Lax representations with non-removable parameter and exotic cohomology of symmetry algebras of PDEs

TL;DR

This paper introduces a method to construct Lax representations with non-removable spectral parameters for PDEs using non-central extensions of their contact symmetry algebras, advancing the understanding of integrability structures.

Contribution

It develops a novel technique for deriving Lax representations with non-removable parameters through non-central extensions of symmetry algebras.

Findings

Applicable to PDEs with contact symmetry algebras

Enables construction of Lax pairs with non-removable spectral parameters

Provides new insights into the cohomology of symmetry algebras

Abstract

This paper develops the technique of constructing Lax representations for PDEs via non-central extensions of their contact symmetry algebras. We show that the method is applicable to the Lax representations with non-removable spectral parameters.

Equations

$$u_{yy}=u_{tx}+u_y{u}_{xx}-u_x{u}_{xy}$$

$$\left\{\begin{array}[]{lcl}v_{t}&=&(\lambda^{2}+\lambda\,u_{x}-u_{y})\,v_{x},\\ v_{y}&=&-(u_{x}+\lambda)\,v_{x},\end{array}\right.$$

$$u_{zz}=u_{tx}+u_x{u}_{yz}-u_z{u}_{xy}$$

$$u_{zz}=u_x{u}_{yz}-u_z{u}_{xy},$$

$$\left\{\begin{array}[]{lcl}v_{t}&=&\lambda^{2}\,v_{x}+(\lambda\,u_{x}+u_{z})\,v_{y},\\ v_{z}&=&\lambda\,v_{x}+u_{x}\,v_{y}\end{array}\right.$$

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

$${\mathbf{E}}_{\phi }=\sum _{\#I\ge 0}\sum _{\alpha =1}^m{D}_I({\phi }^{\alpha })\frac{\partial }{\partial u_I^{\alpha }}$$

$${\ell }_{\mathcal{E}}(\phi )=0,$$

$${\ell }_F=\left(\sum _{\#I\ge 0}\frac{\partial F_r}{\partial u_I^{\alpha }}{D}_I\right).$$

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

$$w_{x^k}^s=T_k^s(x^i,u_I^{\alpha },w^j),$$

$$d{w}^s-\sum _{k=1}^m{T}_k^s(x^i,u_I^{\alpha },w^j)d{x}^k$$

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

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

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

$$d_{c\alpha }\theta =d\theta -c\alpha \wedge \theta .$$

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

$${\varphi }_0(A)=-A{u}_t-\left(x{A}^{\prime }+\frac{1}{2}{y}^2{A}^{\prime \prime }\right)u_x-y{A}^{\prime }{u}_y+u{A}^{\prime }+xy{A}^{\prime \prime }+\frac{1}{6}{y}^3{A}^{\prime \prime \prime },$$

$${\varphi }_1(A)=-y{A}^{\prime }{u}_x-A{u}_y+x{A}^{\prime }+\frac{1}{2}{y}^2{A}^{\prime \prime },$$

$${\varphi }_2(A)=-A{u}_x+y{A}^{\prime },$$

$${\varphi }_3(A)=A,$$

$${\psi }_0=-2x{u}_x-y{u}_y+3u,$$

$${\psi }_1=-y{u}_x+2x$$

$$\left\{\begin{array}[]{lcl}\{\phi_{i}(A),\phi_{j}(B)\}&=&\phi_{i+j}(A\,B^{\prime}-B\,A^{\prime}),\\ \{\psi_{i},\phi_{k}(A)\}&=&-k\,\phi_{k+i}(A),\\ \{\psi_{0},\psi_{1}\}&=&-\psi_{1},\end{array}\right.$$

$$\left\{\begin{array}[]{lcl}d\alpha_{0}&=&0,\\ d\alpha_{1}&=&\alpha_{0}\wedge\alpha_{1},\\ d\Theta&=&\nabla_{1}(\Theta)\wedge\Theta+(h_{2}\,\alpha_{0}+h_{2}^{2}\,\alpha_{1})\wedge\nabla_{2}(\Theta)\end{array}\right.$$

