Deformations of infinite-dimensional Lie algebras, exotic cohomology, and integrable nonlinear partial differential equations

TL;DR

This paper explores how exotic cohomology of infinite-dimensional Lie algebras can be used to identify Lax representations for integrable PDEs, providing a new internal criterion for integrability.

Contribution

It demonstrates that Maurer-Cartan forms of extended Lie algebras with nontrivial exotic cohomology generate Lax representations, linking algebraic cohomology to integrability conditions.

Findings

Maurer-Cartan forms produce Lax representations for known integrable systems

New integrable PDEs are identified through exotic cohomology analysis

Exotic cohomology offers a criterion for integrability based on algebraic properties

Abstract

The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given PDE. The use of the exotic cohomology of the symmetry algebras opens a way to formulate such conditions in internal terms of the PDEs under the study. In this paper we consider certain examples of infinite-dimensional Lie algebras with nontrivial second exotic cohomology groups and show that the Maurer-Cartan forms of the associated extensions of these Lie algebras generate Lax representations for integrable systems, both known and new ones.

Equations

$$D_{x^k}=\frac{\partial }{\partial x^k}+\sum _{\#I\ge 0}\sum _{\alpha =1}^m{u}_{I+1_k}^{\alpha }\frac{\partial }{\partial u_I^{\alpha }},k\in \{1,\dots ,n\},$$

$${\tilde{D}}_{x^k}=D_{x^k}+\sum _{a=0}^{\infty }{T}_k^a(x^i,u_I^{\alpha },w^b)\frac{\partial }{\partial w^a}$$

$$w_{x^k}^a=T_k^a(x^i,u_I^{\alpha },w^b).$$

$${\varpi }^a=d{w}^a-\sum _{k=1}^n{T}_k^a(x^i,u_I^{\alpha },w^b)d{x}^k.$$

$$d{\varpi }^a\equiv \sum _b{\eta }_b^a\wedge {\varpi }^b\mathrm{mod}⟨{\vartheta }_I⟩,$$

$$V=C^0(\mathfrak{g},V)\stackrel{d}{⟶}{C}^1(\mathfrak{g},V)\stackrel{d}{⟶}\dots \stackrel{d}{⟶}{C}^k(\mathfrak{g},V)\stackrel{d}{⟶}{C}^{k+1}(\mathfrak{g},V)\stackrel{d}{⟶}\dots$$

$$d\theta (X_1,...,X_{k+1})=\sum _{q=1}^{k+1}(-1{)}^{q+1}\rho (X_q)(\theta (X_1,...,{\hat{X}}_q,...,X_{k+1}))$$

$$+\sum _{1\le p<q\le k+1}(-1{)}^{p+q}\theta ([X_p,X_q],X_1,...,{\hat{X}}_p,...,{\hat{X}}_q,...,X_{k+1}).$$

$$d_{\lambda \alpha }\theta =d\theta +\lambda \alpha \wedge \theta .$$

$$d_{\lambda \alpha }^2=0.$$

$$C^1(\mathfrak{g},\mathbb{R})\stackrel{d_{\lambda \alpha }}{⟶}\dots \stackrel{d_{\lambda \alpha }}{⟶}{C}^k(\mathfrak{g},\mathbb{R})\stackrel{d_{\lambda \alpha }}{⟶}{C}^{k+1}(\mathfrak{g},\mathbb{R})\stackrel{d_{\lambda \alpha }}{⟶}\dots$$

$$u_{yy}=u_{tx}+u_x{u}_{xx}.$$

$$d\alpha$$

$$d{\Theta }_0$$

$$d{\Theta }_1$$

$$d{\Theta }_2$$

$$d{\Theta }_3$$

$$d{\Theta }_4$$

$${\Theta }_m=\sum _{j=0}^{\infty }\frac{h^j}{j!}{\theta }_{m,j},\nabla {\Theta }_m=\frac{\partial }{\partial h}{\Theta }_m=\sum _{j=0}^{\infty }\frac{h^j}{j!}{\theta }_{m,j+1}$$

$$\alpha$$

$${\theta }_{1,0}$$

$${\theta }_{2,0}$$

$${\theta }_{4,0}$$

$$H^{2}_{\lambda\,\alpha}(Sym(\mathcal{E}_{1}))=\left\{\begin{array}[]{lcl}\mathbb{R}\,[\Omega],&\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\hfil&\lambda=-3,\\ \{0\},&\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\hfil&\lambda\neq-3,\end{array}\right.$$

$$d\omega =3\alpha \wedge \omega +\Omega$$

$$d{\theta }_{3,0}=3\alpha \wedge {\theta }_{3,0}+{\theta }_{3,1}\wedge {\theta }_{0,0}+\frac{2}{3}{\theta }_{2,1}\wedge {\theta }_{1,0}+\frac{1}{3}{\theta }_{1,1}\wedge {\theta }_{2,0}.$$

$${\theta }_{3,0}=p^3\left(dv-v_{x}dx-\left(\frac{1}{3}{v}_x^3-u_x{v}_x-u_y\right)dt-\left(\frac{1}{2}{v}_x^2-u_x\right)dy\right).$$

