN ) and satisfying the following invariance property: Φn+1 = Ad∗lnΦn(λ) = l−1 n Φn(λ)ln (2.22) for any n ∈ ZN . The relationship (2.22) allows to define a function ϕ : GN → C invariant with respect to the adjoint action Gn ×Gn 3 (g, Sn(λ)) → adgSn(λ) = gSn(λ)g−1 ∈ Gn for any n ∈ ZN and such that γ(l) = ϕ[SN (λ)], ΦN = gradϕ[SN (λ)]SN (λ), (2.23) where, by definition, the expression SN (λ) = N∏ j=1 lj [u, ū;λ] (2.24) coincides exactly with the proper monodromy matrix for the linear spectral problem (2.11). Owing to (2.22), the matrices Φn = gradϕ[Sn(λ)]Sn(λ) ∈ G∗n, n ∈ ZN , can be reconstructed from (2.24). Therefore, we have [22, 58] the following Poissonian flow on the matrices Sn(λ) ∈ ∈ Gn, n ∈ ZN : dSn(λ) dt = [R(gradϕ[Sn(λ)]Sn(λ)), Sn(λ)] (2.25) with respect to the invariant Casimir function ϕ ∈ I(G∗n) and the quadratic Poissonian structure {γ1, γ2} := (l, [grad γ1(l),R(l grad γ2(l))] + [R(l grad γ1(l)), grad γ2(l)]) (2.26) for any functionals γ1, γ2 ∈ D(GN ), which is constructed by means of a skew-symmetric R-structureR : G(N)∗ → G(N). In particular, the equality [gradϕ(Sn), Sn] = 0 holds for all n ∈ ZN . ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 236 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI Taking into account (2.23), one can rewrite (2.25) as dSn dt = [R(grad γ(ln)ln), Sn], for all n ∈ ZN . This together with (2.22) makes it possible to retrieve [34, 57] the related evolution of elements ln ∈ Gn, n ∈ ZN : dln dt = pn+1(l)ln − lnpn(l), pn(l) := R(grad γ(ln)ln) (2.27) from the relationships Sn(λ) = ψn(l)SN (λ)ψ−1 n (l), ψn(l) = n∏ j=1 lj [u, v;λ]. The solution f ∈ l∞(Z,C2) to the linear spectral problem (2.11) satisfies the associated temporal evolution equation dfn dt = pn(l)fn (2.28) for any n ∈ Z. It is easy to check that the compatibility condition for the linear equations (2.11) and (2.28) is equivalent to the discrete Lax representation (2.27), which upon reduction on the group manifoldMG, gives rise to the corresponding nonlinear Lax integrable dynamical system on the discrete manifold M (N) m . Hence, all Casimir invariant functions, when reduced on the manifold MG, are in involution [23, 57, 58] with respect to the Poisson bracket (2.26). Since the existence of an infinite hierarchy of mutually commuting conservation laws is a characteristic of the Lax integrability of the nonlinear dynamical system (2.1), this property can be effectively implemented into the scheme of our analysis. Namely, we have the following result. Proposition 2.1. The determiming Lax equation (2.7) allows the following asymptotic (as λ → ∞) periodic solution ϕ(λ) ∈ T ∗(M (N) m ) : ϕn(λ) ∼ an(λ) exp[ω(t;λ)] n∏ j=0 σj(λ), (2.29) where for all n ∈ Z an(λ) := (1, a(1),n[w;λ], a(2),n[w;λ], . . . , a(m−1),n[w;λ])τ , a(k),n(λ) ∼ ∑ s∈Z+ a (s) (k),n[w]λ−s+ã, σj(λ) ∼ ∑ s∈Z+ a (s) j [w]λ−s+σ̃, (2.30) ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 237 1 ≤ k ≤ m − 1 and ω(t; ·) : C → C, t ∈ R, is a dispersion function. Moreover, the functional γ(λ) := ∑ n∈ZN ln(λ−σ̃σn[w;λ]) ∈ D(M (N) m ) is a generating function of conservation laws for the dynamical system (2.1). Proof. Lemma 2.1 and relationship (2.17) imply that the functional (2.15) is a conservation law for our dynamical system (2.1). Whence, expression (2.13) and equation (2.11) lead to the solution representation (2.29) for the Lax equation (2.7). Now, making use of the periodicity of the manifold M (N) m , it follows from the period translation of (2.29) that the functional γ(λ) := ∑ n∈ZN ln(λ−σ̃σn[w;λ]) ∼ ∑ j∈Z+ γjλ −j (2.31) generates an infinite hierarchy of conservation laws to (2.1). Proposition 2.1 is proved. Thus, if we start the Lax integrability analysis of a given nonlinear dynamical system (2.21), it is necessary, as the first step, to study the asymptotic solutions (2.29) to the corresponding Lax equation (2.7). These solutions are then used to construct a related hierarchy of conservation laws in the functional form (2.31), taking into account expansions (2.30). Remark 2.1. It is easy to observe that, owing to the arbitrariness of the period N ∈ Z+ of the manifold M (N) m , all of the finite-sum expressions obtained above can be generalized to the corresponding infinite-dimensional manifold Mm ⊂ l2(Z;Cm), if the associated infinite series are convergent. Since our dynamical system (2.1) induces a bi-Hamiltonian flow on the manifold M(N) under the above circumstances, the next step is to analyze the related compatible Poissonian or symplectic structures, satisfying, respectively, either equality (2.4) or equality (2.5). Before doing this, we shall need the following useful result. Lemma 2.3. All functionals γj ∈ D(M (N) m ) in the expansion (2.31) are mutually with respect to both Poissonian structures ϑ, η : T ∗(M (N) m ) → T (M (N) m ) satisfying the gradient relation- ship (2.32). Proof. It follows from the representations (2.29) and (2.17) that the following asymptotic (as λ → ∞) relationship holds: ln γ̄(λ) ' γ(λ). (2.32) Since the generating function γ̄(λ) ∈ D(M (N) m ) satisfies the commutation relationships (2.19), the same also holds, owing to (2.32), for the generating function γ(λ) ∈ D(M (N) m ). Lemma 2.3 is proved. We proceed now with an analytical approach to construction of the Poissonian structures ϑ, η : T ∗(M (N) m ) → T (M (N) m ) for the dynamical system (2.1). Note that these Poissonian structu- res are also Nötherian for the whole hierarchy of dynamical systems dw dtj := −ϑ grad γj [w], (2.33) where tj ∈ R, j ∈ Z+, are the corresponding evolution parameters, and which, owing to (2.19), commute with each other on the manifold M (N) m . Therefore, it possible to apply Lemma 2.2 to ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 238 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI any one of the dynamical systems (2.33) if the related vector fields commuting with (2.1) are assumed known. To solve equation (2.9) for an element ϕ ∈ T ∗(M (N) m ) one can, in the case of a polynomial dynamical system (2.1), make use of the well-known asymptotic small parameter method [43, 54]. When applying this approach, it is necessary to take into account the following expansions at zero — element (u, ū)ᵀ = 0 ∈ M (N) m with respect to the small parameter µ → 0: w := µw(1), ϕ [ w(1) ] = ϕ(0) + µϕ(1)[w(1)] + µ2ϕ(2)[w(1)] + . . . , d dt = d dt0 + µ d dt1 + µ2 d dt2 + . . . , K [ w(1) ] = µK(1) [ w(1) ] + µ(2)K(2) [ w(1) ] + . . . , K ′ [ w(1) ] = K ′0 + µK ′1 [ w(1) ] + µ2K ′2 [ w(1) ] + . . . , gradL [ w(1) ] = gradL(0) + µgradL(1) [ w(1) ] + µ2gradL(2)[w(1)] + . . . . After solving the corresponding set of linear nonuniform functional equations dϕ(0) dt0 +K ′∗0 ϕ (0) = gradL(0), dϕ(1) dt0 +K ′∗0 ϕ (1) = gradL(1) −K ′∗0 ϕ(0), dϕ(2) dt0 +K ′∗0 ϕ (2) = gradL(2) −K ′∗1 ϕ(1) −K ′∗2 ϕ(0) and so on, using Fourier transforms applied to the suitableN -periodic functions, one can obtain the related Poissonian structure in the series form ϑ−1 = ϕ(0),′ − ϕ(0),′∗ + µ(ϕ(1),′ − ϕ(1),′∗) + . . . and finally set µ = 1. Another direct way of obtaining a Poissonian operator ϑ : T ∗(M (N) m ) → T (M (N) m ) for (2.1) is the following: first reduce the Nötherian equation (2.4) to the set of linear nonuniform equati- ons d dt0 (ϑ0ϕ (0)) = K ′0(ϑ0ϕ (0)), d dt0 (ϑ1ϕ (0)) = K ′0(ϑ1ϕ (0)) + ϑ0K ′,∗ 1 ϕ(0) +K ′1ϑ0ϕ (0), d dt0 (ϑ2ϕ (0)) = K ′0(ϑ2ϕ (0))− ϕ(0)′K1 + ϑ0K ′,∗ 2 