$$H^{2}_{c\,\alpha_{0}}(\mathfrak{b}_{\diamond})=\left\{\begin{array}[]{lcl}\langle[\alpha_{0}\wedge\alpha_{1}]\rangle,&\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\hfil&c=1,\\ \{[0]\},&&c\neq 1.\end{array}\right.$$

$$d\sigma ={\alpha }_0\wedge \sigma +{\alpha }_0\wedge {\alpha }_1.$$

$${\alpha }_0=\frac{dq}{q},{\alpha }_1=qds,\sigma =q(dv+\ln qds),$$

$$\sigma -{\theta }_{2,0}=q\left(dv+\ln qds-a_{0}q\left(dx-(s+u_x)dy+(s^2+s{u}_x-u_y)dt\right)\right)$$

$$\sigma -{\theta }_{2,0}={\mathrm{e}}^{-v_s}\left(dv-v_{s}ds-v_{x}dx+(s+u_x)dy-(s^2+s{u}_x-u_y)dt\right)$$

Full text

1 Introduction

Lax representations, also known as zero-curvature representations, Wahlquist–Estabrook prolongation structures, inverse scattering transformations, or differential coverings [21, 22], are a key feature of integrable partial differential equations (pdes) and a starting setting for a number of techniques of studying them such as Bäcklund transformations, Darboux transformations, recursion operators, nonlocal symmetries, and nonlocal conservation laws. Lax representations with non-removable (spectral) parameter are of special interest in the theory of integrable pdes, see, e.g., [1, 12, 13, 41]. The challenging unsolved problem in this theory is to find conditions that are formulated in inherent terms of a pde under study and ensure existence of a Lax representation for the pde. Recently, an approach to this problem has been proposed in [34, 35], where it was shown that for some pdes their Lax representations can be inferred from the second exotic cohomology of the contact symmetry algebras of the pdes.

The present paper provides an important supplement to the technique of [34, 35]. Namely, we show that Lax representations with a non-removable parameter arise naturally from non-central extensions of the symmetry algebras generated by nontrivial second exotic cohomology groups. We consider here two equations: the hyper-CR Einstein–Weyl equation [24, 29, 39, 15]

[TABLE]

with the Lax representation

[TABLE]

and the four-dimensional equation

[TABLE]

which was introduced in [4]. The 3-dimensional reduction of (3) defined by substitution for $u_{t}=0$ produces the universal hierarchy equation [26, 27]

[TABLE]

therefore we refer equation (3) as the four-dimensional universal hierarchy equation. The Lax representation

[TABLE]

for (3) was found in [40].

The parameters $\lambda$ in (2) and (4) are non-removable. This assertion can be proven by the method of [22, Sections 3.2, 3.6], [18, 17, 28, 9, 10, 11], see Remarks 1 and 3 below.

The following structure distinguishes the contact symmetry algebras for both equations: they are semi-direct products $\mathfrak{s}_{\infty}\rtimes\mathfrak{s}_{\diamond}$ of an (invariantly defined) infinite-dimensional ideal $\mathfrak{s}_{\infty}$ and a non-Abelian finite-dimensional Lie algebra $\mathfrak{s}_{\diamond}$. The second exotic cohomology groups of the finite-dimensional subalgebras $\mathfrak{s}_{\diamond}$ appear to be nontrivial for both equations, and the corresponding nontrivial 2-cocycles produce non-central extensions of their symmetry algebras $\mathfrak{s}_{\infty}\rtimes\mathfrak{s}_{\diamond}$. We show that certain linear combination of the Maurer–Cartan forms of the extension of the symmetry algebra defines the Lax representation (2) for equation (1). For equation (3) we apply the extension procedure twice and show that the Maurer–Cartan forms of the second extension produce the Lax representation (4).