$$\left\{\begin{array}[]{lcl}v_{t}&=&\textstyle{\frac{1}{3}}\,v_{x}^{3}-u_{x}\,v_{x}-u_{y},\phantom{\frac{A}{A}}\\ v_{y}&=&\textstyle{\frac{1}{2}}\,v_{x}^{2}-u_{x}\phantom{\frac{\frac{A}{A}}{\frac{A}{A}}}\end{array}\right.$$

$$d\alpha$$

$$d{\Theta }_k$$

$$d\alpha$$

$$d{\Theta }_k$$

$$[f,g{]}_{\epsilon }=f{g}_t-g{f}_t+\epsilon s(f_s{g}_t-g_s{f}_t).$$

$$\mathfrak{G}(n,\epsilon )=\left({\mathbb{R}}_{n+1}[s]\otimes C^{\omega }(\mathbb{R})\right)⋊\mathbb{R}Y.$$

$${\theta }_{i,j}(X_{k,m})={\delta }_{ik}{\delta }_{jm},{\theta }_{i,j}(Y)=0,\alpha (X_{k,m})=0,\alpha (Y)=1$$

Full text

††journal: Journal of Geometry and Physics

1 Introduction

The existence of a Lax representation is the key property of integrable equations, [34, 7], and a starting setting for a number of techniques to study nonlinear partial differential equations (pdes) such as Bäcklund transformations, nonlocal symmetries and conservation laws, recursion operators, Darboux transformations, etc. Although these structures are of great significance in the theory of integrable pdes, up to now the problem of finding conditions for a pde to admit a Lax representation is open. In [26] we propose an approach for solving this problem in internal terms of the pde under the study. We show there that for some pdes their Lax representations can be derived from the second exotic111Unlike in [26], in this paper we follow [27] and use the term “exotic cohomology” instead of “deformed cohomology”, since here we discuss deformations of Lie algebras which are not related to “deformed cohomology” in the sense of [26]. cohomology of the symmetry pseudogroups of the pdes. The main advantage of this approach is that it allows one to get rid of apriori assumptions about the defining equations of the Lax representation. In this paper we generalize the constructions of [26]. We consider a deformation of the tensor product of the Lie algebra of vector fields on a line and the algebra of truncated polynomials as well as certain extensions of this deformation and show that at some values of the deformation parameter the Maurer–Cartan forms of the obtained Lie algebras produce Lax representations for some known as well as some new integrable systems.

2 Preliminaries

All considerations in this paper are local. All functions are assumed to be real-analytic.

2.1 Coverings of PDEs

The coherent geometric formulation of Lax representations, Wahlquist–Estabrook prolongation structures, Bäcklund transformations, recursion operators, nonlocal symmetries, and nonlocal conservation laws is based on the concept of differential covering of a pde [14, 15]. In this subsection we closely follow [16, 17] to present the basic notions of the theory of differential coverings.

Let $\pi\colon\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$, $\pi\colon(x^{1},\dots,x^{n},u^{1},\dots,u^{m})$ $\mapsto(x^{1},\dots,x^{n})$ be a trivial bundle, and $J^{\infty}(\pi)$ be the bundle of its jets of the infinite order. The local coordinates on $J^{\infty}(\pi)$ are $(x^{i},u^{\alpha},u^{\alpha}_{I})$, where $I=(i_{1},\dots,i_{n})$ is a multi-index, and for every local section $f\colon\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\times\mathbb{R}^{m}$ of $\pi$ the corresponding infinite jet $j_{\infty}(f)$ is a section $j_{\infty}(f)\colon\mathbb{R}^{n}\rightarrow J^{\infty}(\pi)$ such that $u^{\alpha}_{I}(j_{\infty}(f))=\displaystyle{\frac{\partial^{\#I}f^{\alpha}}{\partial x^{I}}}=\displaystyle{\frac{\partial^{i_{1}+\dots+i_{n}}f^{\alpha}}{(\partial x^{1})^{i_{1}}\dots(\partial x^{n})^{i_{n}}}}$. We put $u^{\alpha}=u^{\alpha}_{(0,\dots,0)}$. Also, in the case of $m=1$ and, e.g., $n=4$ we denote $x^{1}=t$, $x^{2}=x$, $x^{3}=y$, $x^{4}=z$, and $u^{1}_{(i,j,k,l)}=u_{{t\dots t}{x\dots x}{y\dots y}{z\dots z}}$ with $i$ times $t$, $j$ times $x$, $k$ times $y$, and $l$ times $z$.

The vector fields

[TABLE]

with $I+1_{k}=(i_{1},\dots,i_{k},\dots,i_{n})+1_{k}=(i_{1},\dots,i_{k}+1,\dots,i_{n})$ are referred to as total derivatives. They commute everywhere on $J^{\infty}(\pi)$: $[D_{x^{i}},D_{x^{j}}]=0$.

A system of pdes $F_{r}(x^{i},u^{\alpha}_{I})=0$, $\#I\leq s$, $r\in\{1,\dots,\sigma\}$, of the order $s\geq 1$ with $\sigma\geq 1$ defines the submanifold $\mathcal{E}=\{(x^{i},u^{\alpha}_{I})\in J^{\infty}(\pi)\,\,|\,\,D_{K}(F_{r}(x^{i},u^{\alpha}_{I}))=0,\,\,\#K\geq 0\}$ in $J^{\infty}(\pi)$.