ϕ(0)+ + ϑ1K ′,∗ 1 ϕ(0) + ϑ2K ′,∗ 0 ϕ(0) +K ′1ϑ1ϕ (0) +K ′2ϑ0ϕ (0), ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 239 and then solve using the above small parameter asymptotics. The analytical expressions for actions ϑj : ϕ(0) → ϑjϕ (0), j ∈ Z+ can now be used to retrieve them in operator form from the expansion ϑ = ϑ0 + µϑ1 + µ2ϑ2 + . . . , by setting µ = 1 at the end of the calculations. Similarly one can also construct the second Poissonian operator η : T ∗(M (N) m ) → T (M (N) m ) for the nonlinear dynamical system (2.1). Now the next result follows directly from all of the above analysis. Proposition 2.2. Let a nonlinear dynamical system (2.1) on a discrete manifold M (N) m admit both a nontrivial symmetric solution ϕ ∈ T ∗(M (N) m ) to the Lax equation (2.7) in the asymptotic as form (2.29) as λ → ∞, generating an infinite hierarchy of nontrivial functionally independent conservation laws (2.31), and compatible nonsymmetric solutions ψ and φ ∈ T ∗(M (N) m ) to the Nöther equations (2.9) and (2.10), respectively. Then this dynamical system is a Lax integrable bi-Hamiltonian flow on M (N) m with respect to two compatible Poissonian structures ϑ, η : T ∗ ( M (N) m ) → T ( M (N) m ) , whose adjoint Lax representation dΛ dt = [ Λ,K ′,∗ ] , (2.34) where Λ := ϑ−1η, is the so-called recursion operator. This operator can be transformed, in virtue of the gradient relationship (2.18), to the standard discrete Lax form dln dt = [pn(l), ln] + (Dnpn(l))ln for some matrix pn(l) ∈ EndCr describing the temporal evolution dfn dt = pn(l)fn related to (2.11), for f ∈ l∞(Z;Cr). Remark 2.2. Inasmuch as all Hamiltonian flows (2.32) commute with each other and the dynamical system (2.1), and since they possess the same Poissonian and compatible (ϑ, η)- pair, the analytical algorithm described above can also be applied to any other flow commuting with (2.1). Solutions to the discrete linear Lax problem (2.11) can be constructed by means of the gradient-holonomic algorithm devised in [11, 32, 54] for studying the integrability of nonlinear dynamical systems on functional manifolds. More specifically, by making use of the preliminary analytical expressions for the related compatible Poissonian structures ϑ, η : T ∗ ( M (N) m ) → → T ( M (N) m ) on the manifold M (N) m and using the fact that the recursion operator Λ := := ϑ−1η : T ∗ ( M (N) m ) → T ∗ ( M (N) m ) satisfies the dual Lax commutator equality (2.34), one can retrieve the standard Lax representation for it in terms of algebraic formulas. As a corollary of ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 240 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI Proposition 2.2 one has the existence of a nontrivial asymptotic (as λ → ∞) solution to the Lax equation (2.7), which provides an effective Lax integrability criterion for a dynamical system (2.1) on the manifold M (N) m . 2.3. The Bogoyavlensky – Novikov finite-dimensional reduction scheme. In this section, we assume that our dynamical system (2.1) on the periodic manifold M (N) m is Lax integrable and possesses two compatible Poissonian structures ϑ, η : T ∗(M (N) m ) → T (M (N) m ). Thus, we have the nonlinear finite-dimensional dynamical system dw dt := Kn[w] = −ϑ gradHn[w] (2.35) for indices n ∈ ZN , owing to its N -periodicity. The finite-dimensional dynamical system (2.35) can be equivalently considered as that on the finite-dimensional space M (N) m ' (Cm)N para- meterized by an integer index n ∈ ZN . The Liouville integrability of this system is our next concern. To study the flow (2.34) on the manifold M(N), we shall make use of the Bogoyavlens- ky – Novikov [13, 50] reduction scheme [9, 50, 51, 54]. Let Λ ( M (N) m ) := ⊗Nj=0Λj ( M (N) m ) be the standard finitely generated Grassmann algebra [6, 11 54] of differential forms on the manifold M(N). Then the differential complex Λ0(M (N) m ) d→ Λ1(M (N) m ) d→ · · · d→ Λj(M (N) m ) d→ Λj+1(M (N) m ) d→ · · · , where d : Λ(M (N) m ) → Λ(M (N) m ) is the exterior differentiation, is finite and exact. Since the di- screte “derivative”Dn := ∆−1 commutes with the differentiation d : Λ ( M (N) m ) → Λ ( M (N) m ) , [Dn, d] = 0 for all n ∈ ZN , and for any element a ∈ Λ0 ( M (N) m ) grad ∑ n∈ZN Dnan[w]  = 0, (2.36) one can formulate the following Gelfand – Dikiy type [26] result. Lemma 2.4. Let L[w] ∈ Λ0 ( M (N) m ) be a Fréchet smooth local Lagrangian functional on the manifold M (N) m . Then there exists a differential 1-form α(1) ∈ Λ1(M (N) m ), such that the equality dLn[w] = 〈gradLn[w], d(w)ᵀ〉+Dnα (1) n [w] (2.37) holds for all n ∈ ZN . Proof. One can easily see that dLn[w] = N−1∑ j=0 〈 ∂Ln[w] ∂wn+j , dwn+j 〉 = N−1∑ j=0 〈 ∂Ln[w] ∂wn+j , ∆jdwn 〉 = = 〈 N−1∑ j=0 ∆−j ∂Ln[w] ∂wn+j , dwn 〉 +Dn N−1∑ j=0 〈pj , dwn+j〉  , ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 241 where pk := N−1∑ j=0 ∆−j ∂Ln[w] ∂wn+j+k+1 for k = 0, . . . , N − 1. Having defined the expression gradLn[w] := N−1∑ j=0 ∆−j ∂Ln[w] ∂wn+j , one obtains the result (2.37), where α(1) n [w] := N−1∑ j=0 〈pj , dwn+j〉 (2.38) is the corresponding differential 1-form on the manifold M (N) m , thereby concluding the proof. Lemma 2.4 is proved. Exterior differentiating expression (2.37), we obtain that −Dnω (2) n [w] = 〈d gradLn[w], ∧dw〉 (2.39) for any n ∈ Z, where the 2-form ω(2)[w] := dα(1)[w] (2.40) is nondegenerate on M (N) m if the Hessian ∂2 nL[w]/∂2w is also nondegenerate. Consider the manifold M̄ (N) m := { gradL(Ñ) n [w] = 0; w ∈ M (N) m } , (2.41) where the Lagrangian functional is defined as L(N̄) := −γN̄ + N̄−1∑ j=0 cjγj , (2.42) with γj ∈ D ( M (N) m ) , j = 0, . . . , N̄ −1, for some N̄ ∈ Z+, being suitable nontrivial conservati- on laws for the dynamical system (2.1) as constructed above. Here cj ∈ C, ≤ j ≤ N̄ − 1, are arbitrary but fixed constants. It follows from (2.41) and (2.39) that the closed 2-form ω(2) ∈ ∈ Λ2 ( M (N) m ) is invariant with respect to the index n ∈ ZN on the manifold M̄ (N) m . Moreover, the submanifold (2.41) is also invariant both with respect to the index n ∈ ZN and the evolution parameter t ∈ R. In fact, for any n ∈ ZN the Lie derivative LKgradL(N̄) = ( gradL(N̄) )′ K +K ′,∗ ( gradL(N̄) ) = 0, ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 242 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI since the functional L(N̄) n [w] ∈ D ( M̄ (N) m ) is a sum of conservation laws for the dynamical system (2.1), whose gradients satisfies the Lax condition (2.7). In addition, it is easy to see that if the Lie derivative LKgradL(N̄) n [w] = 0, n ∈ ZN , at t = 0, then gradL(N̄) n [w] = 0 for all t ∈ R and n ∈ ZN . Thus, the Bogoyavlensky – Novikov reduction of the dynamical system (2.1) upon the invariant submanifold M̄ (N) m is completely invariantly defined. At this point there is a natural question to ask: what is the relationship between the dynami- cal system (2.1) restricted to the submanifold M (N) m and the dynamical system (2.1) reduced on the finite-dimensional submanifold M̄ (N) m ⊂ M (N) m ? To further analyze the reduction, we consi- der the equation 〈 gradL(N̄) n [w],Kn[w] 〉 = −Dnh (t) n [w], (2.43) for a local functional h(t)[w] ∈ Λ0(Mm), which follows from the conditions (2.35) and (2.7): grad 〈 grad L(N̄) n [w],Kn[w] 〉 = ( gradL(N̄) n [w] )′,∗ Kn[w] +K ′,∗n [w]gradL(N̄) n [w] = = ( grad L(N̄) n [w] )′ Kn[w] +K ′,∗n [w]gradL(N̄) n [w] = = LK gradL(N̄) n [w] = 