*2.8. Conclusions*

Within the scope of this Section we have described the main mathematical preliminaries and properties of the quantum mathematics techniques suitable for analytical studying of the important linearization problem for a wide class of nonlinear dynamical systems in partial derivatives in Hilbert spaces. This problem was analyzed in much detail using the Gelfand–Vilenkin representation theory [66] of infinite dimensional groups and the Goldin–Menikoff–Sharp theory [4,5,74] of generating Bogolubov type functionals, classifying these representations. The related problem of constructing cyclic Hilbert space representations and retrieving their creation–annihilation generating structure still needs a deeper investigation within the approach devised. Here we mention only that some aspects of this problem within the so-called Poissonian White noise analysis which was analyzed in a series of works [55,70,107,108], based on some generalizations of the Delsarte type characters technique. The above-stated theorem about the Hamiltonian structure of symmetries of a nonlinear dynamical system on a Hilbert space and their linearization on a suitably constructed Hilbert space presents, from a practical point of view, a strong interest, if the related results, obtained in [14,15,109,110] and devoted to the application of the Hilbert spaces embedding method to finding conservation laws and the so called recursion operators for the well [17,111] known Korteweg–de Vries type nonlinear dynamical systems, are taken into account. Moreover, a development of these results within the modern Lie-algebraic approach, based on the Adler–Kostant–Symes construction and applied to nonlinear dynamical systems on Poissonian functional manifolds, proves to be both unexpected and important for the classification of integrable Hamiltonian flows on Hilbert spaces, and inspires a hope for new investigations of coherent states and their applications.

#### **3. Quantum Current Lie Algebra as a Universal Algebraic Structure of Symmetries of Completely Integrable Nonlinear Dynamical Systems**

*3.1. Quantum Lie Algebra of Currents and Its Vector Field Representations*

We consider the non-relativistic quantum Lie algebra G of currents [4,12,112,113] on the torus T*<sup>n</sup>*, realized by means of the density *ρ*(f) and current *J*(g) operators on the separable Hilbert subspace <sup>Φ</sup>*μ*:

$$\begin{aligned} [\rho(\mathbf{f}\_1), \rho(\mathbf{f}\_2)] &= 0, [\rho(\mathbf{f}), J(\mathbf{g})] = J(\langle \mathbf{g} | \nabla \mathbf{f} \rangle), \\ [J(\mathbf{g}\_1), J(\mathbf{g}\_2)] &= iJ([\mathbf{g}\_2, \mathbf{g}\_1]), \end{aligned} \tag{121}$$

where *ρ*(f) = T*n* <sup>f</sup>(*x*)*ρ*(*x*)*dx*, *J*(g) = T*n g*(*x*)*J*(*x*)*dx* for f, f*j* ∈ *F* - *C* <sup>∞</sup>(T*<sup>n</sup>*; <sup>R</sup>), g, g*j* ∈ *Fn*, *j* = 1, 2. Their representation on the Fock space Φ*F* is given, respectively, by the following operator expressions: *ρ*(*x*) = *a*+(*x*)*a*(*x*) and *J*(*x*) = 1 2*i* [*a*+(*x*)∇*a*(*x*) − <sup>∇</sup>*a*+(*x*)*a*(*x*)], where *a*+(*x*) is the creation and *a*(*x*) is the annihilation operators of boseparticle states at point *x* ∈ T*<sup>n</sup>*, satisfying the canonical commutation relationships:

$$\begin{aligned} [a(\mathfrak{x}), a(\mathfrak{y})] &= 0, [a^+(\mathfrak{x}), a^+(\mathfrak{y})], \\ [a(\mathfrak{x}), a^+(\mathfrak{y})] &= \delta(\mathfrak{x} - \mathfrak{y}) \end{aligned}$$

for all *x*, *y* ∈ T*<sup>n</sup>*. The current Lie algebra (121) is the infinite-dimensional Lie algebra of the semi-direct product *G* := Diff(T*n*) - *F* of the Banach Lie group of currents *G* := Diff(T*n*) and the abelian functional group *F*, where Diff(T*n*) is the topological group of diffeomorphisms [2,20] of the torus T*<sup>n</sup>*. If, to introduce [2,20,22,112,114,115] a family of unitary operators *U*(f) and *<sup>V</sup>*(*ϕ*g*t* ) : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ*, acting on a Hilbert space <sup>Φ</sup>*μ* and defined by the formulas

$$\mathcal{U}I(\mathbf{f}) = \exp[i\rho(\mathbf{f})]\_\prime \, V(\varphi\_t^{\mathbb{R}}) = \exp[itJ(\mathbf{g})]\_\prime \tag{122}$$

where *<sup>d</sup>ϕ*g*t* /*dt* :=g(*ϕ*g*t* ), *t* ∈ R, *ϕ*g*t* ∈ Di*f f*(T*n*) and *ϕ*g*t* |*<sup>t</sup>*=<sup>0</sup> = *x* ∈ T*<sup>n</sup>*, then the following relations

$$\begin{aligned} \mathcal{U}(\mathbf{f\_1})\mathcal{U}(\mathbf{f\_2}) &= \mathcal{U}(\mathbf{f\_1} + \mathbf{f\_2}), \mathcal{V}(\boldsymbol{\varrho})\mathcal{U}(\mathbf{f}) = \mathcal{U}(\mathbf{f} \circ \boldsymbol{\varrho})V(\boldsymbol{\varrho}), \\ \mathcal{V}(\boldsymbol{\varrho\_1})V(\boldsymbol{\varrho\_2}) &= V(\boldsymbol{\varrho\_2} \circ \boldsymbol{\varrho\_1}) \end{aligned} \tag{123}$$

hold for all f, f*j* ∈ *F* and *ϕ*, *ϕj* ∈ Di*f f*(T*<sup>n</sup>*), *j* = 1, 2. As was argued in [2], the various unitary representations of the current group *G* describe different physical systems and their states, and the study of the set of cyclic unitarily irreducible representations of the Banach Lie group relationships (123) is an extremely important and topical problem in the quantum theory of dynamical systems.