Denote $\mathcal{W}=\mathbb{R}^{\infty}$ with coordinates $w^{a}$, $a\in\mathbb{N}\cup\{0\}$. Locally, an (infinite-dimensional) differential covering over $\mathcal{E}$ is a trivial bundle $\tau\colon J^{\infty}(\pi)\times\mathcal{W}\rightarrow J^{\infty}(\pi)$ equipped with the extended total derivatives

[TABLE]

such that $[\tilde{D}_{x^{i}},\tilde{D}_{x^{j}}]=0$ for all $i\not=j$ whenever $(x^{i},u^{\alpha}_{I})\in\mathcal{E}$. For the partial derivatives of $w^{a}$ which are defined as $w^{a}_{x^{k}}=\tilde{D}_{x^{k}}(w^{a})$ we have the system of covering equations

[TABLE]

This over-determined system of pdes is compatible whenever $(x^{i},u^{\alpha}_{I})\in\mathcal{E}$.

Dually the covering with extended total derivatives (1) is defined by the differential ideal generated by the Wahlquist–Estabrook forms, [7, p. 81],

[TABLE]

This ideal is integrable on $\mathcal{E}$, that is,

[TABLE]

where $\eta^{a}_{b}$ are some 1-forms on $\mathcal{E}\times\mathcal{W}$ and $\vartheta_{I}=(du^{\alpha}_{I}-\sum\limits_{k}u^{\alpha}_{I+1_{k}}dx^{k})|_{\mathcal{E}}$.

2.2 Exotic cohomology

Let $\mathfrak{g}$ be a Lie algebra over $\mathbb{R}$ and $\rho\colon\mathfrak{g}\rightarrow\mathrm{End}(V)$ be its representation. Let $C^{k}(\mathfrak{g},V)=\mathrm{Hom}(\Lambda^{k}(\mathfrak{g}),V)$, $k\geq 1$, be the space of all $k$–linear skew-symmetric mappings from $\mathfrak{g}$ to $V$. Then the Chevalley–Eilenberg differential complex

[TABLE]

is is generated by the differential defined by the formula

[TABLE]

[TABLE]

The cohomology groups of the complex $(C^{*}(\mathfrak{g},V),d)$ are referred to as the cohomology groups of the Lie algebra $\mathfrak{g}$ with coefficents in the representation $\rho$. For the trivial representation $\rho_{0}\colon\mathfrak{g}\rightarrow\mathbb{R}$, $\rho_{0}\colon X\mapsto 0$, the complex and its cohomology are denoted by $C^{*}(\mathfrak{g})$ and $H^{*}(\mathfrak{g})$, respectively.

Consider a Lie algebra $\mathfrak{g}$ over $\mathbb{R}$ with non-trivial first cohomology group $H^{1}(\mathfrak{g})$ and take a closed 1-form $\alpha$ on $\mathfrak{g}$. Then for any $\lambda\in\mathbb{R}$ define new differential $d_{\lambda\alpha}\colon C^{k}(\mathfrak{g},\mathbb{R})\rightarrow C^{k+1}(\mathfrak{g},\mathbb{R})$ by the formula

[TABLE]

From $d\alpha=0$ it follows that

[TABLE]

The cohomology groups of the complex

[TABLE]

are referred to as the exotic cohomology groups of $\mathfrak{g}$ and denoted by $H^{*}_{\lambda\alpha}(\mathfrak{g})$.

Remark 1. Cohomology $H^{*}_{\lambda\alpha}(\mathfrak{g})$ coincides with cohomology of $\mathfrak{g}$ with coefficients in the one-dimensional representation $\rho_{\lambda\alpha}\colon\mathfrak{g}\rightarrow\mathbb{R}$, $\rho_{\lambda\alpha}\colon X\mapsto\lambda\,\alpha(X)$. In particular, when $\lambda=0$, cohomology $H^{*}_{\lambda\alpha}(\mathfrak{g})$ coincides with $H^{*}(\mathfrak{g})$. $\diamond$

Remark 2. In all the cases considered in this paper $H^{1}(\mathfrak{g})=Z^{1}(\mathfrak{g})$ due to $C^{0}(\mathfrak{g})=\{0\}$ and $B^{1}(\mathfrak{g})=\{0\}$, so a closed 1-form $\alpha$ can be identified with its cohomology class. $\diamond$

3 The Lax representation for the potential Khokhlov-Zabolotskaya equation

through the second exotic cohomology of the symmetry pseudogroup

A relation between the exotic cohomology of symmetry pseudogroups and Lax representations for integrable systems was established in [26]. In a slightly different notation one of the results of that paper can be presented as follows.

Consider the potential Khokhlov–Zabolotskaya equation $\mathcal{E}_{1}$ (or Lin–Reissner–Tsien equation), [24, 32],

[TABLE]

The infinite normal prolongation, [4, 5, 31, 30], of the structure equations for the pseudogroup $Sym(\mathcal{E}_{1})$ of contact symmetries of this equation has the form

[TABLE]

where

[TABLE]

for $0\leq m\leq 4$, while $dh=0$ and $\theta_{3,0}=0$. We have

[TABLE]

where $p,q,a_{1},a_{2}\in\mathbb{R}$, $p\neq 0$, $q\neq 0$, are parameters.

