Super Lie–Poisson Structures, Their Deformations, and Related New Nonlinear Integrable Super-Hamiltonian Systems

Abstract

Lie-algebraic Poisson structures, related to the superalgebra of super-pseudodifferential operators on the circle over the even component of the ${\mathbb{Z}}_2$-graded Grassmann algebra, have been studied in detail; the corresponding coadjoint orbits, generated by the Casimir invariants, regarding the different superalgebra splittings into the subalgebras, are analyzed. The related Lax-type completely integrable Hamiltonian flows are constructed on suitably defined functional manifolds with respect to the canonical super-Lie–Poisson structures on them. An approach was proposed allowing the extension of the related coadjoint flows by means of the respectively constructed super-evolution flows on the adjoint super-subalgebras, specially deformed by means of super-pseudodifferential operator elements, depending on the generalized eigenfunctions of the corresponding super-linear Lax-type spectral problem. As a consequence, it is stated that all constructed new coadjoint superflows generate on the suitably extended supermanifolds Lax-type integrable Hamiltonian systems. The centrally extended super-Lie-algebraic structures have been analyzed and the related coadjoint orbits described, generated by the corresponding Casimir invariants and coinciding with integrable Hamiltonian systems on suitably defined supermanifolds.

1. Introduction

We deal with the associative algebra ${\mathcal{A}}_0({\mathbb{S}}^1;\partial ),\partial :=\partial /\partial x, x\in {\mathbb{S}}^1,$ of super-pseudo differential operators on the circle ${\mathbb{S}}^1$ over the subalgebra ${\Lambda }_0$ of a ${\mathbb{Z}}_2$-graded Grassmann algebra $\Lambda :={\Lambda }_0\oplus {\Lambda }_1,$ the associated super-Lie algebra ${\mathcal{L}}_0({\mathbb{S}}^1;\partial )$ and its special coadjoint orbits, generated by the corresponding Casimir invariants within the Lie-algebraic ASK-scheme [1,2,3]. As one easily observes, such a Lie algebra, though super-pseudodifferential, behaves very similarly to the ordinary Lie algebra of usual pseudodifferential operators on the circle ${\mathbb{S}}^1,$ in particular, it is also metrized [1,3,4,5] with respect to a suitably defined nondegenerate, ad-invariant and symmetric bilinear form $(·|·):{\mathcal{L}}_0({\mathbb{S}}^1;\partial )\times {\mathcal{L}}_0({\mathbb{S}}^1;\partial )\to {\Lambda }_0,$ generated [1,6,7,8,9,10] by the trace-operation $\mathrm{Tr}:{\mathcal{L}}_0({\mathbb{S}}^1;\partial )\to {\Lambda }_0.$ Taking into account the direct sum splittings of the super-Lie algebra ${\mathcal{L}}_0({\mathbb{S}}^1;\partial )={\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,+}\oplus {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}$, $k=\overline{0,2}$, we considered the coadjoint orbits of an arbitrary elements $l\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,+}, k=\overline{0,2},$ generated by Casimir invariants of the super-Lie algebra ${\mathcal{L}}_0({\mathbb{S}}^1;\partial )$ within the classical [1,2,11,12,13] AKS-framework. The related evolution flows, being involutive to each other, are equivalent to some completely integrable [2,11,12,14] dynamical systems with respect to suitably constructed super-Lie–Poisson bracket on the associated functional supermanifold $M_{l_{\phi }}\subset C^{\infty }({\mathbb{S}}^1;{\Lambda }_0^{p\left(l\right)}), p\left(l_{\phi }\right):=\mathrm{ord}\left(l_{\phi }\right)\in \mathbb{N}$-the differential order of a differential operator $l_{\phi }\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*},$ and which in many cases coincide with some theoretical physics models [15,16,17,18,19] of elementary particles and fields. Taking into account that the eigenvalues ${\lambda }_q, {\lambda }_q^{*}\in {\Lambda }_q, q\in {\mathbb{Z}}_2$, of the associated [20,21,22,23,24] linear eigenvalue problems

$$l_{\phi }(x;\partial ){\phi }_q\left(x\right)={\lambda }_q{\phi }_q\left(x\right),l_{\phi }{(x;\partial )}^{*}{\phi }_q^{*}\left(x\right)={\lambda }_q^{*}{\phi }_q^{*}\left(x\right)$$

(1)

for $l_{\phi }\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}, k=\overline{0,2},$ are invariant with respect the coadjoint evolution flows

$$\partial l_{\phi }/\partial t_{m_q}=[{\left(l_{\phi }^{m/p\left(l\right)}\right)}_{k,+},l_{\phi }]$$

(2)

for all $m\in \mathbb{N},$ one obtains right away that the eigenfunctions ${\phi }_q, {\phi }_q^{*}\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_q),$$q\in {\mathbb{Z}}_2,$ satisfy the following countable hierarchies of linear evolution flows:

$$\partial {\phi }_q/\partial t_m={\left(l_{\phi }^{m/p\left(l\right)}\right)}_{k,+}{\phi }_q,$$

(3)

$$\partial {\phi }_q^{*}/\partial t_m=-{\left(l_{\phi }^{m/p\left(l\right)}\right)}_{k,+}^{*}{\phi }_q^{*}$$

(4)

on the space ${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2),$ with respect to the temporal parameters $t_m\in \mathbb{R}, m\in \mathbb{N}.$

We proved that constructed above evolution flows (2)–(4), if rewritten with respect to the variables $(l_{\phi };{\phi }_q,{\phi }_q^{*})\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}$$\times {\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2),$ as

$$\begin{array}{ccc}& & \partial l_{\phi }/\partial t_m=[{\left({\tilde{l}}_{\phi }^{m/p\left(l\right)}\right)}_{k,+},l_{\phi }],\hfill \\ & & \partial {\phi }_q/\partial t_m={\left(l_{\phi }^{m/p\left(l\right)}\right)}_{k,+}{\phi }_q,\hfill \\ & & \partial {\phi }_q^{*}/\partial t_m=-{\left(l_{\phi }^{m/p\left(l\right)}\right)}_{k,+}^{*}{\phi }_q^{*}\hfill \end{array}$$

(5)

for $k=\overline{0,2}$ and all $m\in \mathbb{N}$, are also completely integrable superflows with respect to the following extended super-Poisson operator

$${\vartheta }_k(l_{\phi };{\overline{\phi }}_q,{\overline{\phi }}_q^{*})=\left(\begin{array}{ccc}[P_{k,+}(·),l_{\phi }]-P_{k,-}^{*}[(·),l_{\phi }]& \underset{q\in {\mathbb{Z}}_2}{\otimes }{(-1)}^q{\partial }^{-k}{(·)}_{q,kx}{\partial }^{-1}{\phi }_q^{*}& {(-1)}^q{\partial }^{-k}{(·)}_{q,kx}{\partial }^{-1}{\phi }_q^{*}\\ {(·)}_{k,+}{\phi }_q& 0& 1\\ {(·)}_{k,+}^{*}{\phi }_q^{*}& -1& 0\end{array}\right).$$

(6)

acting on the gradient-vectors $\mathrm{grad}\gamma (l;{\phi }_q,{\phi }_q^{*})\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\times {\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2)$, and generated by Casimir Hamiltonian functions ${\gamma }_m:={\displaystyle \frac{p\left(l\right)}{p\left(l\right)+m}}\mathrm{Tr}{\left(l_{\phi }\right)}^{(m+p(l\left)\right)/p\left(l\right)}, m\in \mathbb{N},$ where an arbitrary element $l\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,+}^{*}$ is deformed to $l_{\phi }=$$l+$${\sum }_{q\in {\mathbb{Z}}_2}{\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}{\phi }_q^{*}\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}$ and the eigenfunctions $({\phi }_q,{\phi }_q^{*})\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2), q\in {\mathbb{Z}}_2.$ This important result generalizes the ones, before stated in [21,22,23] for the ordinary algebra of pseudodifferential operators on the axis and later in [6,7,8,9,25] for the Lie superalgebras $\mathcal{L}({\mathbb{S}}^{1|r};D_{\theta }),\theta \in {\Lambda }_1^r, r=$$\overline{1,3},$ of super-pseudodifferential operators on the r-dimensional supercircles ${\mathbb{S}}^{1|r}, r=$$\overline{1,3},$ generated by superderivatives $D_{{\theta }_j}=\partial /\partial {\theta }_j+{\theta }_j\partial /\partial x ,j=\overline{1,3}.$

As a simplest example of the construction above for $k=0$ is given by the “superized” Sturm–Liouville type expression

$$l_{\phi }(x;\partial ):={\partial }^2-u-{\phi }_1{\partial }^{-1}{\phi }_1^{*},$$

(7)

on the circle ${\mathbb{S}}^1$, where $u\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_0).$ Having put $q=1$ and $m=2,$ one easily derives that the set of eigenfunctions $({\phi }_1,{\phi }_1^{*})\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_1^2)$ jointly with a coefficient $u\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_0)$ satisfy the system of evolution equations

$$\begin{array}{cc}\hfill u_{t_2}& =2({\phi }_1^{*}{\phi }_{1,x}-{\phi }_1{\phi }_{1,x}^{*}),\hfill \\ \hfill {\phi }_{1,t_2}& ={\phi }_{1,xx}-u{\phi }_1,\hfill \\ \hfill {\phi }_{1,t_2}^{*}& =-{\phi }_{1,xx}^{*}+u{\phi }_1^{*}\hfill \end{array}$$

(8)

with respect to the evolution parameter $t_2\in \mathbb{R}.$ It presents, owing to the expression (6), a completely Hamiltonian system on the functional supermanifold $M_l\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_0\times {\Lambda }_1^2)$ with respect both to the “linear” super-Poisson operator

$${\vartheta }_0(u;{\phi }_1,{\phi }_1^{*})=\left(\begin{array}{ccc}\partial & 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right),$$

(9)

resulting from the R-structure homomorphism $R_0:=\left(P_{0,+}-P_{0,-}\right)/2$ on the super-Lie algebra ${\mathcal{L}}_0({\mathbb{S}}^1;\partial ).$ Similarly, the local R-structure $R_0\left(l_{\phi }\right):=$$\left(P_{0,+}{l}_{\phi }(\circ )-P_{0,-}(·)l_{\phi }\right)/2$ on the Lie superalgebra ${\mathcal{L}}_0({\mathbb{S}}^1;\partial )$ results into the “quadratic” super-Poisson operator

$${\eta }_0(u;{\phi }_1,{\phi }_1^{*})=\left(\begin{array}{ccc}1/2{\partial }^3-u\partial -\partial u& {\phi }_1\partial +1/2\partial {\phi }_1& {\phi }_1^{*}\partial +1/2\partial {\phi }_1^{*}\\ \partial {\phi }_1+1/2{\phi }_1\partial & -3/2{\phi }_1{\partial }^{-1}{\phi }_1& {\partial }^2-u+3/2{\phi }_1{\partial }^{-1}{\phi }_1^{*}\\ \partial {\phi }_1^{*}+1/2{\phi }_1^{*}\partial & -{\partial }^2+u+3/2{\phi }_1^{*}{\partial }^{-1}{\phi }_1& -3/2{\phi }_1^{*}{\partial }^{-1}{\phi }_1^{*}\end{array}\right)$$