2 Preliminaries

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

2.1 Symmetries and differential coverings

The presentation in this subsection closely follows [19, 20], see also [21, 22, 42]. 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 $I=(i_{1},\dots,i_{n})$ are multi-indices, 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, we will simplify notation in the following way, e.g., in the case of $n=4$, $m=1$: 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]

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

The evolutionary vector field associated to an arbitrary vector-valued smooth function $\varphi\colon J^{\infty}(\pi)\rightarrow\mathbb{R}^{m}$ is the vector field

[TABLE]

with $D_{I}=D_{(i_{1},\dots\,i_{n})}=D^{i_{1}}_{x^{1}}\circ\dots\circ D^{i_{n}}_{x^{n}}$.

A system of pdes $F_{r}(x^{i},u^{\alpha}_{I})=0$ of the order $s\geq 1$ with $\#I\leq s$, $r\in\{1,\dots,R\}$ for some $R\geq 1$, 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 $J^{\infty}(\pi)$.

A function $\varphi\colon J^{\infty}(\pi)\rightarrow\mathbb{R}^{m}$ is called a (generator of an infinitesimal) symmetry of equation $\mathcal{E}$ when $\mathbf{E}_{\varphi}(F)=0$ on $\mathcal{E}$. The symmetry $\varphi$ is a solution to the defining system

[TABLE]

where $\ell_{\mathcal{E}}=\ell_{F}|_{\mathcal{E}}$ with the matrix differential operator

[TABLE]

The symmetry algebra $\mathfrak{sym}(\mathcal{E})$ of equation $\mathcal{E}$ is the linear space of solutions to (5) endowed with the structure of a Lie algebra over $\mathbb{R}$ by the Jacobi bracket $\{\varphi,\psi\}=\mathbf{E}_{\varphi}(\psi)-\mathbf{E}_{\psi}(\varphi)$. The algebra of contact symmetries $\mathfrak{sym}_{0}(\mathcal{E})$ is the Lie subalgebra of $\mathfrak{sym}(\mathcal{E})$ defined as $\mathfrak{sym}(\mathcal{E})\cap J^{1}(\pi)$.

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

[TABLE]

such that $[\widetilde{D}_{x^{i}},\widetilde{D}_{x^{j}}]=0$ for all $i\not=j$ whenever $(x^{i},u^{\alpha}_{I})\in\mathcal{E}$. Define the partial derivatives of $w^{s}$ by $w^{s}_{x^{k}}=\widetilde{D}_{x^{k}}(w^{s})$. This yields the system of covering equations

[TABLE]

which is compatible whenever $(x^{i},u^{\alpha}_{I})\in\mathcal{E}$.

Dually, the covering is defined by the Wahlquist–Estabrook forms

[TABLE]

as follows: when $w^{s}$ and $u^{\alpha}$ are considered to be functions of $x^{1}$, … , $x^{n}$, forms (7) are equal to zero whenever system (6) holds.

2.2 Exotic cohomology of Lie algebras

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

[TABLE]

is generated by the differential $d\colon\theta\mapsto d\theta$ such that

[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 coefficients in the representation $\rho$. For the trivial representation $\rho_{0}\colon\mathfrak{g}\rightarrow\mathbb{R}$, $\rho_{0}\colon X\mapsto 0$, the cohomology groups are denoted by $H^{*}(\mathfrak{g})$.

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 $\mathfrak{g}$ such that $[\alpha]\neq 0$. Then for any $c\in\mathbb{R}$ define new differential $d_{c\alpha}\colon C^{k}(\mathfrak{g},\mathbb{R})\rightarrow C^{k+1}(\mathfrak{g},\mathbb{R})$ by the formula

[TABLE]

From $d\alpha=0$ it follows that $d_{c\alpha}^{2}=0$. The cohomology groups of the complex

[TABLE]

are referred to as the exotic cohomology groups [36, 37] of $\mathfrak{g}$ and denoted by $H^{*}_{c\alpha}(\mathfrak{g})$.