Remark 3. All the other forms $\theta_{i,j}$ can be found inductively from the series of equations (5) – (10). For example, the first equation from the series of equations (6), $d\theta_{0,0}=\theta_{0,1}\wedge\theta_{0,0}$, has the form $dq\wedge dt=\theta_{0,1}\wedge q\,dt$, therefore we get $\theta_{0,1}=q^{-1}\,dq+b_{1}\,dt$ with $b_{1}\in\mathbb{R}$. Now forms $\theta_{0,0}$ and $\theta_{0,1}$ in the second equation $d\theta_{1,0}=\theta_{0,2}\wedge\theta_{0,0}$ from the series (6) are known, and we obtain $\theta_{0,2}=db_{1}+b_{2}\,dt$, $b_{2}\in\mathbb{R}$. Likewise, we can compute all the forms $\theta_{0,j}$. When $\theta_{0,j}$ and $\theta_{1,0}$ are known, from (7) it is possible to find $\theta_{1,j}$, etc. $\diamond$ Equations (5) – (10) imply $H^{1}(Sym(\mathcal{E}_{1}))=\mathbb{R}\,\alpha$. The following theorem describes the structure of the second exotic cohomology group for $Sym(\mathcal{E}_{1})$, for the proof of a more general result see Theorem 2 below. Theorem 1.

[TABLE]

where $\Omega=\theta_{3,1}\wedge\theta_{0,0}+\textstyle{\frac{2}{3}}\,\theta_{2,1}\wedge\theta_{1,0}+\textstyle{\frac{1}{3}}\,\theta_{1,1}\wedge\theta_{2,0}$. Corollary. Equation

[TABLE]

is compatible with the structure equations (5) – (10) of $Sym(\mathcal{E}_{1})$. Remark 4. If we rename $\omega=\theta_{3,0}$, then (12) gets the form

[TABLE]

Therefore, if we add $\theta_{3,0}$ as the coefficient at $h^{0}$ in the series for $\Theta_{3}$ in (11), then (9) remains valid. $\diamond$

Since equation (13) is compatible with system (5)–(10), Lie’s third inverse fundamental theorem in Cartan’s form, [4, 5, 31, 30], ensures existence of a solution $\theta_{3,0}$ to (13). We integrate this equation and put $a_{1}=v_{x}$. This yields

[TABLE]

This is the Wahlquist–Estabrook form of the Lax representation, [23, 9, 18, 35],

[TABLE]

for equation (4).

4 A deformation of the tensor product of the Lie algebra of vector fields on a line and the algebra of

truncated polynomials

The structure equations (5)–(10), (13) can be written in the form

[TABLE]

Then the results of Section 3 admit the following generalization. Definition. For $n\in\mathbb{N}$ and $\varepsilon\in\mathbb{R}$ denote by $\mathfrak{G}(n,\varepsilon)$ the Lie algebra with the structure equations

[TABLE]

where equations (11) hold for each $m\in\{0,1,\dots,n\}$.

Remark 5. The Lie algebra $\mathfrak{G}(n,\varepsilon)$ has the following description. Consider the tensor product $\mathbb{R}_{n+1}[s]\otimes C^{\omega}(\mathbb{R})$ of the algebra $\mathbb{R}_{n+1}[s]=\mathbb{R}[s]/\langle s^{n+1}\rangle$ of truncated polynomials of degree less that $n+1$ and the algebra of real-analytic functions of $t\in\mathbb{R}$. This is a vector space over $\mathbb{R}$ generated by functions $X_{k,m}=\frac{1}{m!}\,s^{k}\,t^{m}$, $k\in\{0,\dots,n\}$, $m\in\mathbb{N}\cup\{0\}$. This vector space is equipped with the Lie bracket

[TABLE]

The vector field $Y=s\,\partial_{s}$ is an outer derivative of the above Lie algebra. Then $\mathfrak{G}(n,\varepsilon)$ is the semi-direct sum of $\mathbb{R}_{n+1}[s]\otimes C^{\omega}(\mathbb{R})$ and one-dimensional Lie algebra generated by $Y$:

[TABLE]

This Lie algebra is a deformation of $\mathfrak{G}(n,0)$. For $\varepsilon=0$ the bracket (16) is the Lie bracket in the algebra of analytic vector fields on a line.

Consider 1-forms $\theta_{i,j}$, $\alpha$ dual to $X_{i,j}$, $Y$, that is, 1-forms on $\mathfrak{G}(n,\varepsilon)$ such that equations

[TABLE]

hold for all $i,j,k,m\in\mathbb{N}\cup\{0\}$. Then (17), (2), and (11) imply the structure equations (15). $\diamond$

System (15) is the infinite normal prolongation of the system

[TABLE]

This system is involutive, [4, §6], [28, Def. 11.7], therefore the third inverse fundamental Lie’s theorem in Cartan’s form implies existence of 1-forms $\alpha$, $\theta_{k,j}$, $k\in\{0,\dots,n\}$, $j\in\{0,1\}$, that satisfy (19). Forms $\alpha$, $\theta_{0,0}$, … , $\theta_{n,0}$ define a Lie pseudo-group on $\mathbb{R}^{n+1}$. While for the purposes of the present paper we need explicit expressions for the forms $\theta_{k,0}$ and $\theta_{k,1}$ only, all the other forms $\theta_{k,j}$ can be found inductively by integration of equations (15). We need the whole system (15) to prove Theorem 2 below.