(10)

on the supermanifold $M_{l_{\phi }}\simeq C^{\infty }(S^1;{\Lambda }_0\times {\Lambda }_1^2),$ generating for $m=3$ the interesting integrable Korteweg–de Vries type superflow

$$\begin{array}{cc}\hfill u_t& =u_{3x}+6u{u}_x-6({\phi }_{1,xx}{\phi }_1^{*}-{\phi }_1{\phi }_{1,xx}^{*}),\hfill \\ \hfill {\phi }_{1,t}& =4{\phi }_{1,3x}-6u{\phi }_{1,x}-3u_x{\phi }_1,\hfill \\ \hfill {\phi }_{1,t}^{*}& =4{\phi }_{1,3x}^{*}-6u{\phi }_{1,x}^{*}-3u_x{\phi }_1^{*}.\hfill \end{array}$$

(11)

Moreover, having applied to the super-Poisson operators (9) and (10) the eigenfunction symmetry condition ${\phi }_1={\phi }_1^{*}\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_1)$, we obtain via the well know Dirac reduction scheme [2,3,11,26,27] both the reduced linear super-Poisson structure

$${\vartheta }_0(u;{\phi }_1)=\left(\begin{array}{cc}-\partial & 0\\ 0& -1/4\end{array}\right)$$

(12)

and “quadratic” super-Poisson structure

$${\eta }_0(u;{\phi }_1)=\left(\begin{array}{cc}1/2{\partial }^3-u\partial -\partial u& 1/2(3{\phi }_1\partial +{\phi }_{1,x})\\ 1/2(3\partial {\phi }_1-{\phi }_{1,x})& 1/2(u-{\partial }^2)\end{array}\right)$$

(13)

on the supermanifold $M_{l_{\phi }}$. The latter, respectively, generates on the supermanifold $M_{l_{\phi }}$ the well-known [16,28] integrable super-Korteweg–de Vries system, related to the deformed spectral problem $l_{\phi }f(x;{\lambda }_0)=$${\lambda }_{0}f(x;{\lambda }_0)$ at $(x;{\lambda }_0)\in {\mathbb{S}}^1\times {\Lambda }_0$, for even function $f\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_0)$ and looking as

$$f_{xx}-uf+{\phi }_1{\partial }^{-1}{\phi }_{1}f={\lambda }_{0}f.$$

(14)

The differential relationship (14) can be easily rewritten modulo the new odd functional component $g={\partial }^{-1}\left({\phi }_{1}f\right)\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_1)$ to the equivalent “conformal” spectral problem

$$\left(-D_{\theta }^3+\Phi \right)\psi ={\lambda }_0\theta \psi ,$$

(15)

where, by definition, $\Phi :={\phi }_1+\theta u\in C^{\infty }({\mathbb{S}}^{1|1};{\Lambda }_1)$ and $\psi :=f+\theta g\in C^{\infty }({\mathbb{S}}^{1|1};{\Lambda }_0)$ is a generator [10,16] of the conformal symmetry vector field

$$\psi \partial /\partial x+1/2\left(D_{\theta }\psi \right)D_{\theta }$$

(16)

with respect to the superderivative $D_{\theta }=\partial /\partial \theta +\theta \partial /\partial x$ on the supercircle with coordinates $(x;\theta )\in {\mathbb{S}}^{1|1}.$

As next interesting examples, we considered integrable superflows, generated both by an element $l\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{1,-}^{*},$

$$l\to l_{\phi }={\partial }^2+u\partial +v+{\partial }^{-1}w+{\partial }^{-1}{\phi }_{1,x}{\partial }^{-1}{\phi }_1^{*},$$

(17)

where even coefficients $u,v,w,$ and $v\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_0)$, and by an element

$$l\to l_{\phi }=\partial +u+{\partial }^{-1}v+{\partial }^{-1}{\phi }_{1,x}{\partial }^{-1}{\phi }_1^{*},$$

(18)

deformed by means of an odd functional vector $({\phi }_1,{\phi }_1^{*})\in C^{\infty }({\mathbb{S}}^{1|1};{\Lambda }_1^2).$ They generate, respectively, the following super-Poisson structures on the functional manifolds $M_{\phi }$:

$${\vartheta }_1(u,v,w;{\phi }_1,{\phi }_1^{*})=\left(\begin{array}{ccccc}0& 0& 2\partial & 0& 0\\ 0& 2\partial & {\partial }^2+u\partial & 0& 0\\ 2\partial & -{\partial }^2+\partial u& 0& 0& 0\\ 0& 0& 0& 0& -1\\ 0& 0& 0& -1& 0\end{array}\right)$$

(19)

and

$${\vartheta }_1(u,v;{\phi }_1,{\phi }_1^{*})=\left(\begin{array}{cccc}0& \partial & 0& 0\\ \partial & 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& -1& 0\end{array}\right),$$

(20)

regarding this, the following new Hamiltonian systems

$$\begin{array}{c}\partial u/\partial t=2v_x,\partial v/\partial t=2w_x+v_x+u{v}_x,\\ \partial w/\partial t=-v_{xxx}+{\left(uv\right)}_x-2{\left({\phi }_1^{*}{\phi }_{1,x}\right)}_x,\\ {\phi }_{1,t}={\phi }_{1,xx}+u{\phi }_{1,x},{\phi }_{1,t}^{*}=-{\phi }_{1,xx}^{*}-u{\phi }_{1,x}^{*}\end{array}$$

(21)

and

$$\begin{array}{cc}\hfill \partial u/\partial t& =u_{xx}+2v_x+6u{u}_x,\hfill \\ \hfill \partial v/\partial t& =-v_{xx}+2{\left(uv\right)}_x-2{\phi }_1^{*}{\phi }_{1,x},\hfill \\ \hfill \partial {\phi }_1/\partial t& ={\phi }_{1,xx}+2u{\phi }_{1,x},\partial {\phi }_1^{*}/\partial t={\phi }_{1,xx}^{*}-2{\left(u{\phi }_1\right)}_x\hfill \end{array}$$

(22)

prove to be integrable.

We also considered a generalization of the construction above, using the ${\mathbb{S}}^{1|1}$-parameterizations of the super-Lie algebra ${\mathcal{L}}_0({\mathbb{S}}^1;\partial )$→${\displaystyle \prod _{{\mathbb{S}}^{1|1}}}{\mathcal{L}}_0({\mathbb{S}}^1;\partial ):={\tilde{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )$ and its rigging with a cental extension ${\tilde{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )\to ({\tilde{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )\oplus \Lambda ):={\widehat{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )$ by means of the Maurer–Cartan super-cocycle, determining the super-Lie commutator

$$[(\tilde{a},\alpha ),(\tilde{b},\beta )]:=([\tilde{a},\tilde{b}],\left(\tilde{a}\right|D_{\theta }\tilde{b})),$$

(23)

for any elements $(\tilde{a},\alpha ),(\tilde{b},\beta )\in {\tilde{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial ),$ where $D_{\theta }:=\partial /\partial \theta +\theta \partial /\partial y,(y,\theta )\in {\mathbb{S}}^{1|1},$ is the superderivative on the circle ${\mathbb{S}}^{1|1}.$ As the constructed super-Lie algebra ${\widehat{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )$ proves to be also metrized, we have studied nonlinear integrable Hamiltonian flows on the adjoint space ${\widehat{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )\simeq {\widehat{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial ),$ generated within the Lie-algebraic AKS scheme [1,13,29] by Casimir functionals and related super-Lie–Poisson structures on it. The corresponding superflows on the adjoint spaces ${\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k,-}^{*},k=\overline{0,2},$ generated by Casimir functionals $\gamma \in I\left({\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k,-}^{*}\right)$ with respect to the super-Lie–Poisson structure (23), are easily calculated as

$$\partial \tilde{l}/\partial t=[R_k\mathrm{grad}\gamma \left(\tilde{l}\right),\tilde{l}]-D_{\theta }{R}_k\mathrm{grad}\gamma \left(\tilde{l}\right)$$

(24)

at any point $(\tilde{l},\rho )\in {\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k.-}^{*}$. In particular, the evolution superflows (24) can be equivalently rewritten at $\rho =-1$ as the following commutator equalities

$$[\partial /\partial t-R_k\mathrm{grad}\gamma \left(\tilde{l}\right),D_{\theta }-\tilde{l}]=0$$

(25)

on the adjoint spaces ${\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k,-}^{*},$$k=\overline{0,2},$ coinciding with the classical Zakharov–Shabat type [30,31] representations of nonlinear integrable Hamiltonian systems on the coefficient functional supermanifold $M_{\tilde{l}},$ related to the operator elements $(\tilde{l},1)\in {\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k,-}^{*}$ for $k=\overline{0,2}.$ In addition, these integrable flows were both suitably extended by means of evolutions of their generalized “eigenfunctions” and represented as the bi-Hamiltonian systems with respect to specially constructed Poisson brackets. Having shortly outlined above our main statements and results, we proceed below to a detail presentation of all needed Lie-algebraic constructions and related analytical calculations devised for describing a new wide class of nonlinear super-integrable Hamiltonian systems on functional supermanifolds.

2. Preliminaries

We consider the associative algebra ${\mathcal{A}}_0({\mathbb{S}}^1;\partial ),\partial :=\partial /\partial x,x\in {\mathbb{S}}^1$, of pure pseudo differential operators

$$a(x;\partial )=\sum _{j\gg -\infty }a\left(x\right){\partial }^{-j},$$

(26)

where $x\in {\mathbb{S}}^1,a\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_0)$, ${\Lambda }_0\subset \Lambda$ is the even subalgebra of a ${\mathbb{Z}}_2$-graded Grassmann algebra $\Lambda :={\Lambda }_0\oplus {\Lambda }_1.$ The associative algebra ${\mathcal{A}}_0({\mathbb{S}}^1;\partial )$ transforms into a super-Lie algebra $({\mathcal{L}}_0({\mathbb{S}}^1;\partial ),[·,·])$ with respect to the Lie commutator

$$\left[a\right(x;\partial ),b(x;\partial \left)\right]:=a(x;\partial )\circ b(x;\partial )-b(x;\partial )\circ a(x;\partial )$$

(27)

for arbitrary elements $a(x;\partial ),b(x;\partial )\in {\mathcal{L}}_0({\mathbb{S}}^1;\partial ),x\in {\mathbb{S}}^1,$ satisfying the Jacobi identity. It is important that the Lie algebra ${\mathcal{L}}_0({\mathbb{S}}^1;\partial )$ is metrized [2,3,11] with respect to the following nondegenerate bilinear form $(·|·):{\mathcal{L}}_0({\mathbb{S}}^1;\partial )\times {\mathcal{L}}_0({\mathbb{S}}^1;\partial )\to {\Lambda }_0:$ for any elements $a,b$ and $c\in$${\mathcal{L}}_0({\mathbb{S}}^1;\partial )$

$$\left(a\right|b):=\mathrm{Tr}(a\circ b),(a\left|\right[b,c\left]\right)=\left(\right[a,b\left]\right|c),$$

(28)

where the trace-operation $\mathrm{Tr}:{\mathcal{L}}_0({\mathbb{S}}^1;\partial )\to$${\Lambda }_0$ is defined for arbitrary element $c\in {\mathcal{L}}_0({\mathbb{S}}^1;\partial )$ via the expression

$$\mathrm{Tr}\left(c\right):={\int }_{{\mathbb{S}}^1}{\mathrm{res}}_{{\partial }^{-1}}c(x;\partial )dx.$$

(29)

Take now into account that for integers $k=0,1$ and 2 the Lie algebra ${\mathcal{L}}_0({\mathbb{S}}^1;\partial )$ allows the splitting into the direct sum ${\mathcal{L}}_0({\mathbb{S}}^1;\partial )={\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,+}\oplus {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}$ of subalgebras, consisting of pure differential

$${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,+}=\{a(x;\partial )=\sum _{j=k}^{j\ll \infty }{a}_j\left(x\right){\partial }^j:x\in {\mathbb{S}}^1,a_j\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_0),k\le j\ll \infty \}$$

