1. Introduction
Some twenty years ago, a new class of nonlinear dynamical systems, called ‘dark equations' was introduced by Boris Kupershmidt [1, 2], and shown to possess unusual properties that were not particularly well-understood at that time. Later, in related developments, some Burgers-type [3–5] and also Korteweg–de Vries type [6, 7] dynamical systems were studied in detail, and it was proved that they have a finite number of conservation laws, a linearization and degenerate Lax representations, among other properties. In what follows, we provide a description of a class of self-dual dark-type (or just, dark, for short) nonlinear dynamical systems, which a priori allows their quasi-linearization, whose integrability can be effectively studied by means of a geometrically motivated [8, 9–11] gradient-holonomic approach [12–14]. Moreover, we study a slightly modified form of a self-dual nonlinear dark dynamical system on a functional manifold, whose integrability was recently analyzed in [15]. Not only did this dynamical system appear to be Lax integrable, it also seemed to have a rich mathematical architecture including compatible Poisson structures and an infinite hierarchy of nontrivial mutually commuting conservation laws. In the sequel, we shall prove these properties using the gradient-holonomic integrability scheme devised in our prior work with several collaborators [12–14].
The remainder of this investigation is organized as follows. In section 2 , we describe a rather wide class of self-dual dark quasi-linearized nonlinear dynamical systems on smooth functional manifolds. Then, in section 3 , we study their quasi-linearization property along with the integrability of a new self-dual nonlinear dark dynamical system. Next, in section 4 , we prove the existence of a bi-Hamiltonian structure for the new system, and deduce its complete integrability. Finally, section 5 is devoted to summarizing our results and indicating some possible related future research directions.
2. Self-dual symmetry, quasi-linearization and nonlinear dark dynamical systems
We begin by studying integrability properties of a certain class of nonlinear dynamical systems of the form2.1 ), is uniformly invariant with respect to the following smooth transformation: M ∋ (u, v) → (uϵ, vϵ) ∈ M for any real parameter ϵ ∈ R\{0}, or equivalently, the following relationships2.2 ) with respect to ϵ ∈ R\{0} and taking the limit ϵ → 0, one easily obtains an extended dynamical system on the ex functional manifold of variables $(\alpha =\partial {u}_{\varepsilon }/\partial \varepsilon {| }_{\varepsilon =0},w=\mathrm{ln}v)\in T({M}_{u})\times T({M}_{v})$:2.3 ) make it possible to retrieve the exact form of the dark vector fields (2.1 ) being examined for integrability. Taking the above reasoning into account, we can classify them by means of the reduced integrable quasilinear dynamical systems (2.3 ), equivalently redefined on a suitably modified functional manifold $\tilde{M}$ of smooth mappings $(u,\alpha )\in {M}_{u}\times T({M}_{v})\simeq \tilde{M},$ diffeomorphic to the manifold M.
$\begin{eqnarray}\left.\begin{array}{c}{u}_{t}=P[u,v]\\ {v}_{t}=F[u,v]\end{array}\right\}:= K[u,v]\end{eqnarray}$
on a suitably chosen [8] smooth functional manifold M :=Mu × Mv ⊂ C(R; R2), where t ∈ R is the evolution parameter and K: M → T(M) is a smooth vector field on M, with values in its tangent space T(M), represented by means of polynomial functions on the related jet-space J(R; R2) of a finite order. Moreover, we will assume that the vector field on M satisfies the additional self-dual symmetry: the flow on M, generated by the vector field ( $\begin{eqnarray}{\left({u}_{\varepsilon }\right)}_{t}=P[{u}_{\varepsilon },{v}^{\varepsilon }],\,\,\,\,{\left({v}^{\varepsilon }\right)}_{t}=F[{u}_{\varepsilon },{v}^{\varepsilon }]\end{eqnarray}$
hold for all t ∈ R and all admissible points (u, v) ∈ M. Differentiating the relationships ( $\begin{eqnarray}\left.\begin{array}{c}{\alpha }_{t}={P}_{u}^{{\prime} }[u,1]\alpha +\ {P}_{v}^{{\prime} }[u,1]w\\ {w}_{t}={F}_{u}^{{\prime} }[u,1]\alpha \ +{F}_{v}^{{\prime} }[u,1]w\end{array}\right\},\end{eqnarray}$
where the prime denotes the usual Frechét derivative with respect to the corresponding functional variable. It is now easy to observe that, modulo the expressions for the Frechét derivatives ${P}_{u}^{{\prime} }[u,1],{P}_{v}^{{\prime} }[u,1]$ and ${F}_{u}^{{\prime} }[u,1],{F}_{v}^{{\prime} }[u,1]$ on M, the resulting vector fields (3. Integrability description of dark equations
3.1. Integrability analysis
To proceed with the problem of classifying integrable dark dynamical systems on the functional manifold M, we need from the very beginning to analyze the conditions under which the reduced quasi-linearized system (2.3 ) possesses an infinite hierarchy of suitably ordered conservation laws. To do this, we will make use of the geometrically motivated gradient-holonomic integrability scheme devised in [16] and further developed in [12–14], to first transform the vector fields (2.3 ) to their following equivalent form on the functional manifold $\tilde{M}$:2.1 differential-functional expression. This system is strongly degenerate since it is quasilinear with respect to the variable α ∈ T(Mu), what is very important in that it enables simplification of the classification of the nonlinear dark integrable dynamical systems (3.1 ) on the manifold $\tilde{M},$ which are completely equivalent to the dark systems (2.1 ) on the manifold M. First of all, we need to study asymptotic solutions to the following Lax–Noether equation3.1 ) the solution $\tilde{\varphi }\in {T}^{* }(\tilde{M})$ to the equation (3.2 ) is governed by a special choice of the reduced Frechét derivatives ${P}_{u}^{{\prime} }[u,1],{P}_{v}^{{\prime} }[u,1]$ and ${F}_{u}^{{\prime} }[u,1],{F}_{v}^{{\prime} }[u,1]$ on T(M), depending only on the functional parameter u ∈ Mu. To specify this class of vector fields (3.1 ), one can assume, for instance, that3.1 ) is completely integrable; in particular, possessing a finite, or infinite hierarchy of conservation laws, suitably ordered subject to the degrees of an arbitrarily chosen so-called ‘spectral' parameter λ ∈ R.