We note that system (19) has the following specific structure. For each fixed $k\in\{0,\dots,n\}$ the first $k+1$ equations from (19) satisfy the conditions of the Frobenius theorem, see e.g. [12, Th. 1.3.4], for forms $\theta_{0,0}$, …, $\theta_{k,0}$. We consider $\alpha$ and $\theta_{j,0}$, $\theta_{j,1}$ for $j\in\{0,\dots,k\}$ as 1-forms on $\mathbb{R}^{N}$ for $N\geq 3\,(n+1)+1$. Eq. (18) implies that there is a function $p\neq 0$ on $\mathbb{R}$ such that222Here and below we put $\alpha=dp/p$ instead of the natural choice $\alpha=dp$ to simplify the further computations $\alpha=dp/p$. Then the Frobenius theorem implies existence of independent functions (coordinates) $x_{0}$, … , $x_{k}$ on $\mathbb{R}^{N}$ such that the differential ideal of forms $\theta_{0,0}$, … , $\theta_{k,0}$ is generated algebraically by forms $dx_{0}$, … , $dx_{k}$, that is, $\theta_{0,0}=A_{0}\,dx_{0}$ and $\theta_{i,0}=A_{i}\,\left(dx_{i}+\sum\limits_{j=0}^{i-1}B_{ij}dx_{j}\right)$, $i\in\{1,\dots,k\}$, for some functions $A_{j}\neq 0$, $B_{ij}$ on $\mathbb{R}^{N}$. Substituting these forms into (19) yields a triangular system for 1-forms $\theta_{i,1}$, so it is easy to find them.

Example 1. For $k=2$ from first three equations of (19) we have

[TABLE]

with $a_{0}\neq 0$. Then substituting for $\theta_{3,0}=A_{3}\,(dx_{3}+B_{30}dx_{0}+B_{31}dx_{1}+B_{32}dx_{2})$ into the equation for $d\theta_{3,0}$ from (19) allows one to define coefficients $A_{3}$, $B_{3j}$. After this, the equation for $d\theta_{3,0}$ defines $\theta_{3,1}$ up to adding the form $b_{3}\,dx_{0}$ with the new coordinate $b_{3}$ on $\mathbb{R}^{N}$. $\diamond$

The Lie algebra $\mathfrak{G}(n,\varepsilon)$ has the following important feature: when $\varepsilon=-\nicefrac{{1}}{{r}}$ for $r\in\mathbb{N}$ and $n\geq r$, 1-form $\theta_{r,0}$ has only two entries in the whole system (15). Indeed, the first equation from the series for $d\Theta_{r}$ acquires the form

[TABLE]

while neither 2-form $\Psi_{r}$ nor the other equations from (15) include $\theta_{r,0}$. Therefore in order to generalize results of Section 3 we consider the Lie algebra $\mathfrak{G}(r+1,\nicefrac{{-1}}{{r}})$ and proceed in the following steps. First, we find forms $\theta_{i,0}$ and $\theta_{i,1}$ for $0\leq i\leq r-1$ from equations (15) in the similar way as it was described above. Second, we assume that $\theta_{r+1,0}=A_{r+1}\,(du-u_{x_{0}}\,dx_{0}-\dots-u_{x_{r-1}}\,dx_{r-1})$, $A_{r+1}\neq 0$, that is, we consider $\theta_{r+1,0}$ as the zeroth order contact form on the jet bundle $J^{\infty}(\pi)$ for the bundle $\pi\colon(x_{0},x_{1},\dots,x_{r-1},u)\mapsto(x_{0},x_{1},\dots,x_{r-1})$. This implies some expressions for the parameters $B_{i,j}$ in terms of $u_{x_{0}}$, $\dots$, $u_{x_{r-1}}$. Third, we integrate equation (20) to find $\theta_{r,0}=b_{r}\,(dv-T_{0}\,dx_{0}-\dots-T_{r-1}\,dx_{r-1})$ and then consider the system of pdes $v_{x_{i}}=T_{i}$, $0\leq i\leq r-1$, generated by equation $\theta_{r,0}=0$.

5 Lie algebras $\mathfrak{G}(r+1,\nicefrac{{-1}}{{r}})$, their extensions, and associated

integrable systems

5.1 Integrable systems generated by $\mathfrak{G}(r+1,\nicefrac{{-1}}{{r}})$

In this subsection we present some results of computations discussed in the end of Section 4.