(30)

and mixed pseudodifferential operators

$${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}=\{b(x;\partial )=\sum _{j\in \left\{\overline{-k+1,0}\right\}\cup \mathbb{N}}{b}_j\left(x\right){\partial }^{-j}:x\in {\mathbb{S}}^1,b_j\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_0),j\in \left\{\overline{-k+1,0}\right\}\cup \mathbb{N}\},$$

(31)

that is $[{\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,\pm },{\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,\pm }]\subset {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,\pm }.$ As the super-Lie algebra ${\mathcal{L}}_0({\mathbb{S}}^1;\partial )$ is metrized, its adjoint space ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}$ can be identified with the linear space ${\mathcal{L}}_0({\mathbb{S}}^1;\partial ),$ that is ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}\simeq {\mathcal{L}}_0({\mathbb{S}}^1;\partial ),$ in particular, one has identifications ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,\pm }^{*}\simeq {\partial }^{\mp 2k}{\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,\mp }$ for $k=\overline{0,2}.$

Recall now [2,3,11,12] that the adjoint space ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}$ is endowed with the classical super-Lie–Poisson structure

$$\left\{l\right(a),l(b\left)\right\}:=\left(l\right|[a,b]),$$

(32)

defined at any point $l\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}$ for arbitrary linear functionals $a,b\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{**}\simeq {\mathcal{L}}_0({\mathbb{S}}^1;\partial )$ on ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}$ and generated by the super-Lie commutator structure on ${\mathcal{L}}_0({\mathbb{S}}^1;\partial ).$ Take now into account that within the Lie-algebraic AKS scheme [1,13,29] the linear space ${\mathcal{L}}_0({\mathbb{S}}^1;\partial )$ can be rigged with the so-called $R_k$-structures for $k=\overline{0,2}:$

$${[a,b]}_{R_k}:=[R_{k}a,b]+[a,R_{k}b]$$

(33)

for any $a,b\in {\mathcal{L}}_0({\mathbb{S}}^1;\partial ),$ also satisfying the Jacobi identity. Here $R_k:{\mathcal{L}}_0({\mathbb{S}}^1;\partial )\to {\mathcal{L}}_0({\mathbb{S}}^1;\partial ),$$k=\overline{0,2},$ are linear homomorphisms, defined by the expressions $R_k:=\left(P_{k,+}-P_{k,-}\right)/2,$ where $P_{k,\pm }:{\mathcal{L}}_0({\mathbb{S}}^1;\partial )\to {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,\pm }\subset {\mathcal{L}}_0({\mathbb{S}}^1;\partial )$ are, respectively, projectors on the super-Lie subalgebras ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,\pm }\subset {\mathcal{L}}_0({\mathbb{S}}^1;\partial ).$ The super-Lie algebra $R_k$-structures (33), $k=\overline{0,2},$ generate the "linear" super-Lie–Poisson brackets on ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}$ via the expressions

$${\{l\left(a\right),l\left(b\right)\}}_{R_k}:=\left(l\right|{[a,b]}_{R_k})$$

(34)

for arbitrary linear functionals $a,b\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{**}\simeq {\mathcal{L}}_0({\mathbb{S}}^1;\partial )$ at a point $l\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}.$ Moreover, it is worth mentioning here that the following generalized homomorphisms $R_k\left(l\right):{\mathcal{L}}_0({\mathbb{S}}^1;\partial )\to {\mathcal{L}}_0({\mathbb{S}}^1;\partial ), k=\overline{0,2}$, at an orbit point $l\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*},$ where $R_k\left(l\right)a:=(P_{k,+}\left(la\right)-P_{k,-}\left(al\right))/2$ for any $a\in {\mathcal{L}}_0({\mathbb{S}}^1;\partial ), k=\overline{0,2},$ also define [32,33] local R-structures on the Lie algebra ${\mathcal{L}}_0({\mathbb{S}}^1;\partial ),$ generating the so-called “quadratic” super-Lie–Poisson brackets (34), what makes to state that all constructed this way nonlinear integrable Hamiltonian superflows are, in reality, bi-Hamiltonian and the corresponding super-Poisson structures on related functional supermanifolds are suitably [34] compatible.

As an interesting and instructive for further algebraic structure, let us consider an arbitrary element $l_0\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,+}^{*}\simeq {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,-}$ of the form

$$l_0=\sum _{j\in \mathbb{N}}{u}_j{\partial }^{-j}$$

(35)

and construct on the subspace ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,+}^{*}$ its coadjoint orbit

$$\partial l_0/\partial t=-\left({\mathrm{ad}}_{\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{+}}^{*}{l}_0\right){|}_{-}={[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{+},l_0]}_{-}$$

(36)

with respect to the evolution parameter $t\in \mathbb{R},$ generated by a smooth Casimir invariant ${\gamma }_0\in I\left({\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}\right),$ regarding the super-Lie–Poisson structure (32), that is $\{{\gamma }_0,l\}=0,$ or equivalently, $[\mathrm{grad}{\gamma }_0\left(l\right),l]=0,$ where the Gateau gradient vector $\mathrm{grad}{\gamma }_0\left(l\right)\in {\mathcal{L}}_0({\mathbb{S}}^1;\partial )$ at point $l\in \mathcal{L}{({\mathbb{S}}^1;\partial )}^{*}$ is defined via the infinitesimal condition $(h|\mathrm{grad}{\gamma }_0\left(l\right):={\displaystyle \frac{d}{d\epsilon }}{\gamma }_0(l+\epsilon h){|}_{\epsilon =0}$ for all $h\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}.$ Taking into account that the differentiation $\partial \in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,-}^{*}$ is a character [13] of the super-Lie algebra splitting ${\mathcal{L}}_0({\mathbb{S}}^1;\partial )={\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,+}\oplus {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,-},$ that is $(\partial |[{\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,\pm },{\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,\pm }])=0,$ the flow (36) can be rewritten the following way:

$$\begin{array}{c}\partial (l_0+\partial )/\partial t={[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,+},l_0]}_{-}=\\ ={[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,+}+\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},l_0]}_{-}-\\ -{[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},l_0]}_{-}=\\ ={[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,+}+\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},l_0+\partial ]}_{-}-\\ -{[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},l_0]}_{-}-{[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,+}+\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},\partial ]}_{-}=\\ -{[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},l_0]}_{-}-{[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,+},\partial ]}_{-}-{[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},\partial ]}_{-}=\\ =-{[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},l_0]}_{-}-{[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},\partial ]}_{-}=\\ =-[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},l_0]-[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},\partial ]=\\ =-[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},l_0+\partial ]=[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,+},l_0+\partial ],\end{array}$$

(37)

where we made used of the equality $[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},l_0+\partial ]=-[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,+},l_0+\partial ]$ as well as the conditions ${[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},l_0]}_{0,-}=[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,-},l_0],{[\mathrm{grad}{\gamma }_0{(l_0+\partial )}_{0,+},\partial ]}_{0,-}=0$ for arbitrary $l_0\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,+}^{*}\simeq {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,-}.$ As a result, the relationship (37) takes the standard [35] commutator Lax-type form

$$\partial l_{\partial }/\partial t=[\mathrm{grad}{\gamma }_0{\left(l_{\partial }\right)}_{0,+},l_{\partial }],$$

(38)

where we put, by definition, $l_{\partial }:=l_0+\partial \in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}.$ The representation (38) can be generalized, having assumed, by definition, that the element $l_{\partial }:=l_0+\partial :=l^{1/p\left(l\right)},$ where an element $l\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{-}^{*}$ is of the finite order $p\left(l\right)\in \mathbb{N}$:

$$l=\sum _{j=0}^{p\left(l\right)}{u}_j{\partial }^j.$$

(39)

As any analytical Casimir functional $\gamma \left(l\right)\in I\left({\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,-}^{*}\right)$ the commutator Lax-type representation (38) easily generalizes to the form

$$\partial l/\partial s=[\mathrm{grad}\gamma {\left(l\right)}_{0,+},l]$$

(40)

with respect to the evolution parameter $s\in \mathbb{R},$ coinciding with that of the flow (38), if the Casimir functional $\gamma \left(l_{\partial }^{p\left(l\right)}\right)={\gamma }_0\left(l_{\partial }\right)$$\in I\left({\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,+}^{*}\right).$ If to take now into account that the phase space ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}$ is additionally rigged with the second super-Poisson bracket (34), the generalized statement above can be reformulated as the following useful proposition.

Proposition 1.

Let a smooth functional $\gamma \in I\left(\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*}\right)$ be Casimir at an element $l\in \mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*}$ of the form (39), that is $\left[\mathrm{grad}\gamma \right(l),l]=0.$ Then it generates with respect to the second super-Poisson structure (34) an evolution flow on $\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*},$ exactly coinciding with that of (40).

Proof. 

Let us rewrite equivalently the second super-Poisson structure (34) at a point $l\in \mathcal{L}{({\mathbb{S}}^1;\partial )}_{-}^{*}$ in operator form $\vartheta :\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}\to \mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*},$ acting as

$$\vartheta \left(l\right):\mathrm{grad}\gamma \left(l\right)\to [\mathrm{grad}\gamma {\left(l\right)}_{0,+},l]-{[\mathrm{grad}\gamma \left(l\right),l]}_{0,+}$$

(41)

for any smooth functional $\gamma \in \mathcal{D}\left(\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*}\right).$ Then, since, by definition, the evolution flow, generated by a Casimir functional $\gamma \in I\left(\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*}\right),$ equals

$$\partial l/\partial s:=\vartheta \mathrm{grad}\gamma \left(l\right)=[\mathrm{grad}\gamma {\left(l\right)}_{0,+},l]-{[\mathrm{grad}\gamma \left(l\right),l]}_{0,+}=[\mathrm{grad}\gamma {\left(l\right)}_{0,+},l],$$

(42)

proving the proposition. □

Remark 1.

As mentioned before, the phase space $\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*}$ proves to be endowed also with an additional, so-called “quadratic”, super-Lie–Poisson operator $\eta :\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}\to \mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*}$ at a point $l\in \mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*},$ acting as

$$\eta \left(l\right):\mathrm{grad}\gamma \left(l\right)\to {(l\circ \mathrm{grad}\gamma \left(l\right))}_{0,+}\circ l-l\circ {(\mathrm{grad}\gamma \left(l\right)\circ l)}_{0,+}$$

(43)

for any smooth functional $\gamma \in \mathcal{D}\left(\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*}\right)$.

Consider now, as above, any two smooth functionally independent Casimir functionals $\gamma ,\mu \in \mathcal{D}\left({\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,-}^{*}\right)$ with respect to the first super-Poisson structure (32) on the phase space ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,-}^{*},$ satisfying, by definition, the conditions $\{\gamma ,l\}=0,\{\mu ,l\}=0$ for all $l\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,-}^{*},$ equivalent to the commutator expressions

$$\left[\mathrm{grad}\gamma \right(l),l]=0,\left[\mathrm{grad}\mu \right(l),l]=0.$$

(44)

These Casimir invariants generate with respect to the second super-Lie–Poisson structure (34) the following Hamiltonian flows on ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,-}^{*}$:

$$\partial l/\partial t=[\mathrm{grad}\gamma {\left(l\right)}_{0,+},l],\partial l/\partial s=[\mathrm{grad}\mu {\left(l\right)}_{0,+},l],$$

(45)

where $s,t\in \mathbb{R}$ are the corresponding evolution parameters. The next Adler–Kostant–Symes type theorem [1,13,29] is characteristic regarding the constructed above Hamiltonian flows (44). Specifically, the following proposition holds.