$\begin{eqnarray}\left.\begin{array}{c}{\alpha }_{t}=\ {P}_{u}^{{\prime} }[u,1]\alpha +\ {P}_{v}^{{\prime} }[u,1]\mathrm{ln}v,\\ {v}_{t}={{vF}}_{u}^{{\prime} }[u,1]\alpha +{{vF}}_{v}^{{\prime} }[u,1]\mathrm{ln}v,\end{array}\right\}:= \tilde{K}[\alpha ,v],\end{eqnarray}$
where, by definition, $\alpha := {\xi }_{0}[u]+{\xi }_{1}^{{\prime} }[u]{w}_{x}\in T({M}_{u})\,$ is somewhat compatible with the system $\begin{eqnarray}{\tilde{\varphi }}_{t}+{\tilde{K}}^{{\prime} ,* }[\alpha ,v]\tilde{\varphi }=0\end{eqnarray}$
on the cotangent space ${T}^{* }(\tilde{M})$ to the functional manifold $\tilde{M},$ where ${\tilde{K}}^{{\prime} ,* }[\alpha ,v]$: ${T}^{* }(\tilde{M})\to $ ${T}^{* }(\tilde{M})$ denotes the adjoint Frechét derivative operator of the vector field $\tilde{K}[\alpha ,v]:\tilde{M}\to T(\tilde{M})$ subject to the standard bilinear form $(\cdot | \cdot ):{T}^{* }(\tilde{M})\times T(\tilde{M})\to R$ on the product ${T}^{* }(\tilde{M})\times T(\tilde{M})$. In particular, for the case of the dynamical system ( $\begin{eqnarray}\begin{array}{rcl}{P}_{u}^{{\prime} }[u,1] & := & \sum _{j=0}^{n}{p}_{j}^{(1)}[u]{\partial }^{j},\\ {P}_{v}^{{\prime} }[u,1] & := & \sum _{j=0}^{n}{p}_{j}^{(2)}[u]{\partial }^{j+1},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{F}_{u}^{{\prime} }[u,1] & := & \sum _{k=0}^{n}{f}_{j}^{(1)}[u]{\partial }^{k},\\ {F}_{v}^{{\prime} }[u,1] & := & \sum _{k=0}^{n}{f}_{k}^{(2)}[u]{\partial }^{k+1},\end{array}\end{eqnarray}$
where $\,{p}_{j}^{(1)},{f}_{j}^{(1)}:{J}^{n}(R;R)\to R,j=\overline{0,n},$ and ${p}_{j}^{(2)},{f}_{j}^{(2)}:{J}^{n}(R;R)\to R,k=\overline{0,n},$ are functional expressions on the the corresponding jet-manifolds. These expressions should be determined from the necessary condition that the corresponding quasi-linear dynamical system (To simplify the above scheme, it is enough to choose a priori only the operator expression (3.4 ), imposing no constraints on the expression (3.3 ), and consider the reduced Lax–Noether equation (3.2 ) on the functional manifold Mv:3.6 ) as the one, given a priori in a general form on the functional manifold Mv. Having then imposed on the linear system (3.6 ) the determining condition that it is completely integrable, the latter naturally gives rise to the related evolution properties of the functional parameter u ∈ Mu, thus describing the variety of dynamical systems (3.1 ) completely.
$\begin{eqnarray}{\varphi }_{t}+{R}_{v}^{{\prime} ,* }[u,v]\varphi =0,\end{eqnarray}$
where, by definition, φ ∈ T*(Mv) and $\begin{eqnarray}\begin{array}{rcl}{v}_{t} & = & R[u,v]:= {{vF}}_{u}^{{\prime} }[u,1]{\xi }_{0}[u]\\ & & +v\left({F}_{u}^{{\prime} }[u,1]\circ {\xi }_{1}^{{\prime} }[u]\partial /\partial x+\ {F}_{v}^{{\prime} }[u,1]\right)\mathrm{ln}v,\end{array}\end{eqnarray}$
with a point u ∈ Mu, being assumed as an arbitrary functional parameter. Taking into account that the symmetry element α ∈ T(Mu) is not specified, one can consider the resulting vector field (As a simple example, let us analyze the reduced dynamical system in the form3.7 ) conservation laws determining Lax–Noether equation looks as3.7 ) and a priori imposed its self-duality. The obtained result one can formulate as the following theorem.