Example 2. For $\mathfrak{G}(5,\nicefrac{{-1}}{{4}})$ we have333Here and below we rescale parameters in forms $\theta_{j,0}$ in order to simplify the resulting pde and its Lax representation.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Then we integrate (20) with $r=4$ and put $a=v_{z}$. This gives

[TABLE]

[TABLE]

Then equation $\theta_{4,0}=0$ yields the system

[TABLE]

This system is compatible whenever equations

[TABLE]

hold. System (21) is the second system from the dKP hierarchy, [33, 13, 6, 18, 22, 35, 3, 10]. $\diamond$

Example 3. The Lie algebra $\mathfrak{G}(6,\nicefrac{{-1}}{{5}})$ provides the third system from the dKP hierarchy. In this case integration of (15) gives

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Then from (20) we get

[TABLE]

[TABLE]

[TABLE]

with $a=v_{s}$. This form generates the system

[TABLE]

This is the Lax representation for the third system from the dKP hierarchy

[TABLE]

$\diamond$

5.2 Right extensions of $\mathfrak{G}(n,\nicefrac{{-1}}{{r}})$

While it is natural to expect that Maurer-Cartan forms of $\mathfrak{G}(r+1,\nicefrac{{-1}}{{r}})$ with $r>5$ produce higher elements of the dKP hierarchy, an interesting question is to consider the case $r=2$. But the integration scheme from the end of Section 4 does not give any Lax representation of a pde in the case of $\mathfrak{G}(3,\nicefrac{{-1}}{{2}})$. Therefore we consider an extension of $\mathfrak{G}(n,\varepsilon)$. For $n\geq r\geq 1$ we denote by $\mathfrak{H}(n,r)$ the Lie algebra with the structure equations

[TABLE]

where $k^{\prime}\in\{0,\dots,r-1\}$ and $k^{\prime\prime}\in\{r,\dots,n\}$. Then $\mathfrak{H}(n,r)$ is a one-dimensional right extension, [11, § 1.4.4], of $\mathfrak{G}(n,\nicefrac{{-1}}{{r}})$. Indeed, from the structure equations (22) it follows that the dual element associated to the new 1-form $\beta$ is a derivative $Z\colon\mathfrak{G}(n,\nicefrac{{-1}}{{r}})\rightarrow\mathfrak{G}(n,\nicefrac{{-1}}{{r}})$ such that $Z\colon X_{k,m}\mapsto X_{k+r,m-1}$ for $m\geq 1$ and $k+r\leq n$, $Z\colon X_{k,m}\mapsto 0$ for $m=0$ or $k+r>n$, and $Z(Y(f))-Y(Z(f))=-r\,Y(f)$ for every $f\in\mathfrak{G}(n,\nicefrac{{-1}}{{r}})$, while there is no function $g\in\mathbb{R}_{n+1}[s]\otimes C^{\omega}(\mathbb{R})$ such that $[g,X_{k,m}]_{-1/r}=X_{k+r,m-1}$ for all $m\geq 1$ and $k+r\leq n$.

Example 4. Consider the Lie algebra $\mathfrak{H}(3,2)$. We have

[TABLE]

This assertion can be proven similarly to Theorem 2 below. Integration yields

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Then instead of equation $d\theta_{2,0}=2\,\alpha\wedge\theta_{2,0}+\Phi$ with $\Phi=\theta_{2,1}\wedge\theta_{0,0}+\textstyle{\frac{1}{2}}\,\theta_{1,1}\wedge\theta_{1,0}+\beta\wedge\theta_{0,1}$ from (22) we consider equation

[TABLE]

We rename $q=p\,\mathrm{exp}(v_{x})$ and get

[TABLE]

This form generates the system

[TABLE]

This system is compatible whenever the equation [8, 1, 29]

[TABLE]

holds. Thus system (23) defines an Abelian covering, [2], or a conservation law, for equation (24). To obtain non-Abelian covering for (24) we consider the form

[TABLE]

This form is a solution to the equation $d\omega=2\,\alpha\wedge\omega+\Psi$, where $\Psi=\Phi+2\,\alpha\wedge\beta+d_{-2\alpha}(\theta_{0,0}-\theta_{1,0})$ is $d_{-2\alpha}$–cohomologous to the cocycle $\Phi+2\,\alpha\wedge\beta$. Form (25) generates the Lax representation

[TABLE]

for (24). This system was derived in [29]. $\diamond$

Example 5. In the same way, for the Lie algebra $\mathfrak{H}(4,3)$ we have

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Instead of equation for $d\theta_{3,0}$ from (22) we consider equation

[TABLE]

Then for $q=p\,\mathrm{exp}(v_{x})$ we get the Wahlquist–Estabrook form

[TABLE]

[TABLE]

for the Lax representation

[TABLE]

of the system

[TABLE]

The first and the second equations of this system differ from the potential Khokhlov–Zabolotskaya equation (4) and equation (24), respectively, by notation. $\diamond$

We suppose that for algebras $\mathfrak{H}(r+1,r)$ the same computations will give higher elements of the integrable hierarchy generated by equation (24).

6 Second exotic cohomology group of $\mathfrak{G}(n,\varepsilon)$

To obtain further generalizations of the constructions from Section 5 we study the second exotic cohomology group of the Lie algebra $\mathfrak{G}(n,\varepsilon)$.