For every irreducible cyclic representation of the unitary current group *G* on the separable Hilbert space <sup>Φ</sup>*μ* there exists a unitarily equivalent Hilbert space

$$\Phi\_{\mu} \simeq \int\_{F'}^{\ominus} d\mu(\eta) \Phi\_{(\eta)'} \tag{124}$$

where *μ* is the measure on the space *F* of continuous real linear functionals on *F* and <sup>Φ</sup>(*η*) are complex linear finite-dimensional spaces labeled by the index *η* ∈ *F*. In the case when dim <sup>Φ</sup>(*η*) = 1, <sup>Φ</sup>*μ* - *L*(*μ*) 2 (*F*; C), the space of complex-valued functions on *F*, integrable with respect to the measure *μ* on *F*. Moreover, if an element *ω* ∈ <sup>Φ</sup>*μ*, then for the action of the current group *G* on this element we have the following representations:

$$\mathcal{U}(\mathbf{f})\omega(\boldsymbol{\eta}) = \exp[i(\boldsymbol{\eta}(\mathbf{f}))\omega(\boldsymbol{\eta})] \,\tag{125}$$

$$\mathcal{V}(\boldsymbol{\varrho})\omega(\boldsymbol{\eta}) = \chi\_{\boldsymbol{\varrho}}(\boldsymbol{\eta})\omega(\boldsymbol{\varrho}^\*\boldsymbol{\eta}) \left(\frac{d\mu(\boldsymbol{\varrho}^\*\boldsymbol{\eta})}{d\mu(\boldsymbol{\eta})}\right)^{1/2}$$

where, by definition, *ϕ*<sup>∗</sup>*η*(f) := *η*(<sup>f</sup> ◦ *ϕ*) for all f ∈ *F*, *dμ*(*ϕ*<sup>∗</sup>*η*) *dμ*(*η*) is the corresponding Radon– Nikodym derivative of the measure *μ* ◦ *ϕ*∗ with respect to the measure *μ* on *F* and *χϕ*(*η*) is a complex-valued character of the unit norm, satisfying the relationship

$$
\chi\_{\varphi\_2}(\eta)\chi\_{\varrho\_1}(\varphi\_2^\*\eta) = \chi\_{\varrho\_1 \circ \varrho\_2}(\eta) \tag{126}
$$

for all *ϕj* ∈ Di*f f*(T*<sup>n</sup>*), *j* = 1, 2, *η* ∈ *F*. For the Radon–Nikodym derivative above to exist, the measure *μ* on *F* should be quasi-invariant with respect to the diffeomorphism group Di*f f*(T*<sup>n</sup>*), that is for any measurable set *Q* ⊂ *F* the condition *μ*(*Q*) = 0 if, and only if, *μ*(*ϕ*<sup>∗</sup>*Q*) for arbitrary *ϕ* ∈ Di*f f*(T*<sup>n</sup>*).

In physics applications, the representation (125) is uniquely determined by the measure *μ* on *F*, which in the general case has a very complicated [20,72,114] structure, and its analytic construction is nontrivial. One of the fairly effective approaches to this problem is the quantum method of Bogolubov generating functionals developed in [5–7,20]. Another approach, which is of considerable interest for the theory of dynamical systems, is based on algebraic methods of constructing self-adjoint functional-operator representations of the original current Lie algebra (121). We proceed to its description in the case of the current group *G* = Di*f f*(S<sup>1</sup>) - *F*, where *F* - *C*∞(S1; R) on the circle S1, taking into account results in [12,22,116–118].

We now introduce the following basis operators of the Lie current Lie algebra (121) for*n* = 1:

$$\rho\_{\rangle} := \int\_{\mathbb{S}^1} \exp(i j \mathbf{x} + i \varepsilon \mathbf{x}) \rho(\mathbf{x}) d\mathbf{x}, \qquad J\_k := \int\_{\mathbb{S}^1} \exp(i k \mathbf{x}) f(\mathbf{x}) d\mathbf{x},\tag{127}$$

where *j*, *k* ∈ Z and *ε* ∈ R is a parameter. Then from (121) and (127), we find that

$$\begin{bmatrix} \rho\_j, \rho\_k \end{bmatrix} = 0, \ [l\_k, \rho\_j] = (j+\varepsilon)\rho\_{j+k\prime} \ \ [l\_k, l\_j] = (j-k)l\_{k+j} \tag{128}$$

for all *j*, *k* ∈ Z, that is the set G := { *ρj*, *Jk* : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ* : *j*, *k* ∈ Z} of operators (128) on the representation Hilbert space <sup>Φ</sup>*μ*, is equivalent to the semidirect product G{*J*} - G{*ρ*} of the Lie subalgebra G{*J*} := { *Jk* : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ* : *k* ∈ Z} and the Abelian subalgebra G{*ρ*} := {*ρj* : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ* : *j* ∈ Z} and isomorphic to the current Lie algebra (121) for *n* = 1. It is also worth mentioning [119] that in the case of functional-operator representations, the Lie algebra (128) admits the following central extension by means of the Schwinger cocycle:

$$\mathbb{E}\left[\rho\_{\overline{\cdot}\prime}\rho\_k\right] = \mathbb{E}\rho\_{\overline{\cdot},-k\prime}\text{ [}I\_{k\prime}\rho\_{\overline{\cdot}]} = (j+\varepsilon)\rho\_{\overline{\cdot}+k\prime}\text{ [}I\_{k\prime}I\_{\overline{\cdot}]} = (j-k)I\_{k+\overline{\cdot}} + \nu k(k^2-1)\delta\_{\overline{\cdot},-k\prime}\tag{129}$$

where *j*, *k* ∈ Z and *ζ*, *ν* ∈ R are the Schwinger parameters. The current Lie algebra (129) is called the generalized Virasoro current algebra [44] and has many applications in modern theoretical physics.