Proposition 2.

Let $\gamma ,$$\mu \in I\left(\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,+}^{*}\right)$ be arbitrary smooth functionally independent Casimir invariant functionals on $\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,+}^{*}$ and (45) be the corresponding evolution flows with respect to the parameters $t\in \mathbb{R}$ and $s\in \mathbb{R},$ respectively, generated by the second super-Lie–Poisson bracket (34). Then these flows are commuting to each other on the whole phase space $\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,+}^{*}$ for all parameters $s,t\in \mathbb{R}.$

Proof. 

Proof of the theorem is straightforward, following easily from the condition ${\{\gamma ,\mu \}}_R=0$ for the Casimir functionals $\gamma ,$$\mu \in I\left(\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*}\right),$ which reduces to those $\{\gamma ,l\}=0, \{\mu ,l\}=0$ at a point $l\in \mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*}$, being equivalent to (44). □

As a useful corollary from the proposition above, one can formulate the following theorem.

Theorem 1.

Let the Hamiltonian flow

$$\partial l/\partial t=[\mathrm{grad}\gamma {\left(l\right)}_{0,+},l]$$

(46)

on $\mathcal{L}{({\mathbb{S}}^1;\partial )}_{-}^{*}$ with respect to the evolution parameter $t\in \mathbb{R}$ possess a countable hierarchy of smooth functionally independent Casimir invariants ${\gamma }_j\in I\left(\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*}\right), j\in \mathbb{N}.$ Then the flow (44) presents on the related smooth functional manifold $M_l\simeq l\in \mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*}$ a completely integrable bi-Hamiltonian system.

Proof. 

The complete integrability follows from the commutativity conditions ${\{{\gamma }_j,\gamma \}}_R=0$ and ${\{{\gamma }_j,{\gamma }_k\}}_R=0$ for all $j,k\in \mathbb{N}.$ Moreover, as both super-Poisson operators (41) and (43) are compatible [2,11,12,27], that is the affine operator sum $\left(\eta +\lambda \theta \right):\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}\to \mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*}$ is a super-Poisson operator too for all $\lambda \in {\Lambda }_0,$ the latter easily makes it possible to represent the evolution system (50) as a bi-Hamiltonian flow with respect to both super-Poisson structures on $\mathcal{L}{({\mathbb{S}}^1;\partial )}_{0,-}^{*}$. □

The described above construction of integrable Hamiltonian flows naturally generalizes on the case of superflows on the adjoint spaces $\mathcal{L}{({\mathbb{S}}^1;\partial )}_{k,-}^{*}, k=\overline{0,2},$ if to consider the related super-Lie–Poisson brackets (34), which can be rewritten in the following operator form:

$$\vartheta \left(l\right):\mathrm{grad}\gamma \left(l\right)\to [R_k\mathrm{grad}\gamma \left(l\right),l]+R_k^{*}[\mathrm{grad}\gamma \left(l\right),l]$$

(47)

for any smooth functional $\gamma \in \mathcal{D}\left(\mathcal{L}{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\right),k=\overline{0,2}.$

The following proposition easily holds owing to the basic $R_k$-structure [2,3,11,12] properties.

Proposition 3.

Let $\gamma ,\mu \in I\left(\mathcal{L}{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\right), k=\overline{0,2},$ be Casimir, then they commute with respect to the super-Lie–Poisson brackets (47): ${\{\gamma ,\mu \}}_{\vartheta }=0$ on $\mathcal{L}{({\mathbb{S}}^1;\partial )}_{k,-}^{*}.$ Moreover, the related evolution superflows

$$\partial l/\partial t=[R_k\mathrm{grad}\gamma \left(l\right),l],\partial l/\partial s=[R_k\mathrm{grad}\mu \left(l\right),l]$$

(48)

commute on $\mathcal{L}{({\mathbb{S}}^1;\partial )}_{k,-}^{*}, k=\overline{0,2},$ too for all evolution parameters $s,t\in \mathbb{R}.$

As a simple consequence of the proposition above, one obtains the following theorem.

Theorem 2.

Let the Hamiltonian superflows

$$\partial l/\partial t=[\mathrm{grad}\gamma {\left(l\right)}_{k,+},l]$$

(49)

on $\mathcal{L}{({\mathbb{S}}^1;\partial )}_{k,-}^{*}, k=\overline{0,2},$ with respect to the evolution parameter $t\in \mathbb{R}$ possess a countable hierarchy of smooth functionally independent Casimir invariants ${\gamma }_j\in I\left(\mathcal{L}{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\right),j\in \mathbb{N}.$ Then the flows (49) present on the related smooth functional supermanifolds $M_l\simeq l\in \mathcal{L}{({\mathbb{S}}^1;\partial )}_{k,-}^{*}, k=\overline{0,2},$ completely integrable bi-Hamiltonian systems.

Remark 2.

It is worth observing here that the discussed above super-Lie algebra $({\mathcal{L}}_0({\mathbb{S}}^1;\partial ),[·,·])$ over ${\Lambda }_0$ can be parametrically generalized to a super-Lie algebra $\tilde{\mathcal{L}}({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )$ over ${\mathbb{Z}}_2-$ graded Grassmann algebra $\Lambda =$${\Lambda }_0\oplus {\Lambda }_1,$ as the supercircle ${\mathbb{S}}^{1|1}$-product ${\tilde{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial ):={\displaystyle \prod _{(y,\theta )\in {\mathbb{S}}^{1|1}}}{\mathcal{L}}_0({\mathbb{S}}^1;\partial ),$ and consisting of super-pseudodifferential operator expressions

$$\tilde{a}(x;y,\theta ;\partial )=\sum _{j\gg -\infty }{\tilde{a}}_j(x;y,\theta ){\partial }^{-j}$$

(50)

at $(x;y,\theta )\in {\mathbb{S}}^1\times {\mathbb{S}}^{1|1},$ where coefficients ${\tilde{a}}_j\in C^{\infty }({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\Lambda )$ for all $j\gg -\infty .$ The super-Lie algebra ${\tilde{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )$ is endowed with the usual super-Lie product, which is calculated by means of the point-wise operator compositions as

$$[\tilde{a},\tilde{b}]:=\tilde{a}\circ \tilde{b}-{(-1)}^{\pi \left(\tilde{a}\right)\pi \left(\tilde{b}\right)}\tilde{b}\circ \tilde{a},$$

(51)

satisfying the super-Jacobi identity

$$[\tilde{a},[\tilde{b},\tilde{c}]]=[[\tilde{a},\tilde{b}],\tilde{c}]+{(-1)}^{\pi \left(\tilde{a}\right)\pi \left(\tilde{b}\right)}[\tilde{b},[\tilde{a},\tilde{c}]]$$

(52)

for arbitrary uniform elements $\tilde{a},\tilde{b}$ and $\tilde{c}\in {\tilde{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial ).$ The super-Lie algebra ${\tilde{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )$ proves to be also metrized with respect to the generalized nondegenerate bilinear form

$$\left(\tilde{a}\right|\tilde{b}):=\int dy\int d\theta \mathrm{Tr}\left(\tilde{a}\tilde{b}\right),$$

(53)

that is $\left(\tilde{a}\right|[\tilde{b},\tilde{c}])=\left([\tilde{a},\tilde{b}]\right|\tilde{c}])$ for all $\tilde{a},\tilde{b}$ and $\tilde{c}\in {\tilde{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial ),$ where the following Berezin integration [36,37,38] expressions

$$\int d\theta =0,\int \theta d\theta =1$$

(54)

assumed to be satisfied. Taking now into account that the super-Lie algebra ${\tilde{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )$ is ${\mathbb{S}}^{1|1}$- parameterized, it can be rigged with a central extension ${\tilde{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )\to$${\widehat{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial ):={\tilde{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )\oplus \Lambda$ by means of the following super-Lie commutator expression

$$[\left(\tilde{a},\alpha \right),(\tilde{b},\beta )]:=([\tilde{a},\tilde{b}],\left(\tilde{a}\right|D_{\theta }\tilde{b})),$$

(55)

where $D_{\theta }:=\partial /\partial \theta +\theta \partial /\partial y,(y,\theta )\in {\mathbb{S}}^{1|1}$, is the superderivative on the supercircle ${\mathbb{S}}^{1|1},$ satisfying for arbitrary $\left(\tilde{a},\alpha \right)$ and $(\tilde{b},\beta )\in$${\widehat{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )$ the suitably generalized super-Jacobi identity (52). Moreover, we observe the adjoint space ${\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}^{*}\simeq$${\widehat{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )$ with respect to the nondegenerate bilinear form (53). Now, within the framework of the Lie-algebraic AKS scheme one can proceed to constructing integrable superflows on the adjoint space ${\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}^{*}$, taking into account the super-Lie–Poisson brackets

$${\{\tilde{l}\left(\tilde{a}\right),\tilde{l}\left(\tilde{b}\right)\}}_{R_k}=\left(\tilde{l}\right|[R_k\tilde{a},\tilde{b}]+[\tilde{a},R_k\tilde{b}])+\rho \left(R_k\tilde{a}\right|D_{\theta }\tilde{b})+\rho \left(\tilde{a}\right|R_k{D}_{\theta }\tilde{b})$$

(56)

at any point $(\tilde{l},\rho )\in {\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}^{*},$ where $R_k=\left(P_{k,+}-P_{k,-}\right)/2$ and $P_{k,\pm }{\widehat{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial ):={\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k,\pm }$$\subset {\widehat{\mathcal{L}}}_0({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial ),$$k=\overline{0,2},$ -the corresponding projections on the subspaces of super-differential operators

$${\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k,+}=\{(\tilde{a}=\sum _{j=k}^{j\ll \infty }{\tilde{a}}_j{\partial }^j,\alpha ):{\tilde{a}}_j\in C^{\infty }({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\Lambda ),k\le j\ll \infty \}$$

(57)

and on the subspace of super-pseudodifferential operators

$${\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k,-}=\{(\tilde{b}=\sum _{j\in \{-k+1,0\}\cup \mathbb{N}}{\tilde{b}}_j{\partial }^{-j},\beta ):b_j\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_0),j\in \{-k+1,0\}\cup \mathbb{N}\},$$

(58)

respectively. If a smooth functional $\gamma \in I\left({\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k,-}^{*}\right)$ is Casimir, it satisfies, by definition, the following super-pseudodifferential relationship:

$$[\mathrm{grad}\gamma \left(\tilde{l}\right),\tilde{l},]+\rho D_{\theta }\left(\mathrm{grad}\gamma \left(\tilde{l}\right)\right)=0$$

(59)

at a chosen point $(\tilde{l},\rho )\in {\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k.-}^{*},$$k=\overline{0,2},$ where $\rho \in \Lambda$ is a constant parameter, which can be taken, for brevity, as $\rho =-1.$ Respectively, the corresponding superflow on ${\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k,-}^{*}$ is easily calculated owing to the super-Lie–Poisson structures (56), as

$$\partial \tilde{l}/\partial t=[R_k\mathrm{grad}\gamma \left(\tilde{l}\right),\tilde{l}]-R_k{D}_{\theta }\mathrm{grad}\gamma \left(\tilde{l}\right)$$

(60)

for $k=\overline{0,2}$ with respect to the evolution parameter $t\in \mathbb{R}$ at a point $(\tilde{l},\rho )\in {\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k.-}^{*}$. In particular, the evolution flows (60) can be equivalently rewritten as the following commutator equalities

$$[\partial /\partial t-R_k\mathrm{grad}\gamma \left(\tilde{l}\right),D_{\theta }-\tilde{l}]=0$$

(61)

on the adjoint spaces ${\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k,-}^{*},$$k=\overline{0,2},$ coinciding with the classical Zakharov–Shabat type [30] representations of nonlinear integrable Hamiltonian systems on the coefficient functional supermanifold $M_{\tilde{l}},$ related to the operator elements $(\tilde{l},-1)\in {\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k,-}^{*},$$k=\overline{0,2}.$ Moreover, if $D_{\theta }{R}_k=0, k=\overline{0,2},$ the super-Lie–Poisson brackets (56) can be rewritten in the following operator form:

$$\vartheta \left(\tilde{l}\right):\mathrm{grad}\gamma \left(\tilde{l}\right)\to [R_k\mathrm{grad}\gamma \left(\tilde{l}\right),\tilde{l}]+R_k[\mathrm{grad}\gamma \left(\tilde{l}\right),\tilde{l}]-(R_k^{*}-R_k)D_{\theta }\left(\mathrm{grad}\gamma \left(\tilde{l}\right)\right),$$

(62)

where $\gamma \in \mathcal{D}\left({\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k,-}^{*}\right), k=\overline{0,2},$ is an arbitrary smooth functional on the adjoint space ${\widehat{\mathcal{L}}}_0{({\mathbb{S}}^1\times {\mathbb{S}}^{1|1};\partial )}_{k,-}^{*}$.