Theorem 2. Suppose $n\geq 2$, then for each fixed $r\in\{2,\dots,n\}$

[TABLE]

where

[TABLE]

For other values of $\lambda$ and $\varepsilon$ we have

[TABLE]

Proof. For forms $\eta$, $\omega_{1}$, … , $\omega_{k}$ such that $\mathrm{deg}\,\eta\geq\mathrm{deg}\,\omega_{i}$ denote by $\eta|_{\omega_{1}=0,\dots,\omega_{k}=0}$ the result of replacement of each entry of $\omega_{1}$, … , $\omega_{k}$ in $\eta$ by [math]. The condition $\eta=\eta|_{\omega_{1}=0,\dots,\omega_{k}=0}$ will be denoted as $\eta=o(\omega_{1},\dots,\omega_{k})$.

From (15) it follows that for each $k\geq 0$, $m\geq 0$

[TABLE]

hence for $U=-\varepsilon Y+X_{0,1}\in\mathfrak{G}(n,\varepsilon)$ we have

[TABLE]

Therefore $U$ defines an inner grading in $\mathfrak{G}(n,\varepsilon)$, [11, § 1.5.2], that is,

[TABLE]

where $\mathfrak{g}_{j}=\left\{X\in\mathfrak{G}(n,\varepsilon)\,\,|\,\,[U,X]=j\,X\right\}$ for $j\geq-1$ and $\mathfrak{g}_{j}=\{0\}$ for $j<-1$. We have $\mathfrak{g}_{0}=\langle Y,X_{0,1}\rangle$ and $\mathfrak{g}_{j}=\langle X_{k,j+1}\,\,|\,\,0\leq k\leq n\rangle$ for $j\in\mathbb{N}\cup\{-1\}$, so $\mathrm{dim}\,\mathfrak{g}_{j}\leq n$. From Remark 1 it follows that $H^{*}_{\lambda\alpha}(\mathfrak{G}(n,\varepsilon))$ is the cohomology of $\mathfrak{G}(n,\varepsilon)$ with coefficients in the module $A=\bigcup\limits_{\mu\in\mathbb{R}}A_{\mu}$, where

[TABLE]

For $m\geq 0$ and $\nu\in\mathbb{R}$ denote

[TABLE]

In particular,

[TABLE]

In accordance with [11, Th. 1.5.2a] the inclusion $C_{(0)}^{*}(\mathfrak{G}(n,\varepsilon),A)\rightarrow C^{*}(\mathfrak{G}(n,\varepsilon),A)$ induces a cohomological isomorphism. Hence, from (26) it follows that $H^{2}_{\lambda\alpha}(\mathfrak{G}(n,\varepsilon))$ is finite-dimensional for each fixed pair $\lambda$, $\varepsilon$, and for an arbitrary solution $\Omega$ of equation $d\Omega+\lambda\,\alpha\wedge\Omega=0$ we can assume without loss of generality that $M=\mathrm{min}\,\{j\in\mathbb{N}\cup\{0\}\,\,|\,\,\Omega=o(\theta_{0,j+1},\theta_{1,j+1},\dots,\theta_{n,j+1})\,\}<\infty$. Then $\Omega=\sum\limits_{j=0}^{n}\gamma_{j}\wedge\theta_{j,M}$ with $\gamma_{j}=o(\theta_{j,M},\theta_{j+1,M},\dots\theta_{n,M})$. Suppose $M\geq 1$. From the structure equations (15) it follows that

[TABLE]

Hence $d\Omega+\lambda\,\alpha\wedge\Omega=\gamma_{n}\wedge\theta_{n,M+1}\wedge\theta_{0,0}+o(\theta_{n,M+1})=0$ yields $\gamma_{n}=c\,\theta_{0,0}$ with $c\in\mathbb{R}$. Thus $\Omega-d_{\lambda\,\alpha}(c\,\theta_{n,M-1})=o(\theta_{n,M})$. In other words, we can put $\gamma_{n}=0$ without loss of generality. In the same way we can consequently put $\gamma_{n-1}=0$, then $\gamma_{n-2}=0$, etc., $\gamma_{0}=0$. So we have $M=0$ and

[TABLE]

with $a_{ij},b_{j}\in\mathbb{R}$.

Then we have $d\Omega+\lambda\,\alpha\wedge\Omega=-b_{j}\,\alpha\wedge\theta_{j,1}\wedge\theta_{0,0}+o(\alpha\wedge\theta_{j,1}\wedge\theta_{0,0})$ for each $j\in\{0,1,\dots,n\}$, so $b_{0}=b_{1}=\dots=b_{n}=0$. We assume that $a_{ij}\neq 0$ for some $i$, $j$, then equation $d\Omega+\lambda\,\alpha\wedge\Omega=a_{ij}\,(i+j+\lambda)\,\alpha\wedge\theta_{i,0}\wedge\theta_{j,0}+o(\theta_{i,0}\wedge\theta_{j,0})$ yields $i+j=r$, where $r\in\mathbb{N}$ is a constant, and $\lambda=-r$. Therefore, $\Omega=\sum\limits_{j=0}^{[r/2]}c_{j}\theta_{j,0}\wedge\theta_{r-j,0}$ for some $c_{j}\in\mathbb{R}$.