It is easy to show that the current Lie algebra (128) for *ε* = 0 admits the standard representation in the ring of operators C[*<sup>λ</sup>*, *<sup>λ</sup>*−<sup>1</sup>][*∂*/*∂λ*], *λ* ∈ C*N*, regarded as a Lie subalgebra of the Lie algebra of rational vector fields on C*N*, *N* ∈ N. Namely, if we set

$$\rho\_{\bar{\jmath}} = \sum\_{n=\overline{1,\mathcal{N}}} \lambda\_{n\prime}^{\bar{\jmath}} \quad \mathcal{J}\_{\mathbb{k}} = \sum\_{n=\overline{1,\mathcal{N}}} \lambda\_{n}^{k+1} \partial / \partial \Lambda\_{n\prime} \tag{130}$$

for *j*, *k* ∈ Z, then the current Lie algebra relations (128) are satisfied identically. In this case. if we make the restriction |*<sup>λ</sup>n*| = 1, *λn* = exp(*<sup>i</sup>θn*), *θn* ∈ [0, <sup>2</sup>*π*], *n* = 1, *N*, then for the current algebra operators *ρ*(*x*) and *J*(*x*), *x* ∈ S1, we obtain the expressions

$$\rho(\mathbf{x}) = \sum\_{n=\overline{1,\mathcal{N}}} \delta(\mathbf{x} - \theta\_n), \; f(\mathbf{x}) = \frac{1}{2i} \sum\_{n=\overline{1,\mathcal{N}}} [\delta(\mathbf{x} - \theta\_n)\partial / \partial \theta\_n + \partial / \partial \theta\_n \delta(\mathbf{x} - \theta\_n)].\tag{131}$$

It is readily seen that the operators (131) are *<sup>N</sup>*-particle representations of the current Lie algebra (121) on the circle S1, and that the support of the measure *μ* on *F* in the representation (124) is concentrated on functionals *η* = ∑*<sup>n</sup>*=1,*<sup>N</sup> <sup>δ</sup>*(*x* − *<sup>θ</sup>n*) and the Hilbert space *L*(*μ*) 2 (*F*; C) - *L*(*s*) 2 (T*N*; C), the space symmetric square integrable functions on the torus T*N*. In the general case, the current generalized Lie algebra (129) possesses numerous functional-operator representations by means of vector fields on special infinite dimensional manifolds. As will be shown below, these vector fields are defined on these manifold's so-called completely integrable infinite-dimensional Hamiltonian systems, many of which have applications in theoretical and mathematical physics.

On the infinite-dimensional smooth functional manifold *M* ⊂ *<sup>C</sup>*<sup>∞</sup>(T*<sup>n</sup>*; <sup>R</sup>*<sup>m</sup>*), *n*, *m* ∈ N are finite, we consider a homogeneous autonomous nonlinear dynamical system

$$u\_l = \mathbb{K}[u]\_\prime \tag{132}$$

where *K* : *M* → *T*(*M*) is a Frechet-smooth vector field on *M*, [*u*] ∈ *J*(T*<sup>n</sup>*; R*m*) denotes a point of a finite order [59,120] at the jet-manifold *J*(T*<sup>n</sup>*; R*m*) and *t* ∈ R is the evolution parameter. We assume that the vector field (132) is Hamiltonian, i.e., there exists a skewsymmetric Poissonian [18,59,120,121] operator *ϑ* : *T*∗(*M*) → *T*(*M*) such that condition

$$L\_K \theta = 0 \sim \theta\_t - \theta K'^{"\*} - K' \theta = 0 \tag{133}$$

where *LK* denotes the Lie derivative [11,26,59,120,122,123] along the vector field *K* : *M* → *<sup>T</sup>*(*M*), "*prime*" denotes the usual Frechet derivative of a mapping and "\*" denotes the adjoint mapping subject to the standard bilinear convolution form (·|·) on the product *T*∗(*M*) × *T*(*M*) of the tangent and cotangent spaces over the functional manifold *M*. If the condition (133) holds, there exists such a smooth Hamiltonian functional *Hϑ* ∈ D(*M*) ⊂ *<sup>C</sup>*<sup>∞</sup>(*<sup>M</sup>*; R) that

$$K[\mu] = -\theta \,\mathrm{grad}\,\, H\_{\theta}[\mu] \tag{134}$$

Assume now that the dynamical system (132) possesses one further algebraically independent solution *η* : *T*∗(*M*) → *T*(*M*) to the Equation (133), that is *LKη* = 0, which is Poissonian.

**Definition 12.** *A dynamical system (132) possessing a* (*<sup>ϑ</sup>*, *η*)*-pair of Poissonian operators is said [11,13,18,59] to be bi-Hamiltonian, if for any λ* ∈ R *the pencil* (*ϑλ* + *η*) : *T*∗(*M*) → *T*(*M*) *is also Poissonian. The Poissonian* (*<sup>ϑ</sup>*, *η*)*-pair is called the Magri type compatible.*

**Definition 13.** *If the Poisson operator ϑ* : *T*∗(*M*) → *T*(*M*) *is invertible, the operator* Λ := *<sup>ϑ</sup>*−<sup>1</sup>*η* : *T*∗(*M*) → *T*∗(*M*) *is said to be gradient-recursive and satisfies the Noether–Lax equation*

$$L\_K \Lambda = 0 \sim \Lambda\_t - [\Lambda, \boldsymbol{\mathcal{K}}^{\prime, \*}] = 0. \tag{135}$$

*Similarly, the operator* Φ := *ηϑ*−<sup>1</sup> : *T*(*M*) → *T*(*M*) *is said to be symmetry-recursive and satisfies the Noether–Lax equation*

$$L\_K \Phi = 0 \sim \Phi\_t - [\boldsymbol{\mathsf{K}}', \boldsymbol{\Lambda}] = 0. \tag{136}$$

*The inverse operator ϑ*−<sup>1</sup> : *T*(*M*) → *T*∗(*M*) *is said to be symplectic, and the operator ϑ* : *T*∗(*M*) → *T*(*M*) *itself is often called cosymplectic.*

Yet, if the inverse operator *ϑ*−<sup>1</sup> : *T*(*M*) → *T*∗(*M*) does not exist, the notions of gradient-recursive and symmetry-recursive operators remain the same: *LK*Λ = 0 and *LK*Φ = 0, respectively.