$\begin{eqnarray}{v}_{t}=\ {u}^{-2}{v}_{{xx}}-\ {u}^{-2}{v}^{-1}{v}_{x}^{2},\end{eqnarray}$
on the manifold Mv, parameterized by a functional variable u ∈ Mu. The related to ( $\begin{eqnarray}{\varphi }_{t}+\left[{u}^{-2}{\left({v}_{x}/v\right)}_{x}+{v}^{-1}{\partial }^{2}{u}^{-2}v\right]\varphi =0\ \end{eqnarray}$
and allows the following asymptotic as λ → ∞ solution $\begin{eqnarray}\begin{array}{rcl}\varphi & = & \exp [-{\lambda }^{2}t+{\partial }^{-1}\sigma (x;\lambda )],\\ \sigma (x;\lambda ) & \sim & \sum _{j\in {{\mathbb{Z}}}_{+}\cup \{-1\}}{\lambda }^{-j}\,{\sigma }_{j}[u,v].\end{array}\end{eqnarray}$
From these expressions one ensues the following infinite hierarchy of recurrent differential-algebraic relationships: $\begin{eqnarray}\begin{array}{l}-{\delta }_{j,-2}+{\partial }^{-1}{\sigma }_{j,t}+{u}^{-2}{\sigma }_{j,x}+{u}^{-2}\\ \,\,\times \,\sum _{k\in {{\mathbb{Z}}}_{+}\cup \{-1\}}{\sigma }_{k}{\sigma }_{j-k}+2{u}^{-2}{v}^{-1}{v}_{x}{\sigma }_{j}\\ \,\,+\,2{u}^{-2}{v}^{-1}{v}_{{xx}}{\delta }_{j,0}-{u}^{-2}{v}^{-2}{v}_{x}^{2}{\delta }_{j,0}=0\end{array}\end{eqnarray}$
for j ∈ Z+ ∪ { − 2, − 1}. Whence we obtain those quantities $\begin{eqnarray}\begin{array}{rcl}{\sigma }_{-1} & = & u,\\ {\sigma }_{0} & = & -\displaystyle \frac{1}{2}u{\partial }^{-1}{u}_{t}-{v}_{x}^{-1}{v}_{x}-\displaystyle \frac{1}{2}{u}^{-1}{u}_{x}\\ & = & -1/2{p}_{x}-{v}^{-1}{v}_{x}-1/2{u}^{-1}{u}_{x}\\ & = & -1/2{\left(p+\mathrm{ln}({{uv}}^{2})\right)}_{x},\end{array}\end{eqnarray}$
entailing, in addition, the conditions σj = 0 for all j ∈ N\{0}, if the functional constraint $\begin{eqnarray}{u}_{t}={\left({u}^{-1}{p}_{x}\right)}_{x}\end{eqnarray}$
holds for some smooth mapping p: J2(R; R2) → R, whose evolution is given by the relationship $\begin{eqnarray}\begin{array}{rcl}{p}_{t} & = & -{\left(\mathrm{ln}\left(v/u\right)\right)}_{t}+2\left[{u}^{-2}{\left({v}_{x}/v\right)}_{x}+{v}^{-1}{\left(v/{u}^{2}\right)}_{x}\right]\\ & & -{u}^{-2}{\left[p+\mathrm{ln}(v/u)\right]}_{{xx}}+1/2{\left({[p+\mathrm{ln}(v/u)]}_{x}\right)}^{2},\end{array}\end{eqnarray}$
or, equivalently, satisfies the related nonlinear dynamical system $\begin{eqnarray}\begin{array}{rcl}{p}_{t} & = & -2{u}^{-2}{p}_{{xx}}+1/2{u}^{-2}{p}_{x}^{2}+2{u}^{-3}{u}_{x}{p}_{x}\\ & & -{u}^{-3}{u}_{{xx}}+3/2{u}^{-4}{u}_{x}^{2}.\end{array}\end{eqnarray}$
The latter guarantees the integrability of the nonlinear flow (The nonlinear dynamical system
$\begin{eqnarray}\begin{array}{rcl}{u}_{t} & = & -{u}^{-2}{u}_{x}{p}_{x}+{u}^{-1}\ {p}_{{xx}}:= P[u,v],\\ {v}_{t} & = & {u}^{-2}{v}_{{xx}}-\ {u}^{-2}{v}^{-1}{v}_{x}^{2}:= F[u,v],\end{array}\end{eqnarray}$
where the smooth mapping $p:{J}^{2}({\mathbb{R}};{{\mathbb{R}}}^{2})\to {\mathbb{R}}$ satisfies the compatibility relationship $\begin{eqnarray*}\begin{array}{rcl}{p}_{t} & = & -2{u}^{-2}{p}_{{xx}}+1/2{u}^{-2}{p}_{x}^{2}+2{u}^{-3}{u}_{x}{p}_{x}\\ & & -{u}^{-3}{u}_{{xx}}+3/2{u}^{-4}{u}_{x}^{2},\end{array}\end{eqnarray*}$