0. Since on the submanifold M̄ (N) m the gradient grad L(Ñ) n [w] = 0 for all n ∈ ZN , we deduce from (2.43) that the local functional h(t)[w] ∈ Λ0 ( M̄ (N) m ) does not depend on index n ∈ ZN . The properties of the manifold M̄ (N) m described above, make it possible to consider it as a symplectic manifold endowed with the symplectic structure ω(2) ∈ Λ2 ( M̄ (N) m ) given by expressions (2.38) and (2.40). From this point of view we can study the integrability properties of the dynamical system (2.1) reduced on the invariant finite-dimensional manifold M̄ (N) m ⊂ ⊂ M (N) m . First, we observe that the vector field d/dt on M̄(N) is canonically Hamiltonian [3, 6, 50] with respect to the symplectic structure ω(2) ∈ Λ2(M̃(N)), i.e., −i d dt ω(2)(u, p) = dh(t)(u, p), (2.44) where h(t)(w, p) := h(t)[w], ω(2)(w, p) := ω(2)[w] and (w, p)ᵀ ∈ M̄ (N) m are canonical variables induced on the manifold M̄ (N) m by the Liouville 1-form (2.38). More specifically, from expression (2.43) one obtains that di d dt 〈 gradL(N̄) n [w], dwn 〉 = −Dndh (t) n [w], which together with the identity (2.39) in the form i d dt d 〈 gradL(N̄) n [w], dwn 〉 = −Dni d dt ω(2) n [w], ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 243 leads to d dt 〈 gradL(N̄) n [w], dwn 〉 = −Dn ( dh(t) n [w] + i d dt ω(2) n [w] ) . (2.45) Since gradL(N̄)[w] = 0 = LK gradL[w] identically on M̄ (N) m , from (2.45) one obtains the result (2.44). The same is true of any of the Hamiltonian systems (2.33) commuting with (2.1) on the manifold Mm. Moreover, owing to the functional independence of invariants γj ∈ D ( M (N) m ) , 0 ≤ j ≤ N − 1, in the Lagrangian functional (2.42), we can construct a set of functionally independent functions h(j) ∈ D ( M̄ (N) m ) , j = 0, . . . , N̄ − 1, as follows:〈 gradL(Ñ) n [w], ϑ grad γj,n[w] 〉 = Dnh (j) n [w]. It is easy to check that these functions h(j) ∈ D ( M̄ (N) m ) , 0 ≤ j ≤ N̄ − 1, are invariant with respect to indices n ∈ ZN and commute with each other and the Hamiltonian functi- on h(t) ∈ D ( M̄ (N) m ) with respect to the symplectic structure ω(2) ∈ Λ2 ( M̄ (N) m ) . Thus, if the dimension dim M̃(N) = 2Ñ , the discrete dynamical system (2.1) reduced upon the finite- dimensional submanifold M̄ (N) m ⊂ M (N) m is Liouville integrable. If the set of conservation laws γj ∈ D ( M (N) m ) , j = 0, . . . , N − 1, is functionally dependent on M (N) m , the scheme can be modified using the Dirac reduction technique [3, 9, 54] for determining a regular symplectic structure ω̄(2)[w] ∈ Λ2 ( M̄ (N) m ) on an invariant nonsingular submanifold M̄ (N) m . 3. The discrete nonlinear Schrödinger dynamical system analysis. 3.1. Hamiltonian descrip- tion. In that to follow we proceed to analyzing the properties of discrete approximation for the nonlinear integrable Schrödinger dynamical system (1.2) on a functional manifold M̃ ⊂ ⊂ L2 ( R;C2 ) in the form K̃ [ψ,ψ∗] :=  d dt ψ = iψxx − 2iαψψψ∗, d dt ψ∗ = −iψ∗xx + 2iαψ∗ψψ∗, (3.1) where, by definition (ψ,ψ∗)ᵀ ∈ M̃, α ∈ R is a constant, the subscript “x” means the partial derivative with respect to the independent variable x ∈ R, K̃ : M̃ → T (M̃) is the correspon- ding vector field on M̃ and t ∈ R is the evolution parameter. The system (3.1) possesses a Lax type representation (see [50]) and is Hamiltonian d dt (ψ,ψ∗)ᵀ = −θ̃ grad H̃ [ψ,ψ∗] = K̃ [ψ,ψ∗] (3.2) with respect to the canonical Poisson structure θ̃ and the Hamiltonian function H̃, where θ̃ := ( 0 −i i 0 ) (3.3) ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 244 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI is a nondegenerate mapping θ̃ : T ∗(M̃) → T (M̃) on the smooth functional manifold M̃, and H̃ := 1 2 ∫ R dx [ ψψ∗xx + ψxxψ ∗ − 2α (ψ∗ψ)2 ] , (3.4) is a smooth mapping H̃ : M̃ → C. The corresponding symplectic structure [3, 6, 9, 11] for the Poissonian operator (3.3) is defined by ω̃(2) := − i 2 ∫ R dx 〈 (dψ, dψ∗)ᵀ ,∧θ̃−1 (dψ, dψ∗)ᵀ 〉 = −i ∫ R dx [dψ∗(x) ∧ dψ(x)] , (3.5) which is a nondegenerate and closed 2-form on the functional manifold M̃. The simplest spatial discretizations of the dynamical system (3.1) look as the flows d dt ψn = i h2 (ψn+1 − 2ψn + ψn−1)− 2iαψnψnψ ∗ n, d dt ψ∗n = − i h2 ( ψ∗n+1 − 2ψ∗n + ψ∗n−1 ) + 2iαψ∗nψnψ ∗ n (3.6) and K[ψn, ψ ∗ n] :=  d dt ψn = i h2 (ψn+1 − 2ψn + ψn−1)− iα (ψn+1 + ψn−1)ψnψ ∗ n, d dt ψ∗n = − i h2 ( ψ∗n+1 − 2ψ∗n + ψ∗n−1 ) + iα ( ψ∗n+1 + ψ∗n−1 ) ψnψ ∗ n, (3.7) on some “discrete” submanifold Mh, where, by definition, {(ψn, ψ∗n)ᵀ ∈ C2 : n ∈ Z} ⊂Mh ⊂ ⊂ l2(Z;C2) and K : Mh → T (Mh) is the corresponding vector field on Mh. Definition 3.1. If for a function (ψ,ψ∗)ᵀ ∈ W 2 2 (R;C2) there exists the point-wise limit limh→0(ψn, ψ ∗ n)ᵀ = (ψ(x), ψ∗(x)))ᵀ, where the set of vectors (ψn, ψ ∗ n)ᵀ ∈ C2, n ∈ Z, solves the infinite system of equations (3.7), the set {(ψn, ψ∗n)ᵀ ∈ C2 : n ∈ Z} ⊂l2(Z;C2) will be called an approximate solution to the nonlinear Schrödinger dynamical system (3.1). It is well known [1, 2] that the discretization scheme (3.7) conserves the Lax type integrabi- lity [9, 11, 50] and that the scheme (3.6) does not. The integrability of (3.7) can be easily enough checked by means of either the gradient-holonomic integrability algorithm [11, 51, 53] or the well known [41] symmetry approach. In particular, the discrete dynamical system (3.7) is a Hamiltonian one [3, 6, 9, 51] on the symplectic manifold Mh ⊂ l2(Z;C2) with respect to the noncanonical symplectic structure ω (2) h = − ∑ n∈Z ih 2(1− h2αψ∗nψn) 〈(dψn, dψ∗n)ᵀ ,∧ (dψn, dψ ∗ n)ᵀ〉 (3.8) on Mh and looks as d dt (ψn, ψ ∗ n)ᵀ = −θn gradH [ψn, ψ ∗ n] = K [ψn, ψ ∗ n] , (3.9) ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 245 where the Hamiltonian function H = ∑ n∈Z 1 h ( ψnψ ∗ n+1 + ψn+1ψ ∗ n + 2 αh2 ln ∣∣1− αh2ψ∗nψn ∣∣) (3.10) and the related Poissonian operator θn : T ∗(Mh) → T (Mh) equals θn := ( 0 −ih−1 ( 1− h2αψ∗nψn ) ih−1 ( 1− h2αψ∗nψn ) 0 ) . (3.11) Remark 3.1. For the symplectic structure (3.8) and, respectively, the Hamiltonian function (3.10) to be suitably defined on the manifold Mh ⊂ l2(Z;C2) it is necessary to assume additi- onally that the finite stability condition limN,M→∞ (∏M −N (1− αh2ψ∗nψn) ) 6= 0 holds. The latter is simply reduced as h → 0 to the equivalent integral inequality α ≤ ∫ R (xψ∗ψ)2 dx ∫ R ψ∗ψdx −1 , which will be assumed for further to be satisfied, respectively, the manifold M̃ ⊂ W̃ 2 2 (R;C2), where W̃ 2 2 (R;C2) := W 2 2 (R;C2) ∩ L(1) 2 (R;C2) with the space L (1) 2 (R;C2) := { (ψ,ψ∗)ᵀ ∈ ∈ L2(R;C2) : ∫ R x2(ψ∗ψ)2dx < ∞ } . The symplectic structure (3.8) is well defined on the manifold Mh and tends as h → 0 to the symplectic structure (3.5) on M̃, and respectively the Hamiltonian function (3.10) tends to (3.4). In this work we have investigated the structure of the solution set to the discrete nonlinear Schrödinger dynamical system (3.7) by means of a specially devised analytical approach for invariant reducing the infinite system of ordinary differential equations (3.7) to an equivalent finite one of ordinary differential equations with respect to the evolution parameter t ∈ R. As a result, there was constructed a finite set of recurrent algebraic regular relationships, allo- wing to expand the obtained before finite set of solutions to any discrete order n ∈ Z, which makes it possible to present a wide class of the approximate solutions to the nonlinear Schrödinger dynamical system (3.1). It is worthy here to