**Definition 14.** *The operator* Φ : *T*(*M*) → *T*(*M*) *is said to be hereditary-recursive if the bilinear operator*

$$[\Phi', \Phi]: T(M) \times T(M) \to T(M) \tag{137}$$

*is symmetric.*

It is easy to check that the operator Φ = *ηϑ*−<sup>1</sup> : *T*(*M*) → *T*(*M*) is hereditaryrecursive [11,59,121] if the Poissonian pair (*<sup>ϑ</sup>*, *η*)-pair is compatible. The Poissonian (*<sup>ϑ</sup>*, *η*)- pair is compatible if, and only if, the operator *ηϑ*−<sup>1</sup>*η* : *T*∗(*M*) → *T*(*M*) is Poissonian too. Moreover, the operators *<sup>ϑ</sup>*(*<sup>ϑ</sup>*−<sup>1</sup>*η*)*<sup>n</sup>* : *T*∗(*M*) → *T*(*M*) for all *n* ∈ Z+ are Poissonian also.

**Definition 15.** *A vector field α* : *M* → *T*(*M*) *is called a homogeneous symmetry of the dynamical system (132) if LKα* = 0 ∼ [*<sup>K</sup>*, *α*] = 0. *Respectively, a vector field τ* : *M* → *T*(*M*) *is called an inhomogeneous symmetry of the dynamical system (132), ∂τ*/*∂t* + [*<sup>K</sup>*, *τ*] = 0*.*

It is easy to observe that subsets of homogeneous and inhomogeneous symmetries, respectively, are Lie subalgebras of the symmetry space <sup>Γ</sup>(*M*). Suppose now that for a consistent bi-Hamiltonian dynamical system (132) there exist two nontrivial homogeneous symmetry *α*0 ∈ Γ(*M*) and homogeneous symmetry *τ*0 ∈ <sup>Γ</sup>(*M*), such that

$$\begin{aligned} L\_{\pi\_0} a\_0 &= \varepsilon \mathfrak{a}\_0, \ L\_{\mathfrak{a}\_0} \theta = 0 = L\_{\mathfrak{a}\_0} \eta, \ L\_{\pi\_0} \theta = (\mathfrak{\zeta} - 1/2) \theta, \\ L\_{\pi\_0} a\_0 &= \varepsilon \mathfrak{a}\_0, L\_{\pi\_0} \eta = (\mathfrak{\zeta} + 1/2) \eta, \ L\_{\pi\_0} \Phi = \Phi, \end{aligned} \tag{138}$$

where *ε*, *ξ* ∈ R are certain numerical parameters. Having assumed that the symmetryrecursive operator Φ : *T*(*M*) → *T*(*M*) is invertible, one can construct the following subsets *Q*{*α*} ⊂ Γ(*M*) and *Q*{*τ*} ⊂ <sup>Γ</sup>(*M*), where

$$Q\{\mathfrak{a}\} := \{\mathfrak{a}\_j : \Phi^j \mathfrak{a}\_0 : j \in \mathbb{Z}\}, \quad Q\{\tau\} := \{\tau\_j : \Phi^j \tau\_0 : j \in \mathbb{Z}\}.\tag{139}$$

The following proposition holds.

**Proposition 5.** *The semi-direct product Q* := *Q*{*τ*} - *Q*{*α*} *is a Lie subalgebra of symmetries of the dynamical system (132) isomorphic to the current Lie algebra (128).*

**Proof.** The proof is a direct consequence of the relations (138) and (139).

$$
u\_! = 
u\_{\rm XXx} + 
u 
u\_{\rm x} := 
K[
u],\tag{140}$$

As a simplest example we consider the classical nonlinear Korteweg–de Vries dynamical system on the functional manifold *M* ⊂ *C*∞(S1; <sup>R</sup>), possessing two compatible Poissonian operators

$$
\theta = \partial, \quad \eta = \partial^3 + (u\partial + \partial u)/3,\tag{141}
$$

where *∂* := *∂*/*∂<sup>x</sup>*, *x* ∈ R. Its symmetry-recursive operator equals to the expression

$$
\Phi = \eta \theta^{-1} = \partial^2 + (u + \partial u \partial^{-1})/3,\tag{142}
$$

where *<sup>∂</sup>*−<sup>1</sup>(·) := 1/2 *<sup>x</sup>*0 *dx*(·) <sup>−</sup> <sup>2</sup>*πx dx*(·) is the operator of inverse differentiation, *∂* · *∂*−<sup>1</sup> = *I*. Then, taking into account the homogeneous symmetry, *α*0 = *ux* and inhomogeneous *τ*−1 = 3/2(1 + *tux*) generate, respectively, two subalgebras *Q*{*α*} := {Φ*j<sup>α</sup>*0 : *j* ∈ Z} and *Q*{*τ*} := {Φ*j*+1*τ*−<sup>1</sup> : *j* ∈ <sup>Z</sup>}, whose semidirect product *Q* = *Q*{*τ*} - *Q*{*α*} is isomorphic to the current Lie algebra G (128).