3 Hyper-CR Einstein–Weyl equation

3.1 Contact symmetries

Direct computations111We carried out computations of generators of contact symmetries and their commutator tables in the Jets software [2]. show that the Lie algebra of the contact symmetries $\mathfrak{sym}_{0}(\mathcal{E}_{1})$ of the hyper-CR Einstein–Weyl equation $\mathcal{E}_{1}$ is generated by the functions

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $A=A(t)$ and $B=B(t)$ below are arbitrary functions of $t$. The commutators of the generators are given by equations

[TABLE]

where $\phi_{k}(A)=0$ for $k>3$. From equations (8) it follows that the contact symmetry algebra of equation (1) is the semi-direct product $\mathfrak{sym}_{0}(\mathcal{E}_{1})=\mathfrak{b}_{\infty}\rtimes\mathfrak{b}_{\diamond}$ of the two-dimensional non-Abelian Lie algebra $\mathfrak{b}_{\diamond}=\langle\psi_{0},\psi_{1}\rangle$ and the infinite-dimensional ideal111Here and below we use notation $\mathcal{D}(\mathfrak{g})=\mathcal{D}^{1}(\mathfrak{g})=[\mathfrak{g},\mathfrak{g}]$, $\mathcal{D}^{k+1}(\mathfrak{g})=[\mathcal{D}^{k}(\mathfrak{g}),\mathcal{D}^{k}(\mathfrak{g})]$ for the derived series of a Lie algebra $\mathfrak{g}$. $\mathfrak{b}_{\infty}=\mathcal{D}(\mathfrak{sym}_{0}(\mathcal{E}_{1}))=\langle\phi_{k}(A)\,\,\,|\,\,\,0\leq k\leq 3\rangle$, which, in its turn, is isomorphic to the tensor product $\mathbb{R}_{3}[h]\otimes\mathfrak{w}$ of the (commutative associative) algebra of truncated polynomials $\mathbb{R}_{3}[h]=\mathbb{R}[h]/\langle h^{4}=0\rangle$ and the Lie algebra $\mathfrak{w}=\langle t^{n}\partial_{t}\,\,\,|\,\,\,n\in\mathbb{Z}_{+}\rangle$.

Remark 1. The spectral parameter $\lambda$ in the covering (2) is non-removable, that is, it cannot be eliminated from (2) by a transformation of the covering that is identical on the base equation $\mathcal{E}_{1}$. In accordance with [22, 18, 17], to prove this claim it suffices to note that it is impossible to lift the symmetry $\psi_{1}$ of equation (1) to a symmetry of system (2). Then for the vector field $V(\psi_{1})=y\,\partial_{x}+2\,x\,\partial_{u}$ associated to the generator $\psi_{1}$ and the Wahlquist–Estabrook form $\omega=dv+u_{y}\,v_{x}\,dt-v_{x}\,dx+u_{x}\,v_{x}\,dy$ that defines covering (2) with $\lambda=0$ we have $e^{-\lambda V(\psi_{1})}\omega=dv-(\lambda^{2}+\lambda\,u_{x}-u_{y})\,v_{x}\,dt-v_{x}\,dx+(u_{x}+\lambda)\,v_{x}\,dy$. This is the Wahlquist–Estabrook form of covering (2) with $\lambda\in\mathbb{R}$. $\diamond$

Consider the basis of $\mathfrak{sym}_{0}(\mathcal{E}_{1})$ given by generators $\psi_{0}$, $\psi_{1}$, and $\phi_{k}(t^{n})$ with $k\in\{0,\dots,3\}$, $n\in\mathbb{Z}_{+}$. Define the Maurer–Cartan forms $\alpha_{0}$, $\alpha_{1}$, $\theta_{k,n}$ for $\mathfrak{sym}_{0}(\mathcal{E}_{