With the help of the outlined above preliminaries, we can proceed to describing new classes of pseudodifferential super-Lie algebras and studying related Lie-algebraic structures on them, allowing the construction of a wide class of interesting nonlinear integrable Hamiltonian superflows on functional supermanifolds.

3. Super-Poisson Structures, Their Deformations and Related Integrable Hamiltonian Systems

Let now an integer number $k\in \{0,1,2\}$ be fixed and assume that a smooth functional $\gamma \in I\left({\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\right)$ is Casimir regarding the super-Lie–Poisson structure (32), that is $\left[\mathrm{grad}\gamma \right(l),l]=0$ for $l\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}.$ The related Hamiltonian superflow

$$\partial l/\partial t=[\mathrm{grad}\gamma {\left(l\right)}_{k,+},l]$$

(63)

can be related to the following system of superflows:

$$\begin{array}{cc}\hfill \partial {\phi }_q/\partial t& =\mathrm{grad}\gamma {\left(l\right)}_{k,+}{\phi }_q,\hfill \\ \hfill \partial {\phi }_q^{*}/\partial t& =-\mathrm{grad}\gamma {\left(l\right)}_{k,+}^{*}{\phi }_q^{*},\hfill \end{array}$$

(64)

where elements ${\phi }_q$ and ${\phi }_q^{*}\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_q),$$q\in {\mathbb{Z}}_2$, satisfy the eigenvalue relationships:

$$l(x;\partial ){\phi }_q={\lambda }_q{\phi }_q,l{(x;\partial )}^{*}{\phi }_q^{*}={\lambda }_q^{*}{\phi }_q^{*}$$

(65)

for some constants ${\lambda }_q\in {\Lambda }_q, q\in {\mathbb{Z}}_2.$ It is easy to check that the system of superflows (64) is canonically Hamiltonian with respect to the super-Poisson bracket

$${\vartheta }_0({\phi }_q,{\phi }_q^{*})=\left(\begin{array}{cc}0& \underset{q\in {\mathbb{Z}}_2}{\otimes }{\left(-1\right)}^q\\ {\left(\underset{q\in {\mathbb{Z}}_2}{\otimes }\left(-1\right)\right)}^{⊺}& 0\end{array}\right)$$

(66)

on the functional supermanifold ${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2)$ and the following Hamiltonian function:

$$H=\sum _{q\in {\mathbb{Z}}_2}\left({\phi }_q^{*}\right|\mathrm{grad}\gamma {\left(l\right)}_{k,+}{\phi }_q),$$

(67)

where $(·|·)$ denotes the standard bilinear form on the product ${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2)\times {\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2)$ with values in the ${\mathbb{Z}}_2$-graded Grassmann algebra $\Lambda$:

$$\left(a\right|b):=\sum _{q\in {\mathbb{Z}}_2}{\int }_{{\mathbb{S}}^1}dx{a}_q\left(x\right)b_q\left(x\right)$$

(68)

for arbitrary $a,b\in {\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2).$ Simultaneously, the superflow (63) is Hamiltonian with respect to the super-Poisson operator mapping

$$\vartheta \left(l\right):\mathrm{grad}\gamma \left(l\right)\to [P_{k,+}\mathrm{grad}\gamma \left(l\right),l]-P_{k,-}^{*}[\mathrm{grad}\gamma \left(l\right),l]$$

(69)

and generated by a Hamiltonian functional $\gamma \in I\left({\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\right).$

Now, let us pose the following interesting question.

Question: Can one present a combined system of superflows (63) and (64) as a joint Hamiltonian superflow with respect to some super-Poisson bracket on the combined supermanifold ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\times$${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2)?$

Below, we will show that the solution to this question is positive and based on the functional structure of the Hamiltonian function (67). Specifically, we will demonstrate that the Hamiltonian function (67) as a functional on the supermanifold ${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2)$ lifts to a functional on the combined supermanifold ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\times$${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2)$ and possesses the structure of a trace-functional on the adjoint space ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}.$ Specifically, the following crucial theorem holds.

Theorem 3.

Let $\prod _{q\in {\mathbb{Z}}_2}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2)$ be a supplementing functional superspace. Then the coadjoint orbits on the spaces ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*},$$k=\overline{0,2},$ generated by Lie–Poisson structures (69), extend, respectively, by means of the deformation mappings

$${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\ni l\to l_{\phi }=l+\sum _{q\in {\mathbb{Z}}_2}{(-1)}^q{\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}{\phi }_q^{*}\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}$$

(70)

on the superspaces ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\times$${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2),$ endowed with the suitably extended super-Poisson structures

$${\vartheta }_k(l_{\phi };\phi ,{\phi }^{*})=\left(\begin{array}{ccc}[P_{k,+}(·),l_{\phi }]-P_{k,-}^{*}[(·),l_{\phi }]& \underset{q\in {\mathbb{Z}}_2}{\otimes }{(-1)}^{q+1}{\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}\left(·\right)& \underset{q\in {\mathbb{Z}}_2}{\otimes }{(-1)}^q{\partial }^{-k}{(·)}_{kx}{\partial }^{-1}{\phi }_q^{*}\\ {\left(\underset{q\in {\mathbb{Z}}_2}{\otimes }{(·)}_{k,+}{\phi }_q\right)}^{⊺}& 0& \underset{q\in {\mathbb{Z}}_2}{\otimes }{\left(-1\right)}^q\\ {\left(-\underset{q\in {\mathbb{Z}}_2}{\otimes }{(·)}_{k,+}^{*}{\phi }_q^{*}\right)}^{⊺}& {\left(\underset{q\in {\mathbb{Z}}_2}{\otimes }\left(-1\right)\right)}^{⊺}& 0\end{array}\right).$$

(71)

Proof. 

Really, let us analyze a variation of the Hamiltonian functional (67) at a fixed element $l\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}$:

$$\begin{array}{c}{\delta H|}_{{|}_{\delta l=0}}=\sum _{q\in {\mathbb{Z}}_2}\delta \left({\phi }_q^{*}\right|\mathrm{grad}\gamma {\left(l\right)}_{k,+}{\phi }_q)=\sum _{q\in {\mathbb{Z}}_2}\left(\delta {\phi }_q^{*}\right|\mathrm{grad}\gamma {\left(l\right)}_{k,+}{\phi }_q)+\\ +\sum _{q\in {\mathbb{Z}}_2}\left({\phi }_q^{*}\right|\mathrm{grad}\gamma {\left(l\right)}_{k,+}\delta {\phi }_q)=\sum _{q\in {\mathbb{Z}}_2}{(-1)}^q\mathrm{Tr}\left(\mathrm{grad}\gamma \left(l\right){\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}\delta {\phi }_q^{*}\right)+\\ +\sum _{q\in {\mathbb{Z}}_2}{(-1)}^q\mathrm{Tr}\left(\mathrm{grad}\gamma \left(l\right){\partial }^{-k}\delta {\phi }_{q,kx}{\partial }^{-1}{\phi }_q^{*}\right)=\\ =\sum _{q\in {\mathbb{Z}}_2}{(-1)}^q\left(\mathrm{grad}\gamma \left(l\right)\right|\left({\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}\delta {\phi }_q^{*}+{\partial }^{-k}\delta {\phi }_{q,kx}{\partial }^{-1}{\phi }_q^{*}\right))=\\ =\sum _{q\in {\mathbb{Z}}_2}\left({(-1)}^q\mathrm{grad}\gamma \left(l\right)\right|\delta \left({\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}{\phi }_q^{*}\right))=\left(\mathrm{grad}\gamma \left(l\right)\right|\delta \left(\sum _{q\in {\mathbb{Z}}_2}{(-1)}^q{\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}{\phi }_q^{*}\right))=\\ =\delta \gamma (l+\sum _{q\in {\mathbb{Z}}_2}{(-1)}^q{\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}{\phi }_q^{*}).\end{array}$$

(72)

The result (72) means that we can make the identification $\gamma \to \gamma \left(l_{\phi }\right)\in \mathcal{D}\left({\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\right),$ lifting the Hamiltonian functional (67) from the supermanifold ${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2)$ on the superspace ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*},$ parameterized by means of the deformed element $l_{\phi }=l+{\sum }_{q\in {\mathbb{Z}}_2}$ ${(-1)}^q\left({\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}{\phi }_q^{*}\right)$$\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}.$ Moreover, we can now easily rewrite the superflows (63) and (64) at the deformed element ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\ni l\to l_{\phi }\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}$ as a combined Hamiltonian systems

$${(\partial l/\partial t;\partial {\phi }_q/\partial t,\partial {\phi }_q^{*}/\partial t)}^{⊺}=\vartheta \left(l_{\phi }\right)\otimes \vartheta (\phi ,{\phi }^{*}){({\mathrm{grad}}_l\gamma \left(l_{\phi }\right);{\mathrm{grad}}_{{\phi }_q}\gamma \left(l_{\phi }\right),{\mathrm{grad}}_{{\phi }_q^{*}}\gamma \left(l_{\phi }\right))}^{⊺}$$

(73)

for $q\in {\mathbb{Z}}_2$ on the extended supermanifold ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\times$${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2)$ with respect to the super-Poisson operator $\vartheta \left(l_{\phi }\right)\otimes \vartheta ({\phi }_q,{\phi }_q^{*}),$ equal to the usual tensor product of the Poisson operator (69) at $l_{\phi }=l+{\sum }_{q\in {\mathbb{Z}}_2}{(-1)}^q$$\left({\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}{\phi }_q^{*}\right)$$\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}$ and the Poisson operator (66), generated by a Casimir functional $\gamma \in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}$. As with the results in [11,21,22], the superflows (73) allow a more unified interpretation as a type of the Backlund transformation from the extended supermanifold ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\times$${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2)$ to the extended supermanifold ${\mathcal{L}}_0^{*}({\mathbb{S}}^1;\partial )\times$${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2).$ Specifically, the following mappings

$$\begin{array}{c}{\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\times {\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2)\ni (l;\phi ,{\phi }^{*})\stackrel{B_k}{\to }(l_{\phi }=l+\sum _{q\in {\mathbb{Z}}_2}{(-1)}^q\left({\partial }^{-k}{\overline{\phi }}_{q,kx}{\partial }^{-1}{\overline{\phi }}_q^{*}\right);\\ {\overline{\phi }}_q={\phi }_q,{\overline{\phi }}_q^{*}={\phi }_q^{*},q\in {\mathbb{Z}}_2)\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}^{*}\times {\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2)\end{array}$$