is integrable on the functional manifold $M\times {M}_{p}$ and self-dual with respect to the following symmetry mapping $M\ni (u,v)\to ({u}_{\varepsilon },{v}^{\varepsilon })\in M$ for any $\varepsilon \in R\setminus \{0\},$ where ${{du}}_{\varepsilon }/d\varepsilon {| }_{\varepsilon =0}:= \alpha \in T({M}_{u}),$ ${\alpha }_{t}={P}_{u}^{{\prime} }[u,v]\alpha ,$ ${{dv}}^{\varepsilon }/d\varepsilon {| }_{\varepsilon =0}\,=w\in T({M}_{v}),$ $\ {w}_{t}={u}^{-2}{w}_{{xx}}$.Below we present also two simple enough yet important in modern cosmology so-called Gurevich-Zybin type nonlinear dynamical systems3.1 ) when the expressions ${F}_{u}^{{\prime} }[u,1]=0$ and ${F}_{v}^{{\prime} }[u,1]$ is constant for all u ∈ Mu.
$\begin{eqnarray}\begin{array}{l}\left.\begin{array}{c}{u}_{t}=-{{uu}}_{x}+w\\ {w}_{t}=-{{uw}}_{x}\end{array}\right\},\\ \left.\begin{array}{c}{u}_{t}=-{{uu}}_{x}+{w}_{x}/w\\ {w}_{t}=-{{uw}}_{x}\end{array}\right\}\end{array}\end{eqnarray}$
on the manifold $\tilde{M},$ which are obtained for the special symmetry mapping M ∋ (u, v) → (uvϵ, vϵ) ∈ M for any $\varepsilon \in {\mathbb{R}}\setminus \{0\}\,$ at the choice ${P}_{u}^{{\prime} }[u;1]:= -{{uu}}_{x},{P}_{v}^{{\prime} }[u;1]:= 1,$ ${F}_{v}^{{\prime} }[u;1]:= -u\partial /\partial x,$ ${F}_{u}^{{\prime} }[u;1]\,$ and ${P}_{u}^{{\prime} }[u;1]:= -{{uu}}_{x},{P}_{v}^{{\prime} }[u;1]:= \partial /\partial x,$ ${F}_{v}^{{\prime} }[u;1]:= -u\partial /\partial x,$ ${F}_{u}^{{\prime} }[u;1],$ respectively, and which prove to be completely integrable bi-Hamiltonian systems, actively studied [5, 17–22] during past decades. Some other more nontrivial examples can be taken from the work [4], devoted to description of the integrable hierarchies of modified Burgers type nonlinear dynamical systems. Below we will stop in some detail on a degenerate case of the self-dual quasi-linear dark type dynamical system (3.2. A new degenerate integrable dark type dynamical system
On the functional manifold ${M}_{u}\ \subset {C}^{\infty }(\ {\mathbb{R}};{\mathbb{R}})$ let us consider a degenerate nonlinear dynamical system (2.1 ), depending only on the functional variable u ∈ Mu:
$\begin{eqnarray}\begin{array}{rcl}{u}_{t} & = & {u}_{3x}-3{u}^{-1}{u}_{x}{u}_{{xx}}+2{u}^{-2}{u}_{x}^{3}+{u}_{x}\\ & = & K[u],\end{array}\end{eqnarray}$
with a Frechét smooth [13, 14, 23–26] vector field P: Mu → T(Mu) on the manifold Mu. It is a strongly nonlinear dispersive evolution flow on Mu, whose special perturbations can be eventually used for modeling from a physical point of view [27, 28] interaction of atmospheric magneto-sonic Alfvén plasma waves.The dynamical system (3.17 ), which was recently introduced in [15], appears to be also self-dual, that is invariant with respect to the change of variables u := w −ϵ ∈ Mu for an arbitrary $\,\varepsilon \in {\mathbb{R}}\setminus \{0\}$:3.19 ) seems to be integrable only when n = 0, as was argued in the work [15], we will consider this case, for which we shall pose the direct integrability problem and successively state that the flow (3.17 ) is actually self-dual with respect to the change of variables u := wϵ ∈ Mu for an arbitrary $\,\varepsilon \in {\mathbb{R}}\setminus \{0\}\,$ and show the existence of an infinite hierarchy of conservation laws for the dynamical system (3.17 ) using methods developed in [13, 14, 26], thus presenting another dark nonlinear dynamical system.