#### *3.2. Completely Integrable Hamiltonian Systems and the Current Algebra Symmetry Integrability Criterion*

In analyzing the dynamical system (132) above, we assumed for it the existence of the consistent (*<sup>ϑ</sup>*, *η*)-pair of Poissonian operators, with respect to which it is bi-Hamiltonian. However, if the dynamical system (132) is not bi-Hamiltonian but only Hamiltonian and integrable, then obviously the Noether–Lax Equation (133) has only one solution, which is determined up to multiplication by a constant. On the other hand, if the dynamical system (132) is invariant with respect to the universal Banach Lie group symmetry *G* = Diff(T*n*) - *F*, then for the corresponding Lie algebra of symmetries *Q* = *Q*{*τ*} - *Q*{*α*}, which is isomorphic to the current Lie algebra G, the next conditions should hold:

$$L\_n \theta = 0, \qquad L\_{\tau} \theta = 0 \tag{143}$$

for all *α* ∈ *Q*{*α*} and *τ* ∈ *Q*{*τ*}. In addition, one easily ensues from (128) the following commutation relationships:

$$\begin{aligned} (j+\varepsilon)a\_{j+1} &= [\tau\_1, a\_j]\_\prime \, (j+\varepsilon)a\_{j-1} = [\tau\_{-1}, a\_j]\_\prime \, (j+\varepsilon)a\_{\bar{j}} = [\pi\_0, a\_{\bar{j}}]\_\prime \\ (j-1)\tau\_{\bar{j}+1} &= [\tau\_1, \tau\_{\bar{j}}]\_\prime \, \quad j\tau\_{\bar{j}} = [\tau\_0, \tau\_{\bar{j}}]\_\prime \, (j+1)\tau\_{\bar{j}-1} = [\tau\_{-1}, \tau\_{\bar{j}}]. \end{aligned} \tag{144}$$

Algebraic relationships (144) give rise to the following Lie-algebraic relationships

$$\begin{aligned} L\_{\mathfrak{a}\_{\uparrow}}\theta &= 0 = L\_{\mathfrak{a}\_{\uparrow}}\eta\_{\mathfrak{a}}\ , L\_{\mathfrak{T}\_{\uparrow}}\Lambda = \Lambda^{j+1}\ \_{\mathfrak{a}\_{\uparrow}}\Lambda = 0 = L\_{\mathfrak{a}\_{\uparrow}}\Phi \ \_{\mathfrak{d}\_{\uparrow}}\Phi = \Phi^{j+1}\ , \\ L\_{\mathfrak{T}\_{\uparrow}}\theta &= \left(\mathfrak{J} - j - 1/2\right)\theta\Lambda^{\check{j}} , \ \ L\_{\mathfrak{T}\_{\uparrow}}\eta = \left(\mathfrak{J} - j + 1/2\right)\eta\Lambda^{\check{j}} , \end{aligned} \tag{145}$$

and show that on the basis of the sl(2) Lie subalgebra {*<sup>τ</sup>*−1, *τ*0, *<sup>τ</sup>*1} jointly with the set of initial homogeneous symmetries {*<sup>α</sup>*−1, *<sup>α</sup>*1} for *ε* = 0 or {*<sup>α</sup>*0} for *ε* ∈/ Z, as well as the inhomogeneous symmetries {*<sup>τ</sup>*−2, *<sup>τ</sup>*2}, one can construct recursively an entire infinitehierarchy of symmetries *Q*{*τ*} - *Q*{*α*}, which is isomorphic to the current Lie algebra G (128) by virtue of the construction. In addition, in accordance with the Noether relations (143) there exist two infinite hierarchies of conservation laws to the dynamical system (132), namely, the homogeneous functionals *γj* ∈ <sup>D</sup>(*M*), *j* ∈ Z, and the inhomogeneous *ζj* ∈ <sup>D</sup>(*M*), *j* ∈ Z, satisfying the conditions

$$\begin{aligned} \tau\_{j} &= -\theta \operatorname{grad} \zeta\_{j}, \ a\_{j} = -\theta \operatorname{grad} \gamma\_{j}, \ \{\gamma\_{j}, \gamma\_{k}\} = 0, \\ (j + \varepsilon)\gamma\_{j+k} &= (\operatorname{grad}\,\gamma\_{j}|\tau\_{k}) = \{\gamma\_{j}, \tau\_{k}\}, \\ \Theta \tau\_{k} + \{H, \zeta\_{k}\} &= 0, \ \{H, \gamma\_{j}\} = 0, \ \{\zeta\_{j}, \zeta\_{k}\} = (j - k)\zeta\_{j+k} \end{aligned} \tag{146}$$

for any *j*, *k* ∈ Z, where, by definition, *<sup>K</sup>*[*u*] = −*ϑ* grad *H*, and {·, , ·} := (grad(·)|*<sup>ϑ</sup>* grad(·)) denotes the Poisson bracket on the space of functionals D(*M*) on the functional manifold *M*.

Direct calculations show that the results described above are valid for all the currently known completely integrable nonlinear dynamical systems, including the nonlinear equations of Schrëdinger type [21,59,114,124,125], the Benney—Kaup and Ito equations [114], the Davey—Stewartson and Yajima—Mel'nikov equations [13], and others, defined on infinite-dimensional manifolds, whose symmetry groups are isomorphic to the universal Banach current group Di*f f*(S<sup>1</sup>) - *F* on the circle S1. With regard to "*two- dimensionalized*" integrable dynamical systems of the Kadomtsev–Petviashvily type, it can be asserted that they are closely related [11,13,112,117] to special operator-valued nonlinear integrable dynamical systems, generated by suitably defined iso-spectral Lax type problems [111,126] and which are bi-Hamiltonian with respect to the Poissonian operators on these operatorvalued manifolds.

The analysis made above of the correspondence between the universal Lie algebra of currents (128) and the functional Lie algebras of symmetries of integrable infinitedimensional dynamical systems makes it possible to formulate the following working algorithm as an effective criterion of testing integrability of an arbitrary homogeneous nonlinear dynamical system (132) on the infinite-dimensional manifold *M*.