(74)

for $k=\overline{0,2},$ transform [11] the tensor product $\vartheta \left(l\right)\otimes \vartheta (\phi ,{\phi }^{*})$ of the super-Poisson operators (69) on the supermanifolds ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}$ and (66) on the supermanifold ${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2),$ respectively, into the corresponding super-Poisson structures ${\vartheta }_k(l_{\phi };\overline{\phi },{\overline{\phi }}^{*})$ on the extended supermanifolds ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\times$${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2), k=\overline{0,2}$, via the Backlund type transformation

$${\vartheta }_k(l_{\phi };\overline{\phi },{\overline{\phi }}^{*})=B_k^{\prime }(l;\phi ,{\phi }^{*})\vartheta \left(l\right)\otimes \vartheta ,{\phi }^{*})B_k^{\prime ,*}(l;\phi ,{\phi }^{*}),$$

(75)

where $B_k^{\prime }$ denotes the Frechet derivative of the mapping (74) and $B_k^{\prime *}$ its adjoint with respect to the usual bilinear forms on the extended supermanifolds ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\times$${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2), k=\overline{0,2}.$ Taking into account the operator expressions

$$\begin{array}{ccc}& & B_k^{\prime }=\left(\begin{array}{ccc}1& \underset{q\in {\mathbb{Z}}_2}{\otimes }{(-1)}^{q+1}{\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}(·)& \underset{q\in {\mathbb{Z}}_2}{\otimes }{(-1)}^q{\partial }^{-k}{(·)}_{kx}{\partial }^{-1}{\phi }_q^{*}\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\hfill \\ & & B_k^{\prime ,*}=\left(\begin{array}{ccc}1& 0& 0\\ {\left(\underset{q\in {\mathbb{Z}}_2}{\otimes }{(·)}_{k,+}^{*}{\phi }_q^{*}\right)}^{⊺}& 1& 0\\ {\left(\underset{q\in {\mathbb{Z}}_2}{\otimes }{(·)}_{k,+}{\phi }_q\right)}^{⊺}& 0& 1\end{array}\right),\hfill \end{array}$$

(76)

we easily obtain that

$${\vartheta }_k(l_{\phi };\phi ,{\phi }^{*})=\left(\begin{array}{ccc}[P_{k,+}(·),l_{\phi }]-P_{k,-}^{*}[(·),l_{\phi }]& \underset{q\in {\mathbb{Z}}_2}{\otimes }{(-1)}^{q+1}{\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}\left(·\right)& \underset{q\in {\mathbb{Z}}_2}{\otimes }{(-1)}^q{\partial }^{-k}{(·)}_{kx}{\partial }^{-1}{\phi }_q^{*}\\ {\left(\underset{q\in {\mathbb{Z}}_2}{\otimes }{(·)}_{k,+}{\phi }_q\right)}^{⊺}& 0& \underset{q\in {\mathbb{Z}}_2}{\otimes }{\left(-1\right)}^q\\ {\left(-\underset{q\in {\mathbb{Z}}_2}{\otimes }{(·)}_{k,+}^{*}{\phi }_q^{*}\right)}^{⊺}& {\left(\underset{q\in {\mathbb{Z}}_2}{\otimes }\left(-1\right)\right)}^{⊺}& 0\end{array}\right).$$

(77)

To show this, it is enough to check that the following operator expression

$$\begin{array}{c}-\sum _{q\in {\mathbb{Z}}_2}\left(a\right|[P_{k,+}({\mathrm{grad}}_l{\gamma }_{k,-}+{\left({\nabla }_{\phi }\gamma \right)}_{k,+}),{(-1)}^q{\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}{\phi }_q^{*}])+\\ +\sum _{q\in {\mathbb{Z}}_2}\left(a\right|P_{k,-}^{*}[({\mathrm{grad}}_l{\gamma }_{k,-}+{\left({\nabla }_{\phi }\gamma \right)}_{k,+}),{(-1)}^q{\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}{\phi }_q^{*}])+\\ +\sum _{q\in {\mathbb{Z}}_2}{(-1)}^{q+1}\left(a\right|{\partial }^{-k}{\phi }_{q,kx}{\partial }^{-1}\left({\left({\nabla }_{\phi }\gamma \right)}_{k,+}^{*}{\phi }_q^{*}\right))+\\ +\sum _{q\in {\mathbb{Z}}_2}{(-1)}^q\left(a\right|{\partial }^{-k}{\left({\left({\nabla }_{\phi }\gamma \right)}_{k,+}{\phi }_q\right)}_{kx}{\partial }^{-1}{\phi }_q^{*})\Rightarrow 0\end{array}$$

(78)

should vanish for all operators $a\in {\mathcal{L}}_0({\mathbb{S}}^1;\partial ),$ where we made used [21] of the componentwise splitting of the gradient ${\mathrm{grad}}_{l_{\phi }}\gamma ={\left({\mathrm{grad}}_l\gamma \right)}_{k,-}$$+{\left({\nabla }_{\phi }\gamma \right)}_{k,+}\in {\mathcal{L}}_0({\mathbb{S}}^1;\partial ).$ Taking for brevity the case $k=0$ and $q=0,$ one easily ensues that

$$\begin{array}{c}-(a_{0,+}+a_{0,-}|[{\left({\nabla }_{\phi }\gamma \right)}_{0,+},{\phi }_0{\partial }^{-1}{\phi }_0^{*}])+\left(a_{0,-}\right|[({\mathrm{grad}}_l{\gamma }_{0,-}+{\left({\nabla }_{\phi }\gamma \right)}_{0,+}),{\phi }_0{\partial }^{-1}{\phi }_0^{*}])+\\ -\left(a\right|{\phi }_0{\partial }^{-1}\left({\left({\nabla }_{\phi }\gamma \right)}_{0,+}^{*}{\phi }_0^{*}\right))+\left(a\right|\left({\left({\nabla }_{\phi }\gamma \right)}_{0,+}{\phi }_0\right){\partial }^{-1}{\phi }_0^{*})=\\ =-\left([a_{0,+},{\left({\nabla }_{\phi }\gamma \right)}_{0,+}]\right|{\phi }_0{\partial }^{-1}{\phi }_0^{*})-\left(a_{0,-}\right|[{\left({\nabla }_{\phi }\gamma \right)}_{0,+},{\phi }_0{\partial }^{-1}{\phi }_0^{*}])+\left(a_{0,-}\right|[{\left({\nabla }_{\phi }\gamma \right)}_{0,+},{\phi }_0{\partial }^{-1}{\phi }_0^{*}])-\\ -\underset{\Rightarrow 0}{\underbrace{\left(a_{0,-}\right|[{\mathrm{grad}}_l{\gamma }_{0,-},{\phi }_0{\partial }^{-1}{\phi }_0^{*}])}}-\left(\left(A_{0,+}^{*}{\phi }_0^{*}\right)\right|a_{0,+}{\phi }_0)+\left({\phi }_0^{*}\right|a_{0,+}{\left({\nabla }_{\phi }\gamma \right)}_{0,+}{\phi }_0)=\\ =-\left({\phi }_0^{*}\right|[a_{0,+},{\left({\nabla }_{\phi }\gamma \right)}_{0,+}]{\phi }_0)+\left({\phi }_0^{*}\right|[a_{0,+},{\left({\nabla }_{\phi }\gamma \right)}_{0,+}]{\phi }_0)=0,\end{array}$$

(79)

confirming the statement for arbitrary element $a\in {\mathcal{L}}_0({\mathbb{S}}^1;\partial ),$ thus proving the theorem. □

The constructed above super-Poisson operator (77) generates Hamiltonian systems on the extended supermanifold ${\mathcal{L}}_0({\mathbb{S}}^1;\partial )\times$${\displaystyle {\displaystyle \prod _{q\in {\mathbb{Z}}_2}}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2),$ reducing in the case of the Casimir Hamiltonians $\gamma \to {\gamma }_m={\displaystyle \frac{p\left(l\right)}{m+p\left(l\right)}}\mathrm{T}r{\left(l_{\phi }\right)}^{{\displaystyle \frac{m+p\left(l\right)}{p\left(l\right)}}},m\in {\mathbb{Z}}_{+},$ to the completely integrable evolution superflows

$$\begin{array}{cc}\hfill \partial l_{\phi }/\partial t_{m_q}& =[{\left(l_{\phi }^{m/p\left(l\right)}\right)}_{k,+},l_{\phi }],\partial {\phi }_q/\partial t_m=P_{k,+}\left(l_{\phi }^{m/p\left(l\right)}\right){\phi }_q,\hfill \\ \hfill \partial {\phi }_q^{*}/\partial t_m& =-{\left(l_{\phi }^{m/p\left(l\right)}\right)}_{k,+}^{*}{\phi }_q^{*},q\in {\mathbb{Z}}_2.\hfill \end{array}$$

(80)

Example 1.

Put now $$ and consider the “superized” Sturm–Liouville type expression

$$l_{\phi }(x;\partial ):={\partial }^2-u-{\phi }_1{\partial }^{-1}{\phi }_1^{*},$$

(81)

on the circle ${\mathbb{S}}^1,$ where $u\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_0)$ is an even and ${\phi }_1\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_1)$ is an odd elements of the superalgebra $\Lambda .$ For $m=2$ one easily derives from (80) that the set of odd eigenfunctions $({\phi }_1,{\phi }_1^{*})\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_1^2)$ jointly with a coefficient $u\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_0)$ satisfy the system of evolution equations

$$\begin{array}{cc}\hfill u_{t_2}& =2({\phi }_1^{*}{\phi }_{1,x}-{\phi }_1{\phi }_{1,x}^{*}),\hfill \\ \hfill {\phi }_{1,t_2}& ={\phi }_{1,xx}-u{\phi }_1,\hfill \\ \hfill {\phi }_{1,t_2}^{*}& =-{\phi }_{1,xx}^{*}+u{\phi }_1^{*}\hfill \end{array}$$

(82)

with respect to the evolution parameter $t_2\in \mathbb{R}.$ This system presents a completely Hamiltonian superflow on the functional supermanifold $M_l\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_0\times {\Lambda }_1^2)$ with respect to both the corresponding “linear” super-Poisson operator

$${\vartheta }_0(u;{\phi }_1,{\phi }_1^{*})=\left(\begin{array}{ccc}2\partial & 0& 0\\ 0& 0& -1\\ 0& -1& 0\end{array}\right),$$

(83)

easily resulting from the Hamiltonian superflow

$$\begin{array}{cc}\hfill \partial l/\partial t& =-{[{\mathrm{grad}}_l\gamma ,l]}_{0,+},\hfill \\ \hfill \partial {\phi }_1/\partial t& ={\left({\mathrm{grad}}_{l_{\phi }}\gamma \right)}_{0,+}{\phi }_1-{\mathrm{grad}}_{{\phi }_1^{*}}\gamma ,\hfill \\ \hfill \partial {\phi }_q^{*}/\partial t& =-{\left({\mathrm{grad}}_{l_{\phi }}\gamma \right)}_{0,+}^{*}{\phi }_1-{\mathrm{grad}}_{{\phi }_1}\gamma \hfill \end{array}$$