stress that the problem of construc- ting an effective discretization scheme for the nonlinear Schrödinger dynamical system (3.1) and its generalizations proves to be important both for applications [4, 36, 59] and for deeper understanding the nature of the related algebro-geometric and analytic structures responsible for their limiting stability and convergence properties. From these points of view we would like to mention work [40], where the standard discrete Lie-algebraic approach [9, 10] was recently applied to constructing a slightly different from (3.6) and (3.7) discretization of the nonlinear Schrödinger dynamical system (3.1). As the symplectic reduction method, devised in the present work for studying the solution sets of the discrete nonlinear Schrödinger dynami- cal system (3.7), is completely independent of a chosen discretization scheme, it would be reasonable and interesting to apply it to that of [40] and compare the corresponding results subject to their computational effectiveness. The discrete nonlinear Schrödinger dynamical system (2.2) is defined on the periodic manifold M2 ⊂ l∞(Z;C2). Its Lax type integrability ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 246 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI was proved in [1, 12, 44] making use of the simplest discretization of the standard Zakharov – Shabat spectral problem for the well-known nonlinear Schrödinger equation. We begin this section by applying the gradient-holonomic integrability analysis described above to the di- screte dynamical system (2.2). First, we shall show the existence of an infinite hierarchy of functionally independent conservation laws obtained by solving the determining Lax equation (2.7) in the asymptotic form (2.29). The following is a key result for our analysis. Lemma 3.1. The functional expression ϕn := ( 1 an(λ) ) exp [ it(2− λ− λ−1) ] n∏ j=0 σj(λ), (3.12) where σj(λ) ∼ λ hj [u, ū] 1− ∑ s∈Z+ σ (s) j [u, ū]λ−s−1  , an(λ) ∼ ∑ s∈Z+ a(s) n [u, ū]λ−s, (3.13) is an asymptotic solution to the determining Lax equation dϕn dt +K ′,∗ϕn = 0 (3.14) as λ → ∞ for all n ∈ ZN with the operator K ′,∗ : T ∗(M2) → T ∗(M2) of the form K ′,∗n =  i∆−1D2 n − iūn(un+1 + un−1)− −i(∆ + ∆−1) · ūnun iūn(ūn+1 + ūn−1) −iun(un+1 + un−1) −i∆−1D2 n + iun(ūn+1 + ūn−1)+ +i(∆ + ∆−1) · ūnun  . (3.15) Proof. It suffices to find the corresponding coefficients of the asymptotic expansions (3.13). To do this, we consider the following two equations that can be easily obtained from (3.14), (3.15) and (3.12): D−1 n d dt − lnhn + ln 1− ∑ s∈Z+ σ(s) n λ−s−1 + + iλ h−1 n+1(1− ūnun) 1− ∑ s∈Z+ σ(s) n λ−s−1 − 1 + + i λ (1− ūn−1un−1)hn 1− ∑ s∈Z+ σ(s) n λ−s−1 −1 − 1 − − iūn(un+1 + un−1) + iūn(ūn+1 + ūn−1) ∑ s∈Z+ a(s) n λ−s (3.16) ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 247 and∑ s∈Z+ a(s) n λ−s D−1 n d dt − lnhn + ln 1− ∑ s∈Z+ σ(s) n λ−s−1 + 4i ∑ s∈Z+ a(s) n λ−s + + iλhn+1(ūn+1un+1 − 1) ∑ s∈Z+ a (s) n+1λ −s ∑ s∈Z+ a (s) n+1λ −s − ∑ s∈Z+ a(s) n λ−s + + i λ (ūn−1un−1 − 1) ∑ s∈Z+ a (s) n+1λ −s hn 1− ∑ s∈Z+ σ(s) n λ−s−1 −1 − ∑ s∈Z+ a(s) n λ−s + + d dt ∑ s∈Z+ a(s) n λ−s − iun(un+1 + un−1) + iun(ūn+1 + ūn−1) ∑ s∈Z+ a(s) n λ−s. Now equating the coefficients of (3.16) at the same degrees of the parameter λ ∈ C, we recursi- vely obtain the functional expression expression for hn, σ (s) n and a(s) n , n ∈ Z, s ∈ Z+; namely, hn = (1− u∗nun), a(0) n = 0, a(1) n = β, σ(0) n = u∗n−1(un + un−2)− i∆−1D2 n(lnhn)t, σ(1) n = i d dt σ (0) n−1 + (hn−1hn−2 − 1) + a (1) n−1u ∗ n−1(un + un−2), a(2) n = −3a (1) n−1 + i d dt σ (1) n−1 − ia (1) n−1D −1 n (lnhn−1)t + a(1) n σ(0) n − un−1 ( u∗n + u∗n−2 ) a (1) n−1, dhn dt = iDn(u∗n−1un − u∗nun−1), . . . , whence σ(0) n = − ( u∗nun−1 + u∗n−1un−2 ) , σ(1) n = i d dt σ (0) n−1 + ( 1− u∗n−1un−1 ) ( 1− u∗n−2un−2 ) + βu∗n−1(un + un−2), . . . , and so on. Thus, the corresponding recursion formulas are solvable for all s ∈ Z+, so it follows that the expression (3.12) is a true asymptotic solution to the Lax equation (3.14). Lemma 3.1 is proved. Recalling now that the expression γ(λ) := − N−1∑ n=0 lnhn + N−1∑ n=0 ln 1− ∑ s∈Z+ σ(s) n λ−s−1  ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 248 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI as λ → ∞ is a generating function of conservation laws for the dynamical system (2.2), one finds that functionals γ̄0 = N−1∑ n=0 ln(1− ūnun), γ0 = − N−1∑ n=0 σ(0) n , γ1 = − N−1∑ n=0 ( σ(1) n + 1 2 σ(0) n σ(0) n ) , γ2 = − N−1∑ n=0 ( σ(2) n + 1 3 σ(0) n σ(0) n σ(0) n + σ(0) n σ(1) n ) , . . . , and so on, make up an infinite hierarchy of exact conserved quantities for the discrete nonlinear Schrödinger dynamical system (2.2). A few remarks are in order concerning the complete integrability of the discrete nonlinear Schrödinger dynamical system (2.2). First, we can easily show using the standard asymptotic small parameter approach [11, 32, 54] that the Nöther equation (2.4) on the manifold M (N) 2 possesses [44, 51] the exact Poissonian operator solution ϑn = ( 0 ihn −ihn 0 ) , (3.17) for n ∈ ZN , subject to which the dynamical the dynamical system (2.2) is Hamiltonian via d dt (u, u∗)ᵀ = −ϑ gradHϑ [u, u∗] on the periodic manifold M (N) 2 , where the Hamiltonian function is Hϑ := N∑ n=0 lnh2 n − N∑ n=0 (ūnun+1 − ūnun+1) = 2 ln \|γ0\| − 1 2 (γ0 + γ̄0). Similar, but more cumbersome, calculations can be employed to find a second Poissonian operator solution to the Nöther equation (2.4) in the matrix form: η = ( ( hn − unD−1 n un ) ∆ ( u2 n + unD −1 n un ) ∆−1 u∗nD −1 n u∗n∆ − ( 1 + u∗nD −1 n un ) ∆−1 ) × × ( unD −1 n un ( hn − unD−1 n u∗n ) 1 + u∗nD −1 n un −(u∗n + u∗nD −1 n u∗n) ) . (3.18) where the operation D−1 n (·) := 1 2 [ n−1∑ k=0 (·)k − N−1∑ k=n (·)k ] ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 249 is quasiskew-symmetric with respect to the usual bilinear form on T ∗ ( M (N) 2 ) × T ( M (N) 2 ) , satisfying the operator identity ( D−1 n )∗ = −∆−1D−1 n ∆, n ∈ Z. The Poissonian operators (3.17) and (3.18) are compatible, so we can obtain the related Lax representation for the dynamical system (2.2) by means of the algebraic gradient-holonomic algorithm. The corresponding result is as follows: the discrete linear spectral problem ∆fn = ln[u, u∗;λ]fn, (3.19) where f ∈ l∞(Z;C2) and for n ∈ Z ln[u, u∗;λ] = ( λ un u∗n λ−1 ) , allows the linear Lax isospectral evolution dfn dt = pn(l)fn (3.20) for some matrix pn(l) ∈ End C2, n ∈ Z, which is equivalent to the Hamiltonian flow dfn dt = {Hϑ, fn}ϑ, (3.21) where {·, ·}ϑ is the Poissonian structure on the manifold M (N) 2 corresponding to (3.17). The equivalence of (3.17) and (3.21) can be easily demonstrated by constructing the monodromy matrix Sn(λ), n ∈ ZN , for all λ ∈ C corresponding to (3.19) and calculating the Hamiltonian evolution d dt Sn(λ) = {Hϑ, Sn(λ)}ϑ = [pn(l), Sn(λ)], giving rise to the same matrix pn(l) ∈ EndC2, n ∈ Z, as in equation (3.20). Thus, we have shown that the nonlinear discrete Schrödinger dynamical system (2.2) is a Lax integrable bi-Hamiltonian flow on the manifold M (N) 2 . Since the solution ϕ(λ) ∈ T ∗ ( M (N) 2 ) constructed above satisfies the gradient-like relationship λϑϕ(λ) = ηϕ(λ) for all for λ ∈ C,we showed that the conservation laws are mutually commuting with respect to both Poisson brackets {·, ·}ϑ and {·, ·}η. From whence follows the classical Liouville integrabi- lity [6, 43] of the discrete nonlinear Schrödinger dynamical system (2.2) on the periodic mani- fold M (N) 2 . A detailed analysis of the integrability procedure via the Bogoyavlensky – Novikov reduction [13, 50] and an explicit construction of solutions to the dynamical system (2.2) are planned for a later paper. ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 250 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI 4. Finite dimensional reductions and their exact integrability. 4.1. A class of Hamiltonian discretizations of the NLS dynamical system. The discretizations (3.6) and (3.7) can be extended to a wide classs of Hamiltonian systems, if to assume that the Poissonian structure is given by the local expression θn =  0 −iνn ( gn − h̃2 nαψ ∗ nψn ) iνn ( gn − h̃2 nαψ ∗ nψn ) 0  , (4.1) generalizing (3.11), and the Hamiltonian function is chosen in the form H = ∑ n∈Z hn ( anψnψ ∗ n+1 + bnψnψ ∗ n + cnψnψ ∗ n−1 + 2dn α ln ∣∣∣gn − αh̃2 nψnψ ∗ n ∣∣∣) , (4.2) where hn, h̃n, νn, an, bn, cn, dn and gn ∈ R+, n ∈ Z, are some parameters. The reality condition, imposed on the Hamiltonian function (4.2), yields the relationships cnhn = a∗n−1hn−1, b∗n = bn, d∗n = dn, which should be satisfied for all n ∈ Z. As a result, there is obtained the corresponding gene- ralized discrete nonlinear Schrödinger dynamical system d dt (ψn, ψ ∗ n)ᵀ := −θngradH [ψn, ψ ∗ n] , n ∈ Z, equivalent to the infinite set of ordinary differential equations d dt ψn = iνn ( hn+1cn+1gnψn+1 + ( bngnhn − 2h̃2 nhndn ) ψn + hn−1an−1gnψn−1 ) − − iανnh̃2 n (hn+1cn+1ψn+1 + hnbnψn + hn−1an−1ψn−1)ψnψ ∗ n, (4.3) d dt ψ∗n = −iνn ( hnangnψ ∗ n+1 + ( bngnhn − 2h̃2 nhndn ) ψ∗n + hncngnψ ∗ n−1 ) + + iανnh̃ 2 n ( hnanψ ∗ n+1 + hnbnψ ∗ n + hncnψ ∗ n−1 ) ψnψ ∗ n for all n ∈ Z. In the completely autonomous case, when hn = h, h̃n = h̃, νn = ν, an = a, bn = b, cn = c, dn = d and gn = g ∈ R+ for all n ∈ Z, the Poissonian structure (4.1) becomes θn =  0 −iν ( g − h̃2αψ∗nψn ) iν ( g − h̃2αψ∗nψn ) 0  and the Hamiltonian function (4.2) becomes H = ∑ n∈Z h ( aψnψ ∗ n+1 + bψnψ ∗ n + cψnψ ∗ n−1 + 2d α ln ∣∣∣g − αh̃2ψnψ ∗ n ∣∣∣) . (4.4) The corresponding reality condition for (4.4) reads as c = a∗, b∗ = b, d∗ = d, ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 251 and the related discrete nonlinear Schrödinger dynamical systems reads as a set of the equations d dt ψn = iνh ( cgψn+1 + ( bg − 2h̃2d ) ψn + agψn−1 ) − iανhh̃2 (cψn+1 + bψn + aψn−1)ψnψ ∗ n, d dt ψ∗n = −iνh ( agψ∗n+1 + ( bg − 2h̃2d ) ψ∗n + cgψ∗n−1 ) + iανhh̃2 ( aψ∗n+1 + bψ∗n + cψ∗n−1 ) ψnψ ∗ n for all n ∈ Z. If now to make in (4.3) the substitutions νn = 1 hn , gn = 1, h̃n = hn, an = 1 h2 n , bn = 0, cn = 1 hnhn−1 , dn = 1 h4 n , one obtains the discrete nonlinear Schrödinger dynamical system K(g) n [ψn, ψ ∗ n] :=  d dt ψn = i h2 n ( ψn+1 − 2ψn + hnh −1 n−1ψn−1 ) − −iα ( ψn+1 + hnh −1 n−1ψn−1 ) ψ∗nψn, d dt ψ∗n = − i h2 n ( ψ∗n+1 − 2ψ∗n + hnh −1 n−1ψ ∗ n−1 ) + +iα ( ψ∗n+1 + hnh −1 n−1ψ ∗ n−1 ) ψ∗nψn, (4.5) whose Hamiltonian function equals H(g) = ∑ n∈Z h−1 n ( ψnψ ∗ n+1 + ψn+1ψ ∗ n + 2 αh2 n ln ∣∣1− αh2 nψ ∗ nψn ∣∣) . (4.6) Another substitution, taken in the form c = a 6= 0, νhga = 1 h2 , ( bg − 2h̃2d ) νh = − 2 h2 , νhh̃2(a+ b+ c) = 2, (4.7) is also suitable in the limit h → 0 for discretization the nonlinear Schrödinger dynamical system (3.7). The corresponding discrete nonlinear Schrödinger dynamics takes the form d dt ψn = i h2 (ψn+1 − 2ψn + ψn−1)− 2iα 2 + µ (ψn+1 + ψn−1 + µψn)ψnψ ∗ n, d dt ψ∗n = − i h2 ( ψ∗n+1 − 2ψ∗n + ψ∗n−1 ) + 2iα 2 + µ ( ψ∗n+1 + ψ∗n−1 + µψ∗n ) ψnψ ∗ n (4.8) for all for all n ∈ Z, where µ = b a ∈ R+. Thus we obtained a one-parameter family of Hamil- tonian discretizations of the NLS equation. The set of relationships (4.7) admits a lot of reducti- ons, for instance, one can take ν = 1, g = 1, a = 1 h3 , d = ( µ+ 2 2 )2 1 h5 , h̃2 h2 = 2 2 + µ , ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 252 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI not changing the infinite set of equations (4.8). All of the constructed above discretizations of the nonlinear Schrödinger dynamical system (3.1) on the functional manifold M̃ can be considered as either better or worse from the com- putational point of view. If some of the discretization allows, except the Hamiltonian functi- on, some extra conservation laws, it can be naturally considered as a much more suitable for numerical analysis case, allowing both to control the stability of the solution convergence, as a parameter R+ 3 h → 0, and to make an invariant solution space reduction to a lower effective dimension of the related solution set. It is worthy to observe here that the functional structure of the discretization (3.7) strongly depends both on the manifold M and on the convergent as h → 0 form of the Hamiltonian function (3.11). In particular, the existence of the limit H̃ := lim h→0 ∑ n∈Z 1 h ( ψnψ ∗ n+1 + ψn+1ψ ∗ n + 2 αh2 ln ∣∣1− αh2ψ∗nψn ∣∣) , (4.9) coinciding with the expression (3.4), imposes a strong constraint on the functional space M̃ ⊂ ⊂ L2(R;C2), namely, a vector-function (ψ,ψ∗)ᵀ ∈ W 2 2 (R;C2) ⊂ L2(R;C2), thereby fixing a suitable functional class [7] for which the discretization conserves its physical Hamiltonian system sense. Respectively, the limiting for (4.9) symplectic structure ω̃(2) := − lim h→0 ∑ n∈Z i 2 〈 (dψn, dψ ∗ n)ᵀ ,∧θ−1 n (dψn, dψ ∗ n)ᵀ 〉 = = − lim h→0 i ∑ n∈Z h ( 1− αh2ψ∗nψn )−1 dψ∗n ∧ dψn = −i ∫ R dx [dψ∗(x) ∧ dψ(x)] (4.10) on the manifold M̃ coincides exactly with the canonical symplectic structure (3.5) for the dynami- cal system (3.2). If now to assume that a vector function (ψ,ψ∗)ᵀ ∈ W 1 2 (R;C2) ⊂ L2(R;C2), the Hamiltoni- an function (3.11) can be taken only as H(s) = ∑ n∈Z ( ψnψ ∗ n+1 + ψn+1ψ ∗ n + 2 αh2 ln ∣∣1− αh2ψ∗nψn ∣∣) , (4.11) and the corresponding Poissonian structure as θ(s) n := ( 0 ih−2 ( h2αψ∗nψ − 1 ) ih−2 ( 1− h2αψ∗nψ ) 0 ) . (4.12) The limiting for (4.11) Hamiltonian function H̃(s) := lim h→0 H(s) = ∫ R dx (ψψ∗x + ψxψ ∗) = 0 ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 253 becomes trivial and, simultaneously, the limiting for (4.12) symplectic structure ω̃ (2) (s) := lim h→0 ∑ n∈Z i 2 〈 (dψn, dψ ∗ n)ᵀ ,∧θ(s),−1 n (dψn, dψ ∗ n)ᵀ 〉 = = lim h→0 i ∑ n∈Z h2 ( 1− αh2ψ∗nψn )−1 dψ∗n ∧ dψn = 0 becomes trivial too. Thus, the functional space W 1 2 ( R;C2 ) ⊂ L2 ( R;C2 ) is not suitable for the discretization (3.7) of the nonlinear integrable Schrödinger dynamical system (3.1). It is important here to stress that the discretization parameter h ∈ R+ can be taken as depending on the node n ∈ Z : h → hn ∈ R+, which satisfies the condition supn∈Z hn ≤ ε, where the condition ε → 0 should be later imposed. For instance, one can replace the dynamical system (3.7) by (4.5), the Poissonian structure (3.8) by θ(g) n := ( 0 ih−1 n ( h2 nαψ ∗ nψ − 1 ) ih−1 n ( 1− h2 nαψ ∗ nψ ) 0 ) (4.13) and, respectively, the Hamiltonian function (3.11) becomes exactly (4.6). It is easy to check that the modified discrete dynamical system (4.5) can be equivalently rewritten as d dt (ψn, ψ ∗ n)ᵀ = −θ(g) n gradH(g) [ψn, ψ ∗ n] for all n ∈ Z, meaning, in particular, that the Hamiltonian function (4.6) is conservative. The latter follows from the fact that the skewsymmetric operator (4.13) is Poissonian on the discretized manifold Mh. Moreover, if to impose the constraint that uniformly in n ∈ Z the limit limε→0 ( hnh −1 n−1 ) = 1, the dynamical system (4.5) reduces