Algorithm*: If for the dynamical system ut* = *<sup>K</sup>*[*u*] *on the functional manifold M there exists the nontrivial sl*(2) *Lie subalgebra* {*<sup>τ</sup>*−1, *τ*0, *<sup>τ</sup>*1} *together with a subset of "initial" inhomogeneous symmetries* {*<sup>τ</sup>*−2, *<sup>τ</sup>*2} *and homogeneous* {*<sup>α</sup>*−1, *α*1 : *ε* = 0} *symmetries, satisfying the conditions*

$$\begin{aligned} \left[\pi\_{0}, \pi\_{2}\right] &= 2\pi\_{2}, \left[\pi\_{0}, \pi\_{-2}\right] = -2\pi\_{-2}, \left[\pi\_{1}, \pi\_{-2}\right] = -3\pi\_{-1} \\ \left[\pi\_{-1}, \pi\_{1}\right] &= 0, \left[\pi\_{-1}, \pi\_{2}\right] = 3\pi\_{1}, \left[\pi\_{0}, \pi\_{0}\right] = \varepsilon\mathfrak{a}\_{0} \end{aligned} \tag{147}$$

*then this dynamical system on M possesses an infinite-dimensional Lie algebra of symmetries Q* = *Q*{*τ*} - *Q*{*α*}, *isomorphic to the current Lie algebra G (128) of the Banach group Di*ff(*S*<sup>1</sup>) - *F on the circle S*1, *and if there exists a nontrivial solution of the Noether–Lax equation LKϑ* = 0, *then our dynamical system is an infinite-dimensional completely integrable Hamiltonian flow on the functional manifold M*. *If at the same time the relations (145) are satisfied, then the dynamical system ut* = *<sup>K</sup>*[*u*] *on M is bi-Hamiltonian and possesses an hereditary-recursive operator* Λ = *<sup>ϑ</sup>*−<sup>1</sup>*η*, *where η*(*ξ* − 3/2) = *<sup>L</sup><sup>τ</sup>*1*ϑ*, *and, by virtue of the gradient-holonomic algorithm [11,13,59,116], a standard Lax type representation.*

*If the conditions (143) are not satisfied, the dynamical system does not possess bi- Hamiltonian structure, but there is an additional infinite-dimensional inhomogeneous hierarchy of conservation laws satisfying the conditions (146).*

#### *3.3. Integrable Systems, Their Symmetry Analysis and Structure of the Poissonian Operators*

Suppose we are given the homogeneous nonlinear dynamical system (132) on the functional manifold *M* and pose a question of the existence for this dynamical system of a bi-Hamiltonian structure on *M* and effective methods of determining it in explicit form.

In accordance with the gradient-holonomic algorithm *[11,13,59,116]* for investigating the integrability of nonlinear dynamical systems, we can successively establish in explicit form the presence for our system (132) of an infinite functionally independent and naturally ordered by means of the parameter *λ* ∈ R hierarchy *γj* ∈ <sup>D</sup>(*M*), *j* ∈ Z+, of conservation laws. In addition, by virtue of the homogeneity of the dynamical system (132), it always possesses *a priori* two commuting to each other homogeneous symmetries, which are defined on *M* by the vector fields *d*/*dx* and *d*/*dt*. We can also consider the equivalent realization of these vector fields *d*/*dx* and *d*/*dt* on *M* as Hamiltonian systems [11,17,18,123] on the infinite-dimensional manifold of jets *J*(S1; R*m*) - *M* with respect to a symplectic structure *ω*(2) ∈ Ω<sup>1</sup>(*J*(S1; <sup>R</sup>*<sup>m</sup>*). We denote by *α*(1) ∈ Ω<sup>1</sup>(*J*(S1; R*m*) a Liouville type 1-form, for which *dα*(1) = *ω*(2) and take into account that, by definition, there holds the conditions *id*/*dx ω*(2)[*u*] = −*dγ*[*u*] and *Ld*/*dx ω*(2)[*u*] = 0, where *γ*[*u*] ∈ Ω<sup>0</sup>(*J*(S1; R*m*) denotes the density of the corresponding conservation law *γ* ∈ D(*M*) at point *u* ∈ *M*. Based now on the Cartan representation [127] of the Lie derivative *Ld*/*dx* = *id*/*dxd* + *did*/*dx*, one easily obtains the following general relationship: *γ*[*u*] = *α*(1)[*u*](*d*/*dx*)=( *ψ*[*u*]|*ux*) mod(*d*/*dx*) for some element *ψ* ∈ *T*∗(*M*) at any point *u* ∈ *M*, which should simultaneously satisfy the compatibility condition *d Ld*/*dt ψ* = 0 subject to the vector field *d*/*dt* on *M*. The latter gives rise to the analytical expression that is useful in applications, *ϑ*−<sup>1</sup> = *ψ* [*u*] − *ψ*,<sup>∗</sup>[*u*], for the corresponding cosymplectic operator on the manifold *M*, whose inverse mapping *ϑ* : *T*∗(*M*) → *T*(*M*) is our searched for Poissonian operator for the dynamical system (132).

#### 3.3.1. Two-Dimensional Korteweg–de Vries Type Hydrodynamic System

We consider an example of a nonlinear bi-Hamiltonian Korteweg–de Vries type hydrodynamic system on "*two-dimensionalized*" smooth functional manifold *M* - *J*(T2; R) for which the Noether–Lax property (143), mentioned above, is not satisfied. This system [12,21] has the form

$$
\mu\_t = \mu\_{\rm xxy} + 2\mu\_x \partial\_x^{-1} \mu\_y + 4\mu \mu\_y := K[\mu] \tag{148}
$$

and possesses two algebraically-independent Poissonian operators

$$
\theta = \partial\_{\mathbf{x}} \quad \eta = \partial\_{\mathbf{x}}^3 + 2(\imath \partial\_{\mathbf{x}} + \partial\_{\mathbf{x}} \mu) \tag{149}
$$