(84)

on the extended supermanifold ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,-}^{*}\times$${\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2),$ if to take into account that the following Hamiltonian system

$$\begin{array}{cc}\hfill \partial l_{\phi }/\partial t& =[{\left({\nabla }_{\phi }\gamma \right)}_{0,+},l_{\phi }]-{[{\left({\mathrm{grad}}_l\gamma \right)}_{0,-}+{\left({\nabla }_{\phi }\gamma \right)}_{0,+},l_{\phi }]}_{0,+}+{\phi }_1{\partial }^{-1}{\mathrm{grad}}_{{\phi }_1}\gamma )-{\mathrm{grad}}_{{\phi }_1^{*}}\gamma {\partial }^{-1}{\phi }_1^{*},\hfill \\ \hfill \partial {\phi }_1/\partial t& ={\left({\nabla }_{\phi }\gamma \right)}_{0,+}{\phi }_1-{\mathrm{grad}}_{{\phi }_1^{*}}\gamma ,\partial {\phi }_q^{*}/\partial t=-{\left({\nabla }_{\phi }\gamma \right)}_{0,+}^{*}{\phi }_1-{\mathrm{grad}}_{{\phi }_1}\gamma \hfill \end{array}$$

(85)

for $k=0, q=1$ holds for any Hamiltonian function $\gamma \in \mathcal{D}({\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,-}^{*}\times {\displaystyle \prod _{q\in {\mathbb{Z}}_2}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2)),$ as well as its gradient expression ${\mathrm{grad}}_{l_{\phi }}\gamma =$${\sum }_{1\le j\ll \infty }{\partial }^{-j+1}{\mathrm{grad}}_{u_j}\gamma$$+{\left({\nabla }_{\phi }\gamma \right)}_{0,+}\in {\mathcal{L}}_0({\mathbb{S}}^1;\partial )$ for arbitrary element $l={\sum }_{0\le j\ll \infty }{u}_j{\partial }^j\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{0,-}^{*}$.

It is now worth recalling that the super-Poisson operators (77) correspond within the Lie-algebraic AKS scheme to the basic R-structures on the super-Lie algebra ${\mathcal{L}}_0({\mathbb{S}}^1;\partial )$ by means of the mappings $R_k=\left(P_{k,+}-P_{k,-}\right)/2,k=\overline{0,2}.$ Since the following local expressions $R_k\left(l\right)=\left(P_{k,+}l(·)-P_{k,-}(·)l\right)/2,l\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}, k=\overline{0,2},$ also determine [32] the R-structures on the super-Lie algebra ${\mathcal{L}}_0({\mathbb{S}}^1;\partial ),$ generating the corresponding so-called “quadratic” super-Poisson operators on the extended supermanifold ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{k,-}^{*}\times$${\displaystyle {\displaystyle \prod _{q\in {\mathbb{Z}}_2}}}{C}^{\infty }({\mathbb{S}}^1;{\Lambda }_q^2).$ In particular, for $k=0$ one derives the following super-Lie–Poisson operator

$${\eta }_0(l_{\phi };\phi ,{\phi }^{*})=\left(\begin{array}{ccc}{(l_{\phi }·)}_{0,+}{l}_{\phi }-l_{\phi }{(·l_{\phi })}_{0,+}& \underset{q\in {\mathbb{Z}}_2}{\otimes }{(-1)}^{q+1}{\phi }_q{\partial }^{-1}(·)l_{\phi }& {(-1)}^q{l}_{\phi }(·){\partial }^{-1}{\phi }_q^{*}\\ {(l_{\phi }·)}_{0,+}{\phi }_q& \underset{q\in {\mathbb{Z}}_2}{\otimes }{(-1)}^{q+1}{\phi }_q{\partial }^{-1}({\phi }_q·)& {(-1)}^q{\phi }_q{\partial }^{-1}\left({\phi }_q^{*}(·)\right)+l_{\phi }(·)\\ -{\left((·)l_{\phi }\right)}_{0,+}^{*}{\phi }_q& \underset{q\in {\mathbb{Z}}_2}{\otimes }{(-1)}^{q+1}{\phi }_q^{*}{\partial }^{-1}({\phi }_q·)-l_{\phi }^{*}(·)& {(-1)}^q{\phi }_q^{*}{\partial }^{-1}\left({\phi }_q^{*}(·)\right)\end{array}\right),$$

(86)

compatible with that (77). The super-Lie–Poisson operator (86), if reduced on the orbits of the “superized” Sturm–Liouville type element (81), gives rise to the following super-Lie–Poisson expression

$${\eta }_0(u;{\phi }_1,{\phi }_1^{*})=\left(\begin{array}{ccc}1/2{\partial }^3-u\partial -\partial u& {\phi }_1\partial +1/2\partial {\phi }_1& -{\phi }_1^{*}\partial -1/2\partial {\phi }_1^{*}\\ \partial {\phi }_1+1/2{\phi }_1\partial & -3/2{\phi }_1{\partial }^{-1}{\phi }_1& {\partial }^2-u-3/2{\phi }_1{\partial }^{-1}{\phi }_1^{*}\\ \partial {\phi }_1^{*}+1/2{\phi }_1^{*}\partial & -{\partial }^2+u-3/2{\phi }_1^{*}{\partial }^{-1}{\phi }_1& -3/2{\phi }_1^{*}{\partial }^{-1}{\phi }_1^{*}\end{array}\right)$$

(87)

on the supermanifold $M_{l_{\phi }}\simeq C^{\infty }(S^1;{\Lambda }_0\times {\Lambda }_1^2),$ generating for $m=2$ the slightly generalized integrable Korteweg–de Vries type superflow

$$\begin{array}{cc}\hfill u_t& =u_{3x}+6u{u}_x-6({\phi }_{1,xx}{\phi }_1^{*}-{\phi }_1{\phi }_{1,xx}^{*}),\hfill \\ \hfill {\phi }_{1,t}& =4{\phi }_{1,3x}-6u{\phi }_{1,x}-3u_x{\phi }_1,\hfill \\ \hfill {\phi }_{1,t}^{*}& =4{\phi }_{1,3x}^{*}-6u{\phi }_{1,x}^{*}-3u_x{\phi }_1^{*},\hfill \end{array}$$

(88)

generated by the Hamiltonian functional

$${\gamma }_3=2/7\mathrm{T}r{\left(l_{\phi }\right)}^{{\displaystyle \frac{7}{2}}}=\int dx(u_x^2+2u^3-16{\phi }_{1,x}{\phi }_{1,xx}-24u{\phi }_1{\phi }_{1,x}),$$

(89)

regarding the super-Poisson structure (83), and by the Hamiltonian functional

$${\gamma }_2=2/5\mathrm{T}r{\left(l_{\phi }\right)}^{{\displaystyle \frac{5}{2}}}=\int dx(u^2-4{\phi }_1{\phi }_{1,x}),$$

(90)

regarding the super-Poisson structure (87). Moreover, having applied to the super-Poisson operators (83) and (87) the eigenfunction symmetry condition ${\phi }_1={\phi }_1^{*}\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_1)$, we obtain within the well-known Dirac reduction scheme [2,3,11,26,27] both the reduced linear super-Poisson structure

$${\vartheta }_0(u;{\phi }_1)=\left(\begin{array}{cc}-\partial & 0\\ 0& -1/4\end{array}\right)$$

(91)

and, respectively, the “quadratic” super-Poisson structure

$${\eta }_0(u;{\phi }_1)=\left(\begin{array}{cc}1/2{\partial }^3-u\partial -\partial u& 1/2(3{\phi }_1\partial +{\phi }_{1,x})\\ 1/2(3\partial {\phi }_1-{\phi }_{1,x})& 1/2(u-{\partial }^2)\end{array}\right)$$

(92)

on the supermanifold $M_{l_{\phi }},$ generating on the functional supermanifold $M_{l_{\phi }}$ the well-known integrable super-Korteweg–de Vries dynamical system

$$\begin{array}{cc}\hfill u_t& =u_{3x}+6u{u}_x-12{\phi }_{1,xx}{\phi }_1-{\phi }_1{\phi }_{1,xx}^{*}),\hfill \\ \hfill {\phi }_{1,t}& =4{\phi }_{1,3x}-6u{\phi }_{1,x}-3u_x{\phi }_1.\hfill \end{array}$$

(93)

Observe now that owing to the representations (80), the deformed linear problem $l_{\phi }f(x;{\lambda }_0)=$${\lambda }_{0}f(x;{\lambda }_0)$ at $(x;{\lambda }_0)\in {\mathbb{S}}^1\times {\Lambda }_0$, for even function $f\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_0)$ is isospectral, that is $\partial {\lambda }_0/\partial t=0$ for all $t\in \mathbb{R}.$ The latter makes it possible to look at this linear differential-integral relationship as the linear Lax-type spectral problem

$$f_{xx}-uf-{\phi }_1{\partial }^{-1}{\phi }_{1}f={\lambda }_{0}f.$$

(94)

for the Hamiltonian system (92). This spectral (94) can be easily rewritten modulo the new odd functional component $g=-{\partial }^{-1}\left({\phi }_{1}f\right)\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_1)$ to the equivalent “conformal” spectral problem

$$\left(-D_{\theta }^3+\Phi \right)\psi =-{\lambda }_0\theta \psi ,$$

(95)

where, by definition, $\Phi :={\phi }_1+\theta u\in C^{\infty }({\mathbb{S}}^{1|1};{\Lambda }_1)$ and $\psi :=f+\theta g\in C^{\infty }({\mathbb{S}}^{1|1};{\Lambda }_0)$ is a generator [10,16] of the conformal symmetry vector field $\psi \partial /\partial x+1/2\left(D_{\theta }\psi \right)D_{\theta }\in \mathcal{K}\left({\mathbb{S}}^{1|1}\right)$ with respect to the superderivative $D_{\theta }=\partial /\partial \theta +\theta \partial /\partial x$ on the supercircle with coordinates $(x;\theta )\in {\mathbb{S}}^{1|1}.$

Remark 3.

It is worth remarking here that the slightly generalized spectral problem (81) was recently analyzed in the work [39], devoted to studying a superized completely integrable quasi-classical nonlinear Schrödinger–Davydov dynamical systems, and where, in particular, there was suggested a new suitably deformed spectral problem (95), considered on the supercircle ${\mathbb{S}}^{1|1}.$ Since this picture generalizes on the supercircles ${\mathbb{S}}^{1|N}$ for $N=2$ and $3,$ there arises an interesting question about the construction of such integrable, respectively, superized Schrödinger–Davydov and Korteweg–de Vries dynamical systems.

Example 2.