$\begin{eqnarray}\begin{array}{rcl}{w}_{t} & = & {w}_{3x}-3{w}^{-1}{w}_{x}{w}_{{xx}}+2{w}^{-2}{w}_{x}^{3}+{w}_{x}\\ & = & K[w],\end{array}\end{eqnarray}$
and is a special case at n = 0 of the nonlinear dynamical system $\begin{eqnarray}\begin{array}{rcl}{u}_{t} & = & {u}^{n}{u}_{x}+{u}_{3x}-3{u}^{-1}{u}_{x}{u}_{{xx}}+2{u}^{-2}{u}_{x}^{3}\\ & = & {K}_{n}[u]\end{array}\end{eqnarray}$
on the manifold Mu for $n\ \in {{\mathbb{Z}}}_{+}$. Since the dynamical system (Employing the direct gradient-holonomic integrability scheme [12–14], we first describe asymptotic solutions to the Lax–Noether [29] equation3.17 ), the form3.20 ) allows an asymptotic solution φ ∈ T*(Mu) expressible as3.17 ), ω(λ) is a corresponding dispersion function, and the operation of (indefinite) integration on Mu is defined as
$\begin{eqnarray}{\rm{d}}\varphi /{\rm{d}}t+{K}^{\ ^{\prime} ,* }[u]\varphi =0,\end{eqnarray}$
where φ ∈ T*(Mu), the cotangent space of Mu, the prime represents the Frechét derivative and ‘*' denotes the conjugation via the standard bilinear form ( · ∣ · ) on T*(Mu) × T(Mu), which for any a ∈ T*(Mu) and b ∈ T(Mu) is defined as $\begin{eqnarray}(a| b):= {\int }_{{\mathbb{R}}}a(x)b(x){\rm{d}}x.\end{eqnarray}$
The adjoint operator ${{K}^{{\prime} }}^{,* }[u]:{T}^{* }({M}_{u})\ \to {T}^{* }({M}_{u})$ is linear and differential and has, according to ( $\begin{eqnarray}\begin{array}{rcl}{K}^{\ ^{\prime} ,* }[u] & = & -{\partial }^{3}+3{u}^{-2}{u}_{x}{u}_{{xx}}\\ & & +3\partial \cdot {u}^{-1}{u}_{{xx}}-\ 3{\partial }^{2}\cdot {u}^{-1}{u}_{x}\\ & & -4{u}^{-3}{u}_{x}^{3}-\ 6\partial \cdot {u}^{-2}{u}_{x}^{2}-\partial .\end{array}\end{eqnarray}$
Consequently, the linear differential equation ( $\begin{eqnarray}\ \varphi (x,t;\lambda )=\exp [\omega (\lambda )t+\lambda x+{\partial }^{-1}\sigma (x;\lambda )],\end{eqnarray}$
where $\lambda \in {\mathbb{C}}$ is a complex parameter, the integral ${\int }_{{\mathbb{R}}}\sigma (x;\lambda ){dx}$ is a generating function of conservation laws [12, 14, 29] for the nonlinear dynamical system ( $\begin{eqnarray}{\partial }^{-1}(\cdot ):= \displaystyle \frac{1}{2}[{\int }_{-\infty }^{x}(\cdot ){\rm{d}}x-{\int }_{x}^{\infty }(\cdot ){\rm{d}}x].\end{eqnarray}$
Obviously, the operation (3.24 ) has the necessary property $\partial \cdot {\partial }^{-1}=1,\partial =\partial /\partial x,x\in {\mathbb{R}}$. To find the dispersion term $\omega (\lambda )\in {\mathbb{R}}$ in (3.23 ), we consider (3.22 ) at u = 1, obtaining the equation3.25 ) we find an elementary solution ${\varphi }_{0}=\exp [({\lambda }^{3}+\lambda )t+\lambda x]\in {T}^{* }({M}_{u}),$ giving rise to the dispersion function ω(λ) = λ3 + λ, and the required solution to the equation (3.23 ) can be naturally written as3.27 ) into (3.26 ), from (3.20 ) we obtain the infinite system of recursion relations3.28 ) at j = − 2 yields
$\begin{eqnarray}{\rm{d}}{\varphi }_{0}/{\rm{d}}t+{K}^{{\prime} ,* }[1]{\varphi }_{0}=0,\end{eqnarray}$
where ${K}^{{\prime} ,* }[1]=-{\partial }^{3}-\partial $. From equation ( $\begin{eqnarray}\varphi (x,t;\lambda )=\exp [({\lambda }^{3}+\lambda )t+\lambda x+{\partial }^{-1}\sigma (x;\lambda )],\end{eqnarray}$
where $\begin{eqnarray}\sigma (x,t;\lambda )\simeq \sum _{j\in {{\mathbb{Z}}}_{+}}{\sigma }_{j}[u]{\lambda }^{-j}.\end{eqnarray}$
Substituting ( $\begin{eqnarray*}\begin{array}{rcl}{\partial }^{-1}{\sigma }_{j,t}(x;\lambda ) & = & {\sigma }_{j,{xx}}+3{\sigma }_{j+1,x}\\ & & +3\sum _{k\in {{\mathbb{Z}}}_{+}}{\sigma }_{{jk}}{\sigma }_{k,x}+3{\sigma }_{j+2}+\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}+3\sum _{k\in {{\mathbb{Z}}}_{+}}{\sigma }_{j+1-k}{\sigma }_{k}+\sum _{k\in {{\mathbb{Z}}}_{+}}\sum _{s\in {{\mathbb{Z}}}_{+}}{\sigma }_{{jks}}{\sigma }_{{ks}}{\sigma }_{s}\\ +\,3{u}^{-2}{u}_{x}{u}_{{xx}}{\delta }_{j,0}+3{u}^{-1}{u}_{{xx}}{\delta }_{j,-1}+3{u}^{-1}{u}_{{xx}}{\sigma }_{j}\\ -\,2{u}^{-3}{u}_{x}^{3}{\delta }_{j,0}+3{u}^{-1}{u}_{x}{\sigma }_{j,x}+3{u}^{-1}{u}_{x}{\delta }_{j,-2}\\ +\,6{u}^{-1}{u}_{x}{\sigma }_{j+1}+3{u}^{-1}{u}_{x}\sum _{k\in {{\mathbb{Z}}}_{+}}{\sigma }_{{jk}}{\sigma }_{k}+{\sigma }_{j}.\end{array}\end{eqnarray}$
Hence, ( $\begin{eqnarray*}{\sigma }_{0}=-{u}^{-1}{u}_{x},\end{eqnarray*}$
and at j = − 1, $\begin{eqnarray*}{\sigma }_{1}=0,\end{eqnarray*}$
and at j ≥ 0 we successively obtain $\begin{eqnarray*}{\sigma }_{j+1}=0.\ \end{eqnarray*}$
As it follows from the recurrence formulas (3.28 ) that all local functionals ${\sigma }_{j+1}:{M}_{u}\to {M}_{u},j\in {{\mathbb{Z}}}_{+}\,$ are trivially zero, from (3.26 ) we obtain an exact expression for the element φ ∈ T*(Mu) of the form3.20 ), we immediately obtain the differential relation3.17 ), independent of the parameter $\lambda \in {\mathbb{R}}$.
$\begin{eqnarray}\begin{array}{rcl}\varphi (x,t;\lambda ) & = & \exp [({\lambda }^{3}+\lambda )t+\lambda x+{\partial }^{-1}{\left(-\mathrm{ln}u\right)}_{x}]\\ & = & {u}^{-1}\exp [({\lambda }^{3}+\lambda )t+\lambda x].\end{array}\end{eqnarray}$
Moreover, since φ ∈ T*(Mu) satisfies Lax's equation ( $\begin{eqnarray*}{u}_{t}=u({\lambda }^{3}+\lambda )+{u}^{2}\exp (-\lambda x){K}^{{\prime} ,* }({u}^{-1}{\exp }(\lambda x)),\end{eqnarray*}$