Moreover, it is readily shown that for the dynamical system, (148) allows the representation *<sup>K</sup>*[*u*] = <sup>Φ</sup>*uy*, where Φ := *ηϑ*−<sup>1</sup> = *∂*2 *x* + <sup>2</sup>(*u* + *∂x<sup>u</sup>∂*−<sup>1</sup> *x* ) is the corresponding symmetry-recursive operator on *M*. One can check by direct calculations that the set {*τ*(*x*) −1 = 1/4(1 + *tux*), *τ*(*y*) −1 = 1/4(1 + *tuy*)} consists of inhomogeneous symmetries of the dynamical system (148) and the set {*α*(*x*) 0 = *ux*, *α* (*y*) 0 = *ux*} consists of homogeneous symmetries. From them, one constructs the following hierarchies of symmetries:

$$Q\{a^{(x)}\} := \{a\_j^{(x)} = \Phi^j a\_0^{(x)} : j \in \mathbb{Z}\}, \\ Q\{a^{(y)}\} := \{a\_j^{(y)} = \Phi^j a\_0^{(y)} : j \in \mathbb{Z}\},\tag{150}$$

$$Q\{\tau^{(\mathbf{x})}\} := \{\tau\_j^{(\mathbf{x})} = \Phi^{j+1}\tau\_{-1}^{(\mathbf{x})} : j \in \mathbb{Z}\},\\Q\{\tau^{(y)}\} := \{\tau\_j^{(y)} = \Phi^{j+1}\tau\_{-1}^{(y)} : j \in \mathbb{Z}\}$$

The resulting Lie subalgebras *Q*(*x*) := *Q*{*τ*(*x*)} - *Q*{*α*(*x*)} and *Q*(*y*) := *Q*{*τ*(*y*)} - *Q*{*α*(*y*)} have the following commutation relationships:

$$\begin{aligned} [\mathbf{r}\_j^{(x)}, a\_k^{(y)}] &= k a\_{j+k'}^{(y)} \ [\mathbf{r}\_j^{(y)}, a\_k^{(x)}] &= (k+1/2) a\_{j+k'}^{(x)} \\\\ [\mathbf{r}\_j^{(x)}, \mathbf{r}\_k^{(y)}] &= k \mathbf{r}\_{j+k}^{(y)} - j \mathbf{r}\_{j+k'}^{(x)} \ [a\_j^{(x)}, a\_k^{(y)}] &= 0 \end{aligned} \tag{151}$$

for *j*, *k* ∈ Z, and the Lie subalgebras *Q*(*x*) and *Q*(*y*) are isomorphic to the current Lie algebra G (128) on the circle S1. Taking into account this fact and expressions (151), we readily state that the following sets *Q*{*τ*(+)} and *Q*{*τ*(−)} of symmetries

$$\mathcal{Q}\{\boldsymbol{\tau}^{(+)}\} := \{\boldsymbol{\tau}^{(+)}\_{j} = 1/2(\boldsymbol{\tau}^{(x)}\_{j} + \boldsymbol{\tau}^{(y)}\_{j}) : j \in \mathbb{Z}\},\tag{152}$$

$$Q\{\tau^{(-)}\} := \{\tau\_j^{(-)} = 1/2(\tau\_j^{(x)} - \tau\_j^{(y)}) : j \in \mathbb{Z}\}$$

satisfy for all *j*, *k* ∈ Z the commutation relationships

$$[\mathbf{r}\_j^{(-)}, \mathbf{r}\_j^{(+)}] = 0, [\mathbf{r}\_j^{(+)}, \mathbf{r}\_k^{(+)}] = (k - j)\mathbf{r}\_{j+k'}^{(+)}[\mathbf{r}\_j^{(+)}, \mathbf{a}\_k^{(x)}] = (k + 1/2)a\_{j+k'}^{(x)}\tag{153}$$

$$[\pi\_j^{(-)}, a\_k^{(\chi)}] = 0, [\pi\_j^{(-)}, a\_k^{(y)}] = 0, [\pi\_j^{(+)}, a\_k^{(y)}] = k a\_{j+k'}^{(y)} [\pi\_j^{(+)}, \pi\_k^{(-)}] = k \pi\_{j+k}^{(-)}.$$

The latter make it possible to deduce the direct sum of Lie algebras of commuting to each of the other Abelian symmetries

$$Q\{\mathfrak{a}, \mathfrak{r}^{(-)}\} := Q\{\mathfrak{r}^{(-)}\} \oplus Q\{\mathfrak{a}^{(\mathbf{x})}\} \oplus Q\{\mathfrak{a}^{(\mathbf{x})}\},\tag{154}$$

which jointly with the symmetry Lie subalgebra *Q*{*τ*(+)} constitutes the Lie algebra *Q* constructed above of symmetries to the nonlinear dynamical system (148) as the semidirect product

$$Q = Q\{\tau^{(+)}\} \ltimes Q\{a, \tau^{(-)}\},\tag{155}$$

being fully isomorphic to the current Lie algebra G (128). The latter states the invariance of the nonlinear dynamical system (148) with respect to the current symmetry group *G* = Diff(S<sup>1</sup>) - *F* of the circle S1.

#### 3.3.2. Nonlinear Schrëdinger Type Dynamical System

On a smooth functional manifold *M* ⊂ *<sup>C</sup>*<sup>2</sup>(<sup>R</sup>; C<sup>2</sup>) a nonlinear Schrëdinger type dynamical system, which was first considered in [128], looks as

$$\begin{aligned} \psi\_t &= i\psi\_{xx} + (\psi^2 \psi)\_{x\prime} \\ \bar{\psi}\_t &= -i\bar{\psi}\_{xx} + (\bar{\psi}^2 \psi)\_{x\prime} \end{aligned} \Big| := \mathcal{K}[\psi, \bar{\psi}] \tag{156}$$

and is a bi-Hamiltonian flow with respect to the following two compatible Poisson structures: 

$$
\vartheta = \begin{pmatrix} 0 & \partial\_x \\ \partial\_x & 0 \end{pmatrix}, \quad \eta = \begin{pmatrix} -\psi \partial\_x^{-1} \psi & -i + \psi \partial\_x^{-1} \psi \\ i + \bar{\psi} \partial\_x^{-1} \psi & -\bar{\psi} \partial\_x^{-1} \bar{\psi} \end{pmatrix}. \tag{157}
$$