As a next interesting example, we consider integrable superflows, generated by an element $l\in {\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{1,-}^{*},$ deformed by means of an odd functional vector $({\phi }_1,{\phi }_1^{*})\in C^{\infty }({\mathbb{S}}^{1|1};{\Lambda }_1^2)$:

$$l\to l_{\phi }={\partial }^2+u\partial +v+{\partial }^{-1}w+{\partial }^{-1}{\phi }_{1,x}{\partial }^{-1}{\phi }_1^{*},$$

(96)

where even coefficients $u,v,w,$ and $v\in C^{\infty }({\mathbb{S}}^1;{\Lambda }_0).$ The coadjoint orbit, generated by the element (96) on the space ${\mathcal{L}}_0{({\mathbb{S}}^1;\partial )}_{1,-}^{*},$ is endowed with the super-Poisson operator (83) at $k=1,q=1,$ whose reduction on the functional supermanifold $M_{l_{\phi }}\simeq M_l\times C^{\infty }({\mathbb{S}}^{1|1};{\Lambda }_1^2)$ equals

$$\begin{array}{c}\partial l/\partial t=-{[{\mathrm{grad}}_{l}H,l]}_{1,+},\\ \partial {\phi }_1/\partial t=-{\mathrm{grad}}_{{\phi }_1^{*}}H,\partial {\phi }_1^{*}/\partial t=-{\mathrm{grad}}_{{\phi }_1}H,\end{array}$$

(97)

giving rise to

$${\mathrm{grad}}_{l}H={\partial }^{-2}{\mathrm{grad}}_{u}H+{\partial }^{-1}{\mathrm{grad}}_{v}H+{\mathrm{grad}}_{w}H$$

(98)

to the super-Poisson operator mapping

$${\vartheta }_1(u,v,w;{\phi }_1,{\phi }_1^{*})=\left(\begin{array}{ccccc}0& 0& 2\partial & 0& 0\\ 0& 2\partial & {\partial }^2+u\partial & 0& 0\\ 2\partial & -{\partial }^2+\partial u& 0& 0& 0\\ 0& 0& 0& 0& -1\\ 0& 0& 0& -1& 0\end{array}\right)$$

(99)

on the cotangent space $T^{*}\left(M_{\phi }\right),$ and generating the following Hamiltonian systems:

$$\partial {(u,v,w;{\phi }_1,{\phi }_1^{*})}^{⊺}/\partial t={\vartheta }_1(u,v,w;{\phi }_1,{\phi }_1^{*})\mathrm{grad}H$$

(100)

for arbitrary $H\in D\left(M_{\phi }\right).$ The corresponding Casimir-type Hamiltonian functions read as

$$\begin{array}{c}{\gamma }_1=\int dx(-u^2+4v),{\gamma }_2=\int dxw,{\gamma }_3=\int dx(12u_x^2+3u^4+48v^2-48v{u}_x-\\ -24v{u}^2+96wu-192{\phi }_1^{*}{\phi }_{1,x}),{\gamma }_4=\int dx\left(2wv+2{\phi }_{1,x}^{*}{\phi }_{1,x}-2{\phi }_1^{*}{\phi }_{1,x}u\right),\\ {\gamma }_5=\int dx(80u_{xxx}{u}_x+8u_{xxx}{u}^2+48u_{xx}{u}_{x}u-16u_{xx}{u}^3+80u_x^3+4u_x^2{u}^2-\\ -5u^6-112v_{xx}{u}_x+32v_{xx}{u}^2+96v_x^2-64v_x{u}_{x}u+32v_x{u}^3+32v^3-48v^2{u}^2-\\ -48v{u}_{xx}u-120v{u}_x^2+72v{u}_x{u}^2+30v{u}^4+928w_x{u}_x-128w_x{u}^2-\\ -256w_{x}v+256w^2-128w{u}_{x}u-48w{u}^3+256wvu+\\ +640{\phi }_{1,xx}^{*}{\phi }_{1,x}+768{\phi }_{1,x}^{*}{\phi }_{1,xx}+256{\phi }_{1,x}^{*}{\phi }_{1,x}u+128{\phi }_1^{*}{\phi }_{1,xxx}-128{\phi }_1^{*}{\phi }_{1,xx}u+\\ +320{\phi }_1^{*}{\phi }_{1,x}{u}_x-160{\phi }_1^{*}{\phi }_{1,x}{u}^2-512{\phi }_1^{*}{\phi }_{1,x}v),\dots \end{array}$$

etc., generating, in particular, for $H={\gamma }_4/2$ the following integrable super-Hamiltonian system:

$$\begin{array}{c}\partial u/\partial t=2v_x,\partial v/\partial t=2w_x+v_x+u{v}_x,\\ \partial w/\partial t=-v_{xxx}+{\left(uv\right)}_x-2{\left({\phi }_1^{*}{\phi }_{1,x}\right)}_x,\\ {\phi }_{1,t}={\phi }_{1,xx}+u{\phi }_{1,x},{\phi }_{1,t}^{*}=-{\phi }_{1,xx}^{*}-u{\phi }_{1,x}^{*}\end{array}.$$

(101)

If to apply the Dirac type reduction of the super-Poisson operator (99) and the Hamiltonian flow (101) on the submanifold ${\overline{M}}_{\phi }=\{(u,v,w;{\phi }_1,{\phi }_1^{*})\in M_{\phi }:{\phi }_1={\phi }_1^{*}\},$ then an interesting question arises whether the resulting Hamiltonian system will admit the super-spectral representation like (95)?

Example 3.

In a similar way, having chosen the spectral deformation element as

$$l\to l_{\phi }=\partial +u+{\partial }^{-1}v+{\partial }^{-1}{\phi }_{1,x}{\partial }^{-1}{\phi }_1^{*},$$

(102)

We can suitably construct the corresponding hierarchy of Casimir invariants:

$$\begin{array}{cc}\hfill {\gamma }_{-1}& =-\int dxu,{\gamma }_1=\int dxv,{\gamma }_2=\int dx\left(2vu-2{\phi }_1^{*}{\phi }_{1,x}\right),\hfill \\ \hfill {\gamma }_3& =\int dx\left(3v{u}_x+3v^2+3v{u}^2+3{\phi }_{1,x}^{*}{\phi }_{1,x}-6{\phi }_1^{*}{\phi }_{1,x}u\right),\hfill \\ \hfill {\gamma }_4& =\int dx\left(5v{u}_{xx}-6v_x{u}^2+12v^{2}u+4v{u}^3+12{\phi }_{1,x}^{*}{\phi }_{1,x}u-12{\phi }_1^{*}{\phi }_{1,x}{u}^2-12{\phi }_1^{*}{\phi }_{1,x}v\right),\hfill \\ \hfill {\gamma }_5& =\int dx(13v_{xx}{u}_x-20v_{xx}{u}^2+20v_x^2+10v_x{u}_{x}u-10v_x{u}^3-30v_{x}vu+\hfill \\ \hfill & +10v^3+30v^2{u}^2-7v{u}_{xx}u+5v{u}_x^2+5v{u}^4+20{\phi }_{1,xx}^{*}{\phi }_{1,xx}+40{\phi }_{1,xx}^{*}{\phi }_{1,x}u+\hfill \\ \hfill & +25{\phi }_{1,x}^{*}{\phi }_{1,xxx}+50{\phi }_{1,x}^{*}{\phi }_{1,xx}u-10{\phi }_{1,x}^{*}{\phi }_{1,x}{u}_x+30{\phi }_{1,x}^{*}{\phi }_{1,x}{u}^2+18{\phi }_{1,x}^{*}{\phi }_{1,x}v+\hfill \\ \hfill & +5{\phi }_1^{*}{\phi }_{1,xxxx}+10{\phi }_1^{*}{\phi }_{1,xxx}u-10{\phi }_1^{*}{\phi }_{1,xx}{u}_x-2{\phi }_1^{*}{\phi }_{1,xx}v+2{\phi }_1^{*}{\phi }_{1,x}{u}_{xx}-\hfill \\ \hfill & -20{\phi }_1^{*}{\phi }_{1,x}{u}^3+7{\phi }_1^{*}{\phi }_{1,x}{v}_x-60{\phi }_1^{*}{\phi }_{1,x}vu),\dots \hfill \end{array}$$

(103)

and the super-Poisson operator mapping ${\vartheta }_1(v,w;{\phi }_1,{\phi }_1^{*}):T^{*}\left(M_{\phi }\right)\to T\left(M_{\phi }\right)$ as

$${\vartheta }_1(u,v;{\phi }_1,{\phi }_1^{*})=\left(\begin{array}{cccc}0& \partial & 0& 0\\ \partial & 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& -1& 0\end{array}\right),$$

(104)

generating, in particular, for $H=1/3{\gamma }_3$ the following integrable super-Hamiltonian system reads as

$$\begin{array}{cc}\hfill \partial u/\partial t& =u_{xx}+2v_x+6u{u}_x,\hfill \\ \hfill \partial v/\partial t& =-v_{xx}+2{\left(uv\right)}_x-2{\phi }_1^{*}{\phi }_{1,x},\hfill \\ \hfill \partial {\phi }_1/\partial t& ={\phi }_{1,xx}+2u{\phi }_{1,x},\partial {\phi }_1^{*}/\partial t={\phi }_{1,xx}^{*}-2{\left(u{\phi }_1\right)}_x,\hfill \end{array}$$

(105)

generalizing and correcting, in part, the two-boson flow, constructed in [40]. The latter is a well-known integrable system, which has many different names, and the most popular one is the classical Boussinesq equation. Similarly one can construct a countable hierarchy of integrable Hamiltonian systems generated by deformed super-pseudodifferential element $l_{\phi }=l+{\sum }_{q\in {\mathbb{Z}}_2}{\partial }^{-2}{\phi }_{q,xx}{\partial }^{-1}{\phi }_q^{*}$ with $l\in {\mathcal{L}}_{2,-}^{*}({\mathbb{S}}^1;\partial )$ and the related Casimir functionals with respect to the super-Poisson structure (77) at $k=2$.

It is worth recalling here, as already mentioned before, that all the obtained above results can be generalized on the case of the superflows, generated by coadjoint orbits of the Lie superalgebra $\mathcal{L}({\mathbb{S}}^{1|N};D_{\theta }), \theta \in {\Lambda }_1^{\left(N\right)},$ over the super N-dimensional ${\mathbb{Z}}_2$-graded Grassmann algebra ${\Lambda }^{\left(N\right)}={\Lambda }_0\oplus {\Lambda }_1^{\left(N\right)}$ of the super-pseudodifferential operators [17,18,41,42,43] on the supercircle ${\mathbb{S}}^{1|N},N=\overline{1,3}.$

On these and related aspects of constructed this way integrable dynamical systems on the functional supermanifold $M_{\phi }$, we will not dwell further and plan to stop in more detail in another work under preparation.

4. Conclusions

We have analyzed in detail Lie-algebraic structures related to the superalgebra of pseudodifferential operators on the circle over the even component of the ${\mathbb{Z}}_2$-graded Grassmann algebra, and studied the corresponding coadjoint orbits on it, generated by the corresponding Casimir invariants, regarding the different superalgebra splittings into the subalgebras. These flows coincide with Lax-type completely integrable Hamiltonian flows on suitably defined functional manifolds with respect to the canonical super-Lie–Poisson structures on them. We have proved that all these coadjoint flows can be extended by means of the respectively constructed super-evolution flows on the adjoint super-subalgebras, specially deformed by means of super-pseudodifferential operator elements, depending on the generalized eigenfunctions of the corresponding super-linear Lax-type spectral problem, and generating new integrable Hamiltonian systems on functional supermanifolds. There were also analyzed the centrally extended super-Lie-algebraic structures and the related coadjoint orbits, generated by the corresponding Casimir invariants, and coinciding with integrable Hamiltonian systems on suitably defined supermanifolds. It was also mentioned an interesting yet still not analyzed problem of generalizing the obtained above results on the case of the superflows, generated by coadjoint orbits of the Lie superalgebra $\mathcal{L}({\mathbb{S}}^{1|N};D_{\theta }),\theta \in {\Lambda }_1^{\left(N\right)},$ over the super N-dimensional ${\mathbb{Z}}_2$-graded Grassmann algebra ${\Lambda }^{\left(N\right)}={\Lambda }_0\oplus {\Lambda }_1^{\left(N\right)}$ of the super-pseudodifferential operators, considered before in [14,41,42,43], on the supercircle ${\mathbb{S}}^{1|N}$ for $N=\overline{1,3}.$