which is naturally reduced to the original nonlinear dynamical system (We now show that equation (3.17 ) is, in addition, Lax type linearized [14, 29, 30]. Note that, as follows from (3.29 ), the function φ ∈ T*(Mu) satisfies the linear differential relationship3.30 ), can be interpreted as a suitable linearization of the original dynamical systems (3.17 ). At the same time, φ ∈ T*(Mu) satisfies the following linear evolution relationship, which is compatible with the linear equation (3.30 ):3.32 ) is equivalent to the linear evolution3.30 ) and is equal to the following polynomial in $\lambda \in {\mathbb{R}}$:3.20 ).
$\begin{eqnarray}{\varphi }_{x}=(\lambda -{u}_{x}/u)\varphi ,\end{eqnarray}$
which combined with the function z := uφ ∈ T*(Mu), gives rise to the linearized Korteweg–de Vries (KdV) dynamical system $\begin{eqnarray}{z}_{t}={z}_{3x}+{z}_{x}.\end{eqnarray}$
Hence this, jointly with the relationship ( $\begin{eqnarray}\begin{array}{rcl}{\varphi }_{t} & = & {\varphi }_{{xxx}}+\ {\varphi }_{x}-3{u}^{-2}{u}_{x}{u}_{{xx}}\varphi -\partial \circ [(3{u}^{-1}{u}_{{xx}})\varphi ]\\ & & +{\partial }^{2}\circ [(3{u}^{-1}{u}_{x})\varphi ]+4{u}^{-3}{u}_{x}^{3}\varphi +\partial \circ [6{u}^{-2}{u}_{x}^{2}\varphi ].\end{array}\end{eqnarray}$
It is easy to observe now that the evolution equation ( $\begin{eqnarray*}{\varphi }_{t}=c[u;\lambda ]\varphi ,\end{eqnarray*}$
where the local functional c[u; λ] can be readily calculated using ( $\begin{eqnarray}\begin{array}{rcl}c[u;\lambda ] & = & {\lambda }^{3}+\lambda -2{u}^{-3}{u}_{x}^{3}+3{u}^{-2}{u}_{x}{u}_{{xx}}\\ & & -{u}^{-1}{u}_{x}-{u}^{-1}{u}_{{xxx}}={\lambda }^{3}+\lambda -{u}^{-1}K[u].\end{array}\end{eqnarray}$
Thus, we have obtained a system of a priori compatible linear Lax-type relationships $\begin{eqnarray*}{\varphi }_{x}=(\lambda -{u}_{x}/u)\varphi ,{\varphi }_{t}=({\lambda }^{3}+\lambda -{u}^{-1}K[u])\varphi ,\end{eqnarray*}$
where $\lambda \in {\mathbb{R}}$ is an arbitrary parameter. Whence, we have the next result describing functional solutions to the Lax–Noether equation (The following functional relationships
$\begin{eqnarray}{\varphi }_{n}:= {u}^{-1}{\partial }^{2n}(\mathrm{ln}u)/\partial {x}^{2n}\end{eqnarray}$
for all $n\in {\mathbb{Z}}$ solve the Lax–Noether equation (It is easy to check that for all $n\in {\mathbb{Z}}$ the expressions ${\varphi }_{n}u:= {z}_{n}\in {T}^{* }({M}_{u})$ satisfy the equations (
$\begin{eqnarray}\begin{array}{l}{\partial }^{2n}{\left(\mathrm{ln}u\right)}_{t}/\partial {x}^{2n}\\ =\,{\partial }^{2n}\left[{\left(\mathrm{ln}u\right)}_{3x}\right]/\partial {x}^{2n}+{\partial }^{2n}\left[{\left(\mathrm{ln}u\right)}_{x}\right]/\partial {x}^{2n}=\\ =\,{\partial }^{2n}\left[{\left(\mathrm{ln}u\right)}_{3x}+{\left(\mathrm{ln}u\right)}_{x}\right]/\partial {x}^{2n}.\end{array}\end{eqnarray}$
This follows directly from the fact that, equivalently, the internal equation $\begin{eqnarray}{\left(\mathrm{ln}u\right)}_{t}={\left(\mathrm{ln}u\right)}_{3x}+{\left(\mathrm{ln}u\right)}_{x}\end{eqnarray}$
upon performing simple calculations reduces, owing to the linearity of (As a simple consequence from lemma (3.2 ), we see that the substitution $z:= \mathrm{ln}u\in {M}_{u}$ linearizes the nonlinear self-dual dynamical system (3.17 ), representing it in the following equivalent linear form:3.17 ): $\mathrm{ln}u\to \varepsilon \mathrm{ln}u=\mathrm{ln}{u}^{\varepsilon }$ for any ϵ ∈ R. Therefore, we have solved the direct integrability problem posed for the nonlinear dynamical system (3.17 ) and established the following result.
$\begin{eqnarray}{z}_{t}={z}_{{xxx}}+{z}_{x}:= {K}^{{\prime} }[1]z,\end{eqnarray}$
possessing a trivial symmetry Mu ∋ z → zϵ ∈ Mu and responsible for the self-duality of the dynamical system (The nonlinear dynamical system (
The linearization result (3.37 ) of the dynamical system (3.17 ) can be also easily proved in a completely different way by making use of the self-dual invariance of the dynamical system (3.18 ) with respect to the substitution w := uϵ ∈ Mu for any $\varepsilon \in {\mathbb{R}}\setminus \{0\}$. In particular, let us consider this dynamical system in the following form:3.37 ) upon substitution $\mathrm{ln}u:= z\in {M}_{u},$ thus proving the desired result.
$\begin{eqnarray}\varepsilon {u}^{\varepsilon -1}{u}_{t}=K[{u}^{\varepsilon }]\end{eqnarray}$
and differentiate it with respect to the parameter $\varepsilon \in {\mathbb{R}}\setminus \{0\}$ at ϵ = 0: $\begin{eqnarray}{\left.{u}^{\varepsilon -1}{u}_{t}+\varepsilon {u}^{\varepsilon -1}(\mathrm{ln}u){u}_{t}\right|}_{\varepsilon =0}={\left.{K}^{{\prime} }[{u}^{\varepsilon }]{u}^{\varepsilon }\mathrm{ln}u\right|}_{\varepsilon =0},\end{eqnarray}$
or, equivalently $\begin{eqnarray}{\left(\mathrm{ln}u\right)}_{t}={K}^{{\prime} }[1](\mathrm{ln}u):= ({\partial }^{3}+\partial )(\mathrm{ln}u),\end{eqnarray}$
which coincides exactly with the evolution equation (Taking into account the dispersive structure and the Lax integrability of the dynamical system (3.17 ), it is natural to inquire about its complete integrability; in particular, about the existence of its additional conservation laws and adjoint Poisson structures. These questions are addressed in the next section.
4. Bi-Hamiltonian structure and complete integrability
We now observe now that the functional expressions (3.34 ) satisfy for all $n\in {\mathbb{Z}}$ the Noether self-adjointness conditions:3.17 ), that is ${{\rm{d}}{H}}_{n}/{\rm{d}}{t}{| }_{{u}_{t}=P[u]}=0$ for all $n\in {\mathbb{Z}}$ along the vector field K: Mu → T(Mu) on Mu.