It is easy to check that the following flows on *M*

$$\mathbf{x}\_0 = t\mathbf{K} + (\mathbf{x}\psi\_\mathbf{x} + \psi/2, \mathbf{x}\bar{\psi}\_\mathbf{x} + \bar{\psi}/2)^\mathsf{T},\tag{158}$$

$$\tau\_1 = t\alpha\_3 + \ge \mathcal{K} + (\psi^2 \bar{\psi} + i3/2 \psi\_{\ge \prime} \bar{\psi}^2 \psi - i3/2 \psi / 2)^{\intercal},$$

are nonuniform symmetries of the dynamical system (156), that is

$$\left[\partial\pi\_{\rangle}/\partial t + [K,\pi\_{\rangle}] = 0\tag{159}$$

for *j* = 1, 2, where *α*3 := <sup>Φ</sup><sup>2</sup>(*ψ<sup>x</sup>*, *ψ*¯*x*)- and Φ := *ηϑ*−<sup>1</sup> : *T*(*M*) → *T*(*M*) is the corresponding symmetry-recursive operator. Moreover, the following algebraic relationships hold:

$$L\_K \tau\_0 = -\vartheta, \quad L\_K \tau\_1 = -2\eta,\tag{160}$$

where, as before, *LK* denotes the Lie derivative with respect to the vector field *K* : *M* → *<sup>T</sup>*(*M*). Put now, by definition, *α*0 := (−*iψ*, *iψ*¯)-, and *αj* := <sup>Φ</sup>*j<sup>α</sup>*0, *τj* := Φ*jτ* for *j* ∈ Z. Then the following proposition holds.

**Proposition 6.** *The nonlinear Schrëdinger type dynamical system (156) is a completely integrable bi-Hamiltonian system on the functional manifold M*, *possessing two independent symmetry Lie subalgebras Q*{*τ*} := {*τj* : *j* ∈ Z} *and Q*{*α*} := {*<sup>α</sup>j* : *j* ∈ <sup>Z</sup>}. *Moreover, their semidirect product Q*{*<sup>α</sup>*, *τ*} := *Q*{*τ*} - *Q*{*α*} *is isomorphic to the quantum Lie algebra* G *of currents (128) of the Banach group G* = Diff(S<sup>1</sup>) - *F on the circle* S1.

3.3.3. The Benjamin–Ono Nonlinear Dynamical System

This dynamical system is defined on a functional manifold *M* ⊂ *<sup>C</sup>*<sup>2</sup>(<sup>R</sup>; R) as

$$
\mu\_l = \mathcal{H}\mu\_{\text{xx}} + 2\mu\mu\_{\text{x}} \tag{161}
$$

where *u* ∈ *M* and H : *T*(*M*) → *T*(*M*) is the classical Hilbert transform

$$(\mathcal{H}a)(\mathbf{x}) := \frac{1}{\pi} \int\_{-\infty}^{\infty} dy \frac{a(y)}{y - \mathbf{x}} \tag{162}$$

for any *α* ∈ *<sup>T</sup>*(*M*). The Hilbert transform (162) satisfies the following algebraic properties: H<sup>2</sup> = −1, H∗ = −H subject to the standard bilinear convolution form on the product *T*∗(*M*) × *<sup>T</sup>*(*M*). It is easy to check that the dynamical system (161) is Hamiltonian [129] with respect to the Poisson operator

$$
\boldsymbol{\theta} = \boldsymbol{\partial}/\partial \mathbf{x},\tag{163}
$$

that is *ut* = −*ϑ* grad *H*, where the Hamiltonian function *H* <sup>=</sup> <sup>∞</sup>−<sup>∞</sup> *dx*(*u*3/3 + *<sup>u</sup>*H*ux*). Simple calculations make it possible to state [113] that the following functional expressions

$$\begin{aligned} \pi\_{-1} &= 1 + t u\_{\text{x}}, \pi\_{0} = x u\_{\text{x}} + u + t K, \\ \pi\_{1} &= t u\_{2} + x K + u^{2} - 3/2 \mathcal{H} u\_{\text{x}}, \\ \pi\_{2} &= [2u^{3} + 3H(u u\_{\text{x}}) + 3u H u\_{\text{x}} - 2u\_{\text{xx}}]\_{\text{x}} \end{aligned} \tag{164}$$

are symmetries of the Benjamin–Ono nonlinear dynamical system (161). Moreover, since there hold algebraic relationships *<sup>L</sup><sup>τ</sup>j<sup>ϑ</sup>* = 0 for *j* = −1, 1, we can state that this dynamical system is not bi-Hamiltonian on the functional manifold *M*, as owing to the relationships (145) we should have *<sup>L</sup><sup>τ</sup>*−<sup>1</sup>*ϑ* = (*ξ* + 1/2)*<sup>ϑ</sup>*Λ−<sup>1</sup> = 0, *<sup>L</sup><sup>τ</sup>*0*ϑ* = (*ξ* − 1/2)*ϑ* = 0 and *<sup>L</sup><sup>τ</sup>*1*ϑ* = (*ξ* − 3/2)*ϑ*Λ = 0, whose common solution is *ξ* = 1/2 and *η* = 0. The latter means that the Benjamin–Ono nonlinear dynamical system (161) is not bi-Hamiltonian on the functional manifold *M*, albeit it proves to be bi-Hamiltonian [129] on an extended spatially two-dimensional operator manifold *M* ˆ , being equivalent to a respectively defined Hilbert– Schmidt operator algebra, whose theory was previously developed in [11,20,21,130,131] and applied to other nonlinear dynamical systems such as Devey–Stewartson, Kadomtsev– Petviashvily, etc.