to (3.1) and the corresponding limiting symplectic structure ω̃ (2) (g) := − lim ε→0 ∑ n∈Z i 2 〈 (dψn, dψ ∗ n)ᵀ ,∧θ(g),−1 n (dψn, dψ ∗ n)ᵀ 〉 = = − lim ε→0 i ∑ n∈Z hn ( 1− αh2 nψ ∗ nψn )−1 dψ∗n ∧ dψn = −i ∫ R dx [dψ∗(x) ∧ dψ(x)] , coincides exactly with the symplectic structure (4.10). Remark 4.1. It is, by now, a not solved, but interesting, problem whether the modified discrete Hamiltonian dynamical system (4.5) sustains to be Lax type integrable. It is left for studying in a separate work. 4.2. Conservation laws for the integrable discrete NLS system. Taking into account that the discrete dynamical system (3.7) is well posed in the space Mh := w2 h,2 ( Z;C2 ) ⊂ l2 ( Z;C2 ) , suitably approximating the Sobolev space of functions W 2 2 ( R;C2 ) , we can go further and to approximate the space w2 h,2 ( Z;C2 ) by means of an infinite hierarchy of strictly invariant finite dimensional subspaces M (N) h ' w̄2 h,2 ( Z(N);C2 ) , N ∈ Z+. In particular, as it was before shown both in [1, 2] by means of the inverse scattering transform method [1, 50] and in [12, 51, 53] by ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 254 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI means of the gradient-holonomic approach [54], the discrete nonlinear Schrödinger dynamical system (3.7) possesses on the manifoldMh an infinite hierarchy of the functionally independent conservation laws: γ̄0 = 1 αh3 ∑ n∈Z ln ∣∣1− αh2ψ∗nψn ∣∣ , γ0 = ∑ n∈Z+ σ(0) n , γ1 = ∑ n∈Z ( σ(1) n + 1 2 σ(0) n σ(0) n ) , γ2 = ∑ n∈Z ( σ(2) n + 1 3 σ(0) n σ(0) n σ(0) n + σ(0) n σ(1) n ) , . . . , where the quantities σ(j) n , n ∈ Z, j ∈ Z+, are defined as follows: σ(0) n = − 1 αh2 ( ψ∗nψn−1 + ψ∗n−1ψn−2 ) , (4.14) σ(1) n = i d dt σ (0) n−1 + ( 1− αh2ψ∗n−1ψn−1 ) ( 1− αh2ψ∗n−2ψn−2 ) + + β α h2 ψ∗n−1 (ψn + ψn−1) , . . . , and β ∈ R is an arbitrary constant parameter. As a result of (4.14) one finds the following infinite hierarchy of smooth conservation laws: H̄0 = ∑ n∈Z ln ∣∣1− αh2ψ∗nψn ∣∣ , H0 = ∑ n∈Z ψ∗nψn+1, H∗0 = ∑ n∈Z ψnψ ∗ n+1, H1 = ∑ n∈Z ( 1 2 ψ2 nψ ∗,2 n−1 + ψnψn+1ψ ∗ n−1ψ ∗ n − ψnψ ∗ n−2 αh2 ) , H∗1 = ∑ n∈Z ( 1 2 ψ2 n−1ψ ∗,2 n + ψn−1ψnψ ∗ n+1ψ ∗ n − ψn−2ψ ∗ n αh2 ) , (4.15) H2 = ∑ n∈Z [ 1 3 ψ3 nψ ∗,3 n−1 + ψnψn+1ψ ∗ n−1ψ ∗ n ( ψnψ ∗ n−1 + ψn+1ψ ∗ n + ψn+2ψ ∗ n+1 ) − − ψnψ ∗ n−1 αh2 ( ψnψ ∗ n−2 + ψn+1ψ ∗ n−1 ) − − ψnψ ∗ n αh2 ( ψn+1ψ ∗ n−2 + ψn+2ψ ∗ n−1 ) + ψnψ ∗ n−3 α2h4 ] , ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 255 H∗2 = ∑ n∈Z [ 1 3 ψ∗,3n ψ3 n−1 + ψ∗nψ ∗ n+1ψn−1ψn ( ψ∗nψn−1 + ψ∗n+1ψn + ψ∗n+2ψn+1 ) − − ψ∗nψn−1 αh2 ( ψ∗nψn−2 + ψ∗n+1ψn−1 ) − − ψnψ ∗ n αh2 ( ψ∗n+1ψn−2 + ψ∗n+2ψn−1 ) + ψ∗nψn−3 α2h4 ] , and so on. Taking into account the functional structure of the equations (3.6) or (3.7), one can define the space D(Mh) of smooth functions γ : Mh → C on Mh as that invariant with respect to the phase transformation C2 3 (ψn, ψ ∗ n) → (eαψn, e −αψ∗n) ∈ C2 for any n ∈ Z and α ∈ C. Equivalently, a function γ ∈ D(Mh) iff the condition∑ n∈Z 〈grad γ [ψn, ψ ∗ n] , (ψn,−ψ∗n)ᵀ〉 = 0 (4.16) holds on Mh. Note that conserved quantities (4.15) belong to D(Mh). The conservation law H̄0 ∈ D(Mh) is a Casimir function for the Poissonian structure (3.8) on the manifold Mh, that is for any γ ∈ D(Mh) the Poisson bracket {γ, H̄0} := ∑ n∈Z 〈 grad γ [ψn, ψ ∗ n] , θn grad H̄0 [ψn, ψ ∗ n] 〉 = = iαh ∑ n∈Z 〈grad γ [ψn, ψ ∗ n] , (ψn,−ψ∗n)〉 = 0, (4.17) owing to the condition (4.16). The Hamiltonian function (3.11) is obtained from the first three invariants of (4.15) as H = 2 αh3 H̄0 + 1 h (H0 +H∗0 ) . Remark 4.2.. Similarly to the limiting condition (4.9), the same limiting expression one obtains from the discrete invariant function H(w) = 1 2αh3 H̄0 − αh 4 (H1 +H∗1 ) , that is lim h−0 H(w) = H̃ := 1 2 ∫ R dx [ ψψ∗xx + ψxxψ ∗ − 2α (ψ∗ψ)2 ] . As one can observe, some combinations of the discrete conservation laws allow well defined and finite limiting expressions in the functional form, coinciding with the corresponding conservati- on laws of the continuous nonlinear Schrödinger dynamical system (3.1), yet almost all other ones fail to possess such well defined limiting functional expressions. This phenomenon appears to be strictly connected with the mathematical properties of the basic manifioldM (N) 2 , on which ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 256 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI these disscrete conservation laws are defined. As in general, the sequences of these conservati- on laws are often not bounded as the discretization parameter h → 0, their limiting functional expression naturally does not exists. Another effect can happen when these sequences are in reality bounded as h → 0, yet compiling only compact subsets, possessing having no limiting functional expression. In general, all these phenomena are deeply related with the well known mathematical fact, which states that a given continuous function possesses a not countable set of different discrete approximations, and many of them can be not convergable as h → 0 to the function under regard. Based on these results, one can apply to the discrete dynamical system (3.7) the Bogo- yavlensky – Novikov type reduction scheme, devised before in [50, 53] and obtain a completely Liouville integrable finite dimensional dynamical system on the manifold M (N) h . Namely, we consider the critical submanifold M (N) h ⊂ Mh of the following real-valued action functional: L(N) h := ∑ n∈Z L(N) h [ψn, ψ ∗ n] = c̄0(h)H̄0 + N∑ j=0 cj(h) ( Hj +H∗j ) , where, by definition, c̄0, cj : R+ → R, j = 0, N, are suitably defined functions for arbitrary but fixed N ∈ Z+, and M (N) h := { (ψ,ψ∗)ᵀ ∈ Mh : gradL(N) h [ψn, ψ ∗ n] = 0, n ∈ Z } . As one can easily show, the submanifold M (N) h ⊂ Mh is finite-dimensional and for any N ∈ ∈ Z+ is invariant with respect to the vector field K : Mh → T (Mh). This property makes it possible to reduce it on the submanifoldM (N) h ⊂ Mh and to obtain a resulting finite-dimensional system of ordinary differential equations on M (N) h , whose solution manifold coincides with an subspace of exact solutions to the initial dynamical system (3.7). The latter proves to be canonically Hamiltonian on the manifold M (N) h and, moreover, completely Liouville – Arnold integrable. If the mappings c̄0, cj : R+ → R, j = 0, N, are chosen in such a way that the flow (3.7), invariantly reduced on the finite dimensional submanifold M (N) h ⊂ Mh, is nonsingular as h → 0 and complete, then the corresponding solutions to the discrete dynamical system (3.7) will respectively approach those to the nonlinear integrable Schrödinger dynamical sys- tem (3.1). Below we will proceed to realizing this scheme for the most simple cases N = 1 and N = = 2. Another way of analyzing the discrete dynamical system (3.7), being interesting enough, consists in applying the approaches recently devised in [15, 45] and based on the long-time behavior of the chosen discretization subject to a fixed Hamiltonian function structure. ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 257 4.3. The finite dimensional reduction scheme: the case N = 1. Consider the following non- degenerate action functional: L(1) h = ∑ n∈Z c̄0(h) ln ∣∣1− αh2ψ∗nψn ∣∣+ ∑ n∈Z c0(h) ( ψ∗nψn+1 + ψnψ ∗ n+1 ) + + ∑ n∈Z c1(h) ( 1 2 ψ2 nψ ∗,2 n−1 + ψnψn+1ψ ∗ n−1ψ ∗ n − ψnψ ∗ n−2 αh2 + + 1 2 ψ2 nψ ∗,2 n+1 + ψnψn+1ψ ∗ n+1ψ ∗ n+2 − ψn−1ψ ∗ n+1 αh2 ) with mappings c̄0, cj : R+ → R, j = 0, 1, taken as c̄0(h) = 4ξ + 1 2αh3 , c0(h) = ξ h , c1(h) = αh 4 , and being easily determined for any ξ ∈ R from the condition for existence of a limit as h → 0: L̃(1) := lim h→0 L(1) h . The corresponding invariant critical submanifold M (1) h := { (ψ,ψ∗)ᵀ ∈ Mh : gradL(1) h [ψn, ψ ∗ n] = 0, n ∈ Z } is equivalent to the following system of discrete up-recurrent relationships with respect to indi- ces n ∈ Z : ψn+2 = − −c̄0(h)/c1(h)ψn ( 1 αh2 − ψn+1ψ∗n+1)( 1 αh2 − ψnψ∗n) + + 2ψn−1c0(h)/c1(h) + ψn(ψn+1ψ ∗ n−1 + ψn−1ψ ∗ n+1) ( 1 αh2 − ψn+1ψ∗n+1) + + (ψ2 n+1 + ψ2 n−1)ψ∗n − ψn−2( 1 αh2 − ψn−1 ψ ∗ n−1) ( 1 αh2 − ψn+1 ψ∗n+1) := := Φ+(ψn+1, ψ ∗ n+1;ψn, ψ ∗ n;ψn−1, ψ ∗ n−1), (4.18) ψ∗n+2 = − −c̄0(h)/c1(h)ψ∗n ( 1 αh2 − ψn+1ψ∗n+1)( 1 αh2 − ψnψ∗n) + + 2ψ∗n−1c0(h)/c1(h) + ψ∗n (ψn+1ψ ∗ n−1 + ψn−1ψ ∗ n+1) ( 1 αh2 − ψn+1 ψ∗n+1) + + (ψ∗,2n+1 + ψ∗,2n−1)ψn − ψ∗n−2( 1 αh2 − ψn−1ψ ∗ n−1) ( 1 αh2 − ψn+1ψ∗n+1) := := Φ∗+(ψn+1, ψ ∗ n+1;ψn, ψ ∗ n;ψn−1, ψ ∗ n−1). ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 258 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI The latter can be also rewritten as the system of down-recurrent mappings ψn−2 = − −c̄0(h)/c1(h)ψn ( 1 αh2 − ψn−1ψ∗n−1)( 1 αh2 − ψnψ∗n) + + 2ψn−1c0(h)/c1(h) + ψn(ψn+1ψ ∗ n−1 + ψn−1ψ ∗ n+1) ( 1 αh2 − ψn−1ψ∗n−1) + + (ψ2 n+1 + ψ2 n−1)ψ∗n − ψn+2( 1 αh2 − ψn+1ψ ∗ n+1) ( 1 αh2 − ψn−1ψ∗n−1) := := Φ−(ψn−1, ψ ∗ n−1;ψn, ψ ∗ n;ψn+1, ψ ∗ n+1), (4.19) ψ∗n−2 = − −c̄0(h)/c1(h)ψ∗n ( 1 αh2 − ψn−1ψ∗n−1)( 1 αh2 − ψnψ∗n) + + 2ψ∗n−1c0(h)/c1(h) + ψ∗n(ψn+1ψ ∗ n−1 + ψn−1ψ ∗ n+1) ( 1 αh2 − ψn−1ψ∗n−1) + + (ψ∗,2n+1 + ψ∗,2n−1)ψn − ψ∗n+2( 1 αh2 − ψn+1ψ ∗ n+1) ( 1 αh2 − ψn−1ψ∗n−1) := := Φ∗−(ψn−1, ψ ∗ n−1;ψn, ψ ∗ n;ψn+1, ψ ∗ n+1), which also hold for all n ∈ Z. The relationships (4.18) (or, the same, relationships (4.19)) mean that the whole submanifold M (1) h ⊂ Mh is retrieved by means of the initial values ( ψ̄−1, ψ̄ ∗ −1; ψ̄0, ψ̄ ∗ 0; ψ̄1, ψ̄ ∗ 1; ψ̄2, ψ̄ ∗ 2 )ᵀ ∈ M (1) h ' C8. Thereby, the submanifold M (1) h ⊂ M8 h is naturally diffeomorphic to the finite dimensional complex space M8 h . Taking into account the canonical symplecticity [51, 53] of the submani- fold M (1) h ' M8 h and its invariance with respect to the vector field (3.7) one can easily reduce it on this submanifold M (1) h ' M8 h and obtain the following equivalent finite dimensional flow on the manifold M8 h : d dt ψ2 = i h2 [Φ+ (ψ2, ψ ∗ 2;ψ1, ψ ∗ 1;ψ0, ψ ∗ 0)− 2ψ2 + ψ1]− − iα [Φ+ (ψ2, ψ ∗ 2;ψ1, ψ ∗ 1;ψ0, ψ ∗ 0) + ψ1]ψ2ψ ∗ 2, d dt ψ1 = i h2 [ψ2 − 2ψ1 + ψ0]− iα (ψ2 + ψ0)ψ1ψ ∗ 1, (4.20) d dt ψ0 = i h2 [ψ1 − 2ψ0 + ψ−1]− iα (ψ1 + ψ−1)ψ0ψ ∗ 0, ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 259 d dt ψ−1 = i h2 [ ψ0 − 2ψ−1 + Φ− ( ψ−1, ψ ∗ −1;ψ0, ψ ∗ 0;ψ1, ψ ∗ 1 )] − − iα [ ψ0 + Φ− ( ψ−1, ψ ∗ −1;ψ0, ψ ∗ 0;ψ1, ψ ∗ 1 )] ψ−1ψ ∗ −1, and d dt ψ∗−1 = − i h2 [ ψ∗0 − 2ψ∗−1 + Φ∗− ( ψ−1, ψ ∗ −1;ψ0, ψ ∗ 0;ψ1, ψ ∗ 1 )] + + iα [ ψ∗0 + Φ∗− ( ψ−1, ψ ∗ −1;ψ0, ψ ∗ 0;ψ1, ψ ∗ 1 )] ψ−1ψ ∗ −1, d dt ψ∗0 = − i h2 [ ψ∗1 − 2ψ∗0 + ψ∗−1 ] + iα ( ψ∗1 + ψ∗−1 ) ψ0ψ ∗ 0, (4.21) d dt ψ∗1 = − i h2 [ψ∗2 − 2ψ∗1 + ψ∗0] + iα (ψ∗2 + ψ∗0)ψ1ψ ∗ 1, d dt ψ∗2 = − i h2 [ Φ∗+ (ψ2, ψ ∗ 2;ψ1, ψ ∗ 1;ψ0, ψ ∗ 0)− 2ψ∗2 + ψ∗1 ] + + iα [ Φ∗+ (ψ2, ψ ∗ 2;ψ1, ψ ∗ 1;ψ0, ψ ∗ 0) + ψ∗1 ] ψ2ψ ∗ 2. The next proposition, characterizing the Hamiltonian structure of the reduced dynamical system (4.20) and (4.21), holds. Proposition 4.1. The eight-dimensional complex dynamical system (4.20) and (4.21) is Ha- miltonian on the manifold M (1) h ' M8 h with respect to the canonical symplectic structure ω (2) h = ∑ j=−2,1 ( dp−j ∧ dψ−j + dp∗−j ∧ dψ∗−j ) , (4.22) where, by definition, p−j := L(1)′,∗ h,ψn−j+1 [ψn, ψ ∗ n] · 1, p∗−j := L(1)′,∗ h,ψ∗n−j+1 [ψn, ψ ∗ n] · 1 (4.23) for j = −2, 1 modulo the constraint gradL(1) h [ψn, ψ ∗ n] = 0, n ∈ Z, on the submanifold M (1) h ' ' M8 h , and the sign ′′′, ∗′′ means the corresponding discrete Frechét up-directed derivative and its natural conjugation with respect to the convolution mapping on T ∗(M (1) h )× T (M (1) h ). Proof. The symplectic structure (4.22) easily follows [11, 51, 53] from the discrete version of the Gelfand – Dikiy [26] differential relationship: dL(1) h [ψn, ψ ∗ n] = 〈 gradL(1) h [ ψn−1, ψ ∗ n−1 ] , ( dψn−1, dψ ∗ n−1 )ᵀ〉 + + d dn α (1) h ( ψn−1, ψ ∗ n−1;ψn, ψ ∗ n;ψn+1, ψ ∗ n+1;ψn+2, ψ ∗ n+2 ) , whereα(1) h ∈ Λ1(M (1) h ) is, owing to the condition gradL(1) h [ψn, ψ ∗ n] = 0, n ∈ Z, on the submani- ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 260 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI fold M (1) h , not depending on the index n ∈ Z and suitably defined one-form on the manifold M8 h , allowing the following canonical representation: α (1) h = ∑ j=−2,1 ( p−j(h)dψ−j + p∗−j(h)dψ∗−j ) with functions p−j , p∗−j : M (1) h ×R → C, j = −2, 1. The latter, being defined by the expressions (4.23), compile jointly with variables ψ−j , ψ∗−j : M (1) h × R → C, j = −2, 1, the global coordi- nates on the finite dimensional symplectic manifold M8 h , proving the proposition. Proposition 4.1 is proved. The dynamical system (4.20) and (4.21) on the manifoldM8 h possesses, except its Hamiltoni- an function, additionally exactly four mutually commuting functionally independent conservati- on laws Hk,H∗k : M8 h → R, k = 0, 1, and one Casimir function H̄0 : M8 h → R, which can be calculated [53] from the following functional relationships: 〈gradHk [ψn, ψ ∗ n] ,K [ψn, ψ ∗ n]〉 :=− d dn Hk ( ψn−1, ψ ∗ n−1;ψn, ψ ∗ n;ψn+1, ψ ∗ n+1;ψn+2, ψ ∗ n+2 ) , 〈gradH∗k [ψn, ψ ∗ n] ,K [ψn, ψ ∗ n]〉 :=− d dn H∗k ( ψn−1, ψ ∗ n−1;ψn, ψ ∗ n;ψn+1, ψ ∗ n+1;ψn+2, ψ ∗ n+2 ) , (4.24) 〈 gradH̄0 [ψn, ψ ∗ n] ,K [ψn, ψ ∗ n] 〉 :=− d dn H̄0 ( ψn−1, ψ ∗ n−1;ψn, ψ ∗ n;ψn+1, ψ ∗ n+1;ψn+2, ψ ∗ n+2 ) , for k = 0, 1 modulo the constraint gradL(1) h [ ψn−2, ψ ∗ n−2 ] = 0, n ∈ Z, on the manifold M (1) h ' ' M8 h , where d dn := ∆ − 1 is a discrete analog of the differentiation and the shift operator ∆ acts as ∆fn := fn+1, n ∈ Z, for any mapping f : Z → C. From (4.24) one can obtain by menas of simple but tedious calculations analytical expressions for the invariantsH∗k : M8 h → R, which give rise to the corresponding Hamiltonian function for the dynamical system (4.20) and (4.21), owing to the relationship (4.17): H = 2 αh3 H̄0 + 1 h (H0 +H∗0) , satisfying the following canonical Hamiltonian system with respect to the symplectic structu- re (4.22): dψ−j dt = ∂H ∂p−j , dψ∗−j dt = ∂H ∂p∗−j , dp−j dt = − ∂H ∂ψ−j , dp∗−j dt = − ∂H ∂ψ∗−j , where j = −2, 1, which is a Liouville – Arnold integrable on the symplectic manifold M8 h . Remark 4.3. The same way on can construct the finite dimensional reduction of the discrete Schrödinger dynamical system (3.7) in the case N = 2. Making use of the calculated before ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 261 conservation laws (4.15), one can take the corresponding action functional as L(2) h = c̄0(h) ∑ n∈Z ln ∣∣1− αh2ψ∗nψn ∣∣+ c0(h) ∑ n∈Z ( ψ∗nψn−1 + ψnψ ∗ n−1 ) + + c1(h) ∑ n∈Z ( 1 2 ψ2 nψ ∗,2 n−1 + ψnψ ∗ n ( ψn+1ψ ∗ n−1 + ψn−1ψ ∗ n+1 ) + + 1 2 ψ2 n−1ψ ∗,2 n − ψnψ ∗ n−2 αh2 − ψn−2ψ ∗ n αh2 ) + + c2(h) ∑ n∈Z ( 1 3 ψ3 nψ ∗,3 n−1 + ψnψn+1ψ ∗ n−1ψ ∗ n ( ψnψ ∗ n−1 + ψn+1ψ ∗ n + ψn+2ψ ∗ n+1 ) − − ψnψ ∗ n−1 αh2 ( ψnψ ∗ n−2 + ψn+1ψ ∗ n−1 ) − ψnψ ∗ n αh2 ( ψn+1ψ ∗ n−2 + ψn+2ψ ∗ n−1 ) + + ψnψ ∗ n−3 α2h4 + 1 3 ψ∗,3n ψ3 n−1 + ψ∗nψ ∗ n+1ψn−1ψn ( ψ∗nψn−1 + ψ∗n+1ψn + ψ∗n+2ψn+1 ) − −ψ ∗ nψn−1 αh2 ( ψ∗nψn−2 + ψ∗n+1ψn−1 ) − ψnψ ∗ n αh2 ( ψ∗n+1ψn−2 + ψ∗n+2ψn−1 ) + ψ∗nψn−3 α2h4 ) with mappings c̄0, cj : R+ → R, j = 0, 2, defined as before from the condition that there exists the limit L̃(2) := lim h→0 L(2) h . The respectively defined critical submanifold M (2) h := { (ψ,ψ∗)ᵀ ∈ Mh : gradL(2) h [ψn, ψ ∗ n] = 0, n ∈ Z } becomes diffeomorphic to a finite dimensional canonically symplectic manifold M12 h on whi- ch the suitably reduced discrete Schrödinger dynamical system (3.7) becomes a Liouville – Arnold integrable Hamiltonian system. The details of the related calculations are planned to be presented in a separate work under preparation. 