$\begin{eqnarray}{\varphi }_{n}^{{\prime} }[u]={\varphi }_{n}^{{\prime} ,* }[u],\end{eqnarray}$
meaning that there exist smooth functionals γn, $n\in {\mathbb{Z}},$ such that $\mathrm{grad}$ γn[u] = φn[u] ∈ T(M), and which can be easily retrieved via the classical homotopy formula as $\begin{eqnarray}{\gamma }_{n}:= {\int }_{0}^{1}({\varphi }_{n}[{su}]| u){\rm{d}}{s}=\displaystyle \frac{1}{2}{\int }_{{\mathbb{R}}}{\left[{\partial }^{n}(\mathrm{ln}u)/\partial {x}^{n}\right]}^{2}{\rm{d}}{x}\end{eqnarray}$
for all $n\in {\mathbb{Z}}$. Thus the corresponding expressions $\begin{eqnarray}{H}_{n}:= \displaystyle \frac{1}{2}{\int }_{{\mathbb{R}}}{[{\partial }^{n}({u}_{x}/u)/\partial {x}^{n}]}^{2}{\rm{d}}{x}\end{eqnarray}$
are conservation laws for our dynamical system (In view of the results above, one can ask about the existence of the Hamiltonian representation for our self-dual dynamical system (3.17 ). In order to answer to this question, we show that the system (3.17 ) is Hamiltonian, proving the existence of a suitable skew-symmetric Poisson structure ϑ: T*(Mu) → T(Mu) on the manifold Mu, subject to which the representation
$\begin{eqnarray}{u}_{t}:= \{H,u\}{}_{\vartheta }=-\vartheta \mathrm{grad}H[u]\end{eqnarray}$
holds on the whole manifold Mu for some smooth functional $H:{M}_{u}\to {\mathbb{R}}.$ Here, by definition, the gradient function $\mathrm{grad}H[u]\in {T}^{* }({M}_{u})$ is defined via the differential relationship $\tfrac{{\rm{d}}}{{\rm{d}}\varepsilon }H[u+\varepsilon \alpha ]{| }_{\varepsilon =0}=(\mathrm{grad}H[u]| \alpha ),$ satisfied for all α ∈ T(Mu), and the Poisson bracket { · , · } on Mu is defined for any smooth functionals $F,G:{M}_{u}\to {\mathbb{R}}$ as {F, G}ϑ := $(\mathrm{grad}F[u]| \vartheta \mathrm{grad}G[u]),$ which must, as usual, satisfy [13, 14, 30] the classical Jacobi identity.Now we return to the Lax–Noether equation (3.20 ) and recall [12–14, 29] that it is satisfied by $\tilde{\varphi }:= \mathrm{grad}\tilde{H}[u]\in {T}^{* }({M}_{u})$ for an arbitrary conservation law $\tilde{H}:{M}_{u}\to {\mathbb{R}}$. Moreover, if we assume the existence of a functional $\tilde{H}:{M}_{u}\to {\mathbb{R}},$ representable as (ψ∣ut) = H for some element ψ ∈ T*(Mu), solving the modified Lax–Noether equation
$\begin{eqnarray}{\psi }_{t}+{K}^{{}^{{\prime} ,* }}[u]\psi =\mathrm{grad}L[u]\end{eqnarray}$
for some smooth functional $L:{M}_{u}\to {\mathbb{R}},$ then one has the following result.Let the condition (
$\begin{eqnarray}\vartheta ={({\psi }^{{\prime} }-{\psi }^{{\prime} ,* })}^{-1}\end{eqnarray}$
and the corresponding Hamiltonian functional $H=$ $\tilde{H}-L$.Actually, we essentially need only calculate the gradient function
$\begin{eqnarray}\begin{array}{rcl}\ \mathrm{grad}\tilde{H}[u] & = & \mathrm{grad}(\psi | {u}_{t})=\mathrm{grad}(\psi | P)[u]\\ & = & -({\psi }^{{\prime} }-{\psi }^{{\prime} ,* })P[u]+({\psi }_{t}+{P}^{{}^{{\prime} },* }[u]\psi )=\\ & = & -\vartheta \ P[u]+\mathrm{grad}L[u]\end{array}\end{eqnarray}$
at $u\in {M}_{u}$. The expression ( $\begin{eqnarray}{u}_{t}=P[u]=-\vartheta \mathrm{grad}(\tilde{H}-L)[u]:= -\vartheta \mathrm{grad}H[u],\end{eqnarray}$
where $H:= $ $\tilde{H}-L:{M}_{u}\to {\mathbb{R}}$. Recalling the representation (Using proposition 4.1 , one can readily construct Poisson operators for our dynamical system (3.17 ), where instead of the basic vector field ut = P[u] on the manifold Mu, we use the related vector field uτ = ux that shifts its orbits along the independent variable x ∈ ${\mathbb{R}}$. For the conservation law $\tilde{H}:= {H}_{-1}=\tfrac{1}{2}{\int }_{{\mathbb{R}}}{(\mathrm{ln}u)}^{2}{\rm{d}}{x},$ one easily obtains that4.5 ). This means that the operator3.17 ) is Hamiltonian.
$\begin{eqnarray}\begin{array}{rcl}{H}_{-1} & = & \displaystyle \frac{1}{2}{\displaystyle \int }_{{\mathbb{R}}}{(\mathrm{ln}u)}^{2}{\rm{d}}{x}\\ & = & (\mathrm{ln}u| \mathrm{ln}u)/2=(\mathrm{ln}u| {\partial }^{-1}\partial \left(\mathrm{ln}u\right))/2\\ & = & -({\partial }^{-1}(\mathrm{ln}u)| \partial \left(\mathrm{ln}u\right))/2\\ & = & -({\partial }^{-1}(\mathrm{ln}u)| {u}_{x}/u)/2\\ & = & -({u}^{-1}{\partial }^{-1}(\mathrm{ln}u)/2| {u}_{x}):= ({\psi }_{-1}| {u}_{x}),\end{array}\end{eqnarray}$
where ${\psi }_{-1}:= -{u}^{-1}{\partial }^{-1}(\mathrm{ln}u)/2\in {T}^{* }({M}_{u})\,$ satisfies the modified Noether–Lax equation ( $\begin{eqnarray}\begin{array}{rcl}{\vartheta }_{-1} & := & -{\left({\psi }_{-1}^{{\prime} }-{\psi }_{-1}^{{\prime} ,* }\right)}^{-1}\\ & = & -{\left(-\displaystyle \frac{1}{2u}{\partial }^{-1}\displaystyle \frac{1}{u}-\displaystyle \frac{1}{u}{\partial }^{-1}\displaystyle \frac{1}{2u}\right)}^{-1}\\ & = & u\partial u\end{array}\end{eqnarray}$
is Poisson on the functional manifold Mu, for which the dynamical system (Similarly, one can construct another Poisson operator using the conservation law $\tilde{H}:= {H}_{0}=\tfrac{1}{2}{\int }_{{\mathbb{R}}}{({u}_{x}/u)}^{2}{\rm{d}}{x},$ representable as H0 = (ux/(2u2)∣ux) := (ψ0∣ux), where the element ψ0 := ux/(2u2) ∈ T*(Mu) satisfies the modified Noether–Lax equation (4.5 ) and generates the second Poisson operator4.10 ) and (4.11 ) are characterized as follows:
$\begin{eqnarray}\begin{array}{rcl}{\vartheta }_{0} & := & {\left({\psi }_{0}^{{\prime} }-{\psi }_{0}^{{\prime} ,* }\right)}^{-1}\\ & = & {\left({u}^{-2}/2\partial +\ \partial {u}^{-2}/2\right)}^{-1}\\ & = & {\left({u}^{-1}\partial {u}^{-1}\right)}^{-1}=u{\partial }^{-1}u\end{array}\end{eqnarray}$
on Mu. These two Poisson operators (The Poisson operators (
As is well known [12, 14, 32], it suffices to check that the operator $\tilde{\vartheta }:= {\vartheta }_{0}{\vartheta }_{-1}^{-1}{\vartheta }_{0}$ is also Poisson. This is readily verified since $\tilde{\vartheta }:= {\left({\psi }_{1}^{{\prime} }-{\psi }_{1}^{{\prime} ,* }\right)}^{-1}=u{\partial }^{-3}u\,$ is, by construction, a priori Poisson, where the element ${\psi }_{1}:= -\tfrac{1}{2u}{\left(\mathrm{ln}u\right)}_{3x}$ factorizes the conservation law ${H}_{1}=({\psi }_{1}| {u}_{x})=\tfrac{1}{2}{\int }_{{\mathbb{R}}}{\left[{\left({u}^{-1}{u}_{x}\right)}_{x}\right]}^{2}{\rm{d}}{x}$. Thus the proof is complete.