We claim that $r\leq n$. Indeed, assume $r>n$ and put $r_{0}=\mathrm{min}\,\{j\in\mathbb{N}\,\,|\,\,\Omega=o(\theta_{j+1,0})\}$. Since $r_{0}\leq n$, we have $r-r_{0}\geq 1$. Then from $\Omega=c_{r-r_{0}}\,\theta_{r-r_{0},0}\wedge\theta_{r_{0},0}+o(\theta_{r_{0},0})$ we have $c_{r-r_{0}}\neq 0$. But $d\Omega-r\,\alpha\wedge\Omega=-c_{r-r_{0}}\,\theta_{r-r_{0},0}\wedge\theta_{r_{0},1}\wedge\theta_{0,0}+o(\theta_{r_{0},1})$ implies $c_{r-r_{0}}=0$. The contradiction proves the claim.

Consider the case of even $r$ and put $r=2\,m$ for $m\in\mathbb{N}$. Then $\Omega=\sum\limits_{j=0}^{m-1}c_{j}\theta_{j,0}\wedge\theta_{2\,m-j,0}$ and

[TABLE]

So we get $c_{j}=c_{0}\,(1+\varepsilon\,j)$ and thus

[TABLE]

Finally we have

[TABLE]

and hence $\varepsilon=-\frac{1}{m}=-\frac{2}{r}$. Substituting this into (27) with $c_{0}=-r$ yields $\Omega=\Phi_{r}$.

In the case $r=2\,m+1$ the proof is similar. $\square$

Corollary. For each $r\in\{2,\dots,n\}$ the equation

[TABLE]

is compatible with the structure equations of $\mathfrak{G}(n,-\nicefrac{{2}}{{r}})$.

Example 6. In the case of $n=r=3$ integration of the structure equations of $\mathfrak{G}(3,\nicefrac{{-2}}{{3}})$ gives

[TABLE]

[TABLE]

[TABLE]

[TABLE]

while equation (28) acquires the form

[TABLE]

We integrate this to obtain the form $\omega$, then rename $q=p^{-1}\,v_{x}^{-1}$ and consider

[TABLE]

This form defines the Lax representation

[TABLE]

for the equation

[TABLE]

This Lax representation was found in [25]. $\diamond$

Example 7. In the case $n=r=4$ instead of $\mathfrak{G}(4,\nicefrac{{-1}}{{2}})$ we consider the Lie algebra $\mathfrak{H}(4,2)$. We have

[TABLE]

where

[TABLE]

Consider the case $\lambda=-4$. From the structure equations of $\mathfrak{H}(4,2)$ we obtain

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Then we integrate the equation

[TABLE]

and get

[TABLE]

After the change of notation $q=p^{-1}\,v_{z}^{-1}$, $w=v_{x}+u_{z}\,v_{z}$ we have

[TABLE]

[TABLE]

This is the Wahlquist–Estabrook form for the Lax representation

[TABLE]

of the equation

[TABLE]

[TABLE]

$\diamond$

7 Concluding remarks

In the present paper we have shown that both known (Examples 2, 3, 4, and 6) and new (Examples 5 and 7) Lax representations for integrable pdes can be derived from the second exotic cohomology group of symmetry pseudo-groups of the pdes under the study. This approach gives the solution to the problem of existence of a Lax representation in internal terms of the pde and allows one to eliminate apriori assumptions about the possible form of the Lax representation. The approach is universal and can be used to analyze a lot of equations or Lie algebras with nontrivial second exotic cohomology group. Quite natural this leads to the question of generalization of the Lie algebras considered above. In particular, it seems to be important to study the adjoint cohomology group $H^{1}(\mathfrak{G}(n,\varepsilon),\mathfrak{G}(n,\varepsilon))$ in order to describe right extensions of $\mathfrak{G}(n,\varepsilon)$ and to generalize the Lie algebras $\mathfrak{H}(n,r)$. Another challenge is to replace the Lie algebra of vector fields on $\mathbb{R}$ by Lie algebras of vector fields on $\mathbb{R}^{n}$ in the constructions of $\mathfrak{G}(n,\varepsilon)$ and $\mathfrak{H}(n,r)$, to consider exotic cohomology of the resulting Lie algebras and to study whether there exist related integrable systems. For example, the symmetry pseudo-groups of the heavenly equations and their deformations are right extensions of Lie algebras of the form $\mathcal{A}\otimes\mathfrak{h}$, where $\mathfrak{h}$ is the Lie algebra of Hamiltonian vector fields on $\mathbb{R}^{2}$ and $\mathcal{A}$ are algebras of truncated polynomials, [19, 20]. Other interersting examples include equations discussed in [21]. It is natural to elucidate the relationship among the Lax representations and the structure of symmetry pseudogroups for the above-mentioned integrable equations. We intend to address this issue in our future work.

Acknowledgments

This research was supported in part by the Polish Ministry of Science and Higher Education. I am grateful to the organizers of the conference “Geometry and Algebra of PDEs” held at the Arctic University of Norway, Tromsø, 6–10 June 2017, for financial support.

I am pleased to thank V.V. Lychagin, I.S. Krasil*′*shchik, P.J. Olver, B.S. Kruglikov, D.V. Alekseevsky, B.A. Khesin and P. Zusmanovich for useful and stimulating discussions, and to V.S. Gerdjikov for pointing out the reference [8] and providing the copy thereof.