4.4. The Fourier analysis of the integrable discrete NLS system. It easy to observe that the linearized Schrödinger system (3.1) admits the following Fourier type solution: ψ(x, t) = ∫ R dsξ(s, t) exp(ixs), ψ∗(x, t) = ∫ R dsξ∗(s, t) exp(−ixs) (4.25) for all x, t ∈ R, where dξ dt = −is2ξ, dξ∗ dt = is2ξ∗, i. e., ξ(s, t) = ξ̄(s)e−is 2t, ξ∗(s, t) = ξ̄∗(s)eis 2t ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 262 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI and ξ̄, ξ̄∗ : R → C are prescribed functions (the Fourier transforms of initial conditions). Likewise, the linearized discrete Schrödinger dynamical system (3.7) allows the following ge- neral discrete Fourier type solution: ψn = ∫ R dsξh(s, t) exp(ihns), ψ∗n = ∫ R dsξ∗h(s, t) exp(−ihns) (4.26) for all n ∈ Z, where the evolution parameter t ∈ R, (ψn, ψ∗n)ᵀ ∈ w2 h,2(Z;C2) and ξh(s, t) = ξ̄h(s) exp ( −i 4t h2 sin2 sh 2 ) , ξ∗h(s, t) = ξ̄∗h(s) exp ( i 4t h2 sin2 sh 2 ) . Here the function ( ξ̄h, ξ̄ ∗ h )ᵀ ∈ W 2 h,2 ( R;C2 ) ⊂ L2 ( R;C2 ) , where the functional space W 2 h,2 ( R;C2 ) is yet to be determined. From the boundary condition (ψn, ψ ∗ n)ᵀ ∈ w2 h,2 ( Z;C2 ) it follows that expressions 1 2π ∑ n∈Z ψ∗nψn = ∫ R dsξ∗h(s)ξh(s) < ∞, 1 2π ∑ n∈Z ( ψ∗n+1ψn + ψ∗nψn+1 ) = 2 ∫ R ds cos(hs)ξ∗h(s)ξh(s) < ∞, ensure the boundedness of the Hamiltonian function (3.11), thereby determining a functional space W 2 h,2 ( R;C2 ) to which belong the vector function (ξh, ξ ∗ h)ᵀ ∈ L2 ( R;C2 ) . However the discrete evolution is not following along the continuous trajectory. Being motivated by works [14, 16], we modify the discrete system as folows in order to obtain the exact discretization: d dt ψn = i δ2 (ψn+1 − 2ψn + ψn−1)− iα (ψn+1 + ψn−1)ψnψ ∗ n, d dt ψ∗n = − i δ2 ( ψ∗n+1 − 2ψ∗n + ψ∗n−1 ) + iα ( ψ∗n+1 + ψ∗n−1 ) ψnψ ∗ n. (4.27) Substituting (4.26) into the linearization of (4.27) we obtain ξh(s, t) = ξ̄h(s) exp ( −i 4t δ2 sin2 sh 2 ) , ξ∗h(s, t) = ξ̄∗h(s) exp ( i 4t δ2 sin2 sh 2 ) . Therefore, linearization of the discretization (4.27) is exact (i.e., ψ(nh, t) = ψn(t), n ∈ Z, if we assume δ = 2 s sin hs 2 for any h ∈ R. Thus, the parameter δ > 0 depends on s ∈ R yet for small h → 0 one gets δ = h(1 +O(h2s2). ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 THE DISCRETE NONLINEAR SCHRÖDINGER TYPE HIERARCHY . . . 263 The nonlinear case is more difficult. In the continuous nonlinear case (4.25) we have dξ dt = −is2ξ − 2iαβ[s; ξ], dξ∗ dt = is2ξ∗ + 2iαβ∗ [s; ξ∗] , (4.28) where the functionals β, β∗ : R×L2(R;C) → L2(R;C), determined as β[s; ξ] := ∫ R2 ds′ds′′ξ ( s+ s′ − s′′ ) ξ ( s′′ ) ξ∗ ( s′ ) , β∗[s; ξ] := ∫ R2 ds′ds′′ξ∗ ( s+ s′ − s′′ ) ξ∗ ( s′′ ) ξ ( s′ ) , depend both on s ∈ R and on the element ξ ∈ L2(R;C), as well as depends parametrically on the evolution parameter t ∈ R through (4.28). In the nonlinear discrete case (4.27) we, respectively, obtain dξh dt = −iξh 4 δ2 sin2 sh 2 + 2iαβh[s; ξh], dξ∗h dt = iξ∗h 4 δ2 sin2 sh 2 + 2iαβ∗h[s; ξh], where the functionals βh, β∗h : R×L2(R;C) → L2(R;C) are determined as βh[s; ξh] := ∫ R2 ds′ds′′ cos [ h ( s+ s′ − s′′ )] ξh ( s+ s′ − s′′ ) ξh ( s′′ ) ξ∗h ( s′ ) , β∗h[s; ξh] := ∫ R2 ds′ds′′ cos [ h ( s+ s′ − s′′ )] ξ∗h ( s+ s′ − s′′ ) ξ∗h ( s′′ ) ξh(s′) for any s ∈ R. To the regret, proceeding further with the truly nonlinear case still presists to be a nontrivial problem, yet we hope to obtain a suitable procedure analogous to that of [15, 17]. Instead of it one can analyze the related functional space constraints on the space of functi- ons (ξ̄h, ξ̄ ∗ h)ᵀ ∈ W 2 h,2(R;C2), representing solutions to the discrete nonlinear equation (3.7) via the expressions (4.26), being imposed by the corresponding finite dimensional reduction scheme of Section 4.3. This procedure actually may be realized, if to consider the derived before recurrence relationships (4.18) (or similarly, (4.19)) allowing to obtain the related constraints on the space of functions ( ξ̄h, ξ̄ ∗ h )ᵀ ∈ W 2 h,2 ( R;C2 ) , but the resulting relationships prove to be much complicated and cumbersome expressions. Thus, one can suggest the following practical numerical-analytical scheme of constructi- ng solutions to the discrete nonlinear Schrödinger dynamical system (3.7): first to solve the Cauchy problem to the finite-dimensional system of ordinary differential equations (4.20) and (4.21), and next to substitute them into the system of recurrent algebraic relationships (4.18) and (4.19), obtaining this way the whole infinite hierarchy of the sought for solutions. 5. Conclusion. Within the presented investigation of solutions to the discrete nonlinear Schrödinger dynamical system (3.7) we have succeeded in two important points. First, we have developed an effective enough scheme of invariant reducing the infinite system of ordi- nary differential equations (3.7) to an equivalent finite one of ordinary differential equations ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 264 A. K. PRYKARPATSKI, J. L. CIEŚLIŃSKI with respect to the evolution parameter t ∈ R. Second, we constructed a finite set of recurrent algebraic regular relationships, allowing to expand the obtained before solutions to any discrete order n ∈ Z and giving rise to the sought for solutions of the system (3.7). It is important to mention here that within the presented analysis we have not used the Lax type representation for the discrete nonlinear Schrödinger dynamical system (3.7), whose existence was stated many years ago in [1] and whose complete solution set analysis was done in works [1, 2, 12, 50] by means of both the inverse scattering transform and the algebraic- geometric methods. Concerning the set of recurrent relationships for exact solutions to the finite-dimensional reduction of the discrete nonlinear Schrödinger dynamical system (3.7), ob- tained both in the presented work and in work [12], based on the corresponding Lax type rep- resentation, an interesting problem of finding between them relationship arises, and an answer to it would explain the hidden structure of the complete Liouville – Arnold integrability of the related set of the reduced ordinary differential equations. 6. Acknowledgements. A. K. Prykarpatski is thankful both to Prof. D. Blackmore for va- luable discussions and remarks during the Nonlinear Mathematical Physics Conference held in the Sophus Lie Center, Nordfjordeid (Norway) and to Prof. Kamal Soltanov, the Hacettepe University of Ankara, Turkey, for the nice hospitality and financial support within the visiting grant TUBITAK- 2221- Fellowships. J. L. 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The discrete nonlinear Schrödinger type hierarchy, its finite dimensional reduction analysis and numerical integrability scheme

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