As a natural corollary of proposition 4.2 , we show that the dynamical system (3.17 ) is bi-Hamiltonian and all operators4.10 ) and (4.11 ) the following functional relationships3.3 .
$\begin{eqnarray*}{\vartheta }_{n}:= {\vartheta }_{0}{\left({\vartheta }_{-1}^{-1}{\vartheta }_{0}\right)}^{n}\end{eqnarray*}$
for all integers $n\in {\mathbb{Z}}$ are Poissonian on Mu. Moreover, owing to the compatibility of the Poisson operators ( $\begin{eqnarray}{\vartheta }_{0}\mathrm{grad}{H}_{n}[u]={\vartheta }_{-1}\mathrm{grad}{H}_{n+1}[u]\end{eqnarray}$
hold on Mu for all $n\in {\mathbb{Z}},$ giving rise, in particular, to the following commuting relationships: ${\{{H}_{p},{H}_{m}\}}_{{\vartheta }_{n}}=0\,$ for all n, m and $p\in {\mathbb{Z}}$. The results above can be now restated in the following final form that extends theorem The nonlinear dynamical system (
Recall now [12, 32, 33] that the adjoint recursion operator ${{\rm{\Lambda }}}^{* }:= {\vartheta }_{-1}{\vartheta }_{0}^{-1}=u{\partial }^{2}{u}^{-1}:T({M}_{u})\to T({M}_{u})$ maps the symmetry space to the dynamical system (3.17 ) to itself, and apply it to the simplest ‘shifting' symmetry α := ux ∈ T(Mu):3.17 ) . Moreover, one can easily observe that the symmetry recursion operator Λ* = Φ2, where the operator Φ = u∂u−1: T(Mu) →T(Mu) proves, evidently, to be also a symmetry recursion one, and makes it possible to construct an infinite hierarchy of the self-dual nonlinear dynamical system in the form3.17 ). The latter makes it possible to apply to it the above-constructed symmetry recursion operator Φ = u∂u−1: T(Mu) → T(Mu) and obtain the following new self-dual spatially two-dimensional dynamical systems
$\begin{eqnarray}\begin{array}{rcl}{u}_{\tau } & := & \left[{\left({{\rm{\Lambda }}}^{* }\right)}^{1}+{\left({{\rm{\Lambda }}}^{* }\right)}^{0}\right]{u}_{x}\\ & = & {u}_{3x}-3{u}^{-1}{u}_{x}{u}_{{xx}}+2{u}^{-2}{u}_{x}^{3}+{u}_{x},\end{array}\end{eqnarray}$
coinciding exactly with the self-dual dynamical system ( $\begin{eqnarray}{u}_{{t}_{n}}:= {{\rm{\Phi }}}^{n}{u}_{x}\end{eqnarray}$
for integers n ∈ Z. In particular, for n = 1 one obtains an interesting nonlinear self-dual dissipative dynamical system of the Burgers type: $\begin{eqnarray}{u}_{{t}_{1}}={u}_{{xx}}-{u}^{-1}{u}_{x}^{2}.\end{eqnarray}$
In addition, having assumed that the functional manifold Mu ⊂ C∞(Rx × Ry; R), is rigged with a new spatial variable y ∈ R, we can observe that the element uy ∈ T(Mu) is also a symmetry of the self-dual dynamical system ( $\begin{eqnarray}\begin{array}{rcl}{u}_{{\tau }_{1}} & := & {\rm{\Phi }}{u}_{y}={u}_{{xy}}-{u}^{-1}{u}_{x}{u}_{y},\\ {u}_{{\tau }_{2}} & := & {{\rm{\Phi }}}^{2}{u}_{y}\\ & = & {u}_{{xxy}}-{u}^{-1}{u}_{{xx}}{u}_{y}+2{u}_{x}{u}^{-1}({u}_{x}{u}_{y}{u}^{-1}-{u}_{{xy}})\ \end{array}\end{eqnarray}$
on the manifold Mu, whose solutions possess already much more nontrivial properties, and can suitably fit for modeling some physically interesting phenomena.It is worth mentioning here that the integrability technique and results obtained above for the self-dual nonlinear dynamical system (3.17 ) on the functional manifold Mu can be effectively used via the corresponding Bogoyavlensky–Novikov reduction scheme [12–14, 34] to describe a wide class of its both solitonic and finite-zoned quasiperiodic solutions in exact analytic form, which is now a topic of a work in progress.
5. Concluding remarks
We analyzed a new self-dual nonlinear dark dynamical system (on a functional manifold), which is a mathematical model of the interaction of atmospheric magneto-sonic Alfvén plasma waves. Using a gradient-holonomic approach, we were able to prove that the dynamical system is a completely integrable bi-Hamiltonian system, possessing an infinite hierarchy of nontrivial mutually commuting conservation laws with respect to the corresponding pair of compatible Poisson structures. It was noted that the gradient-holonomic approach, introduced by some of us and our collaborators, together with Bogoyavlensky–Novikov reduction, can apparently be used to construct a wide class of both solitonic and finite-zoned quasiperiodic solutions of the dark system in exact analytic form, and this is a goal that we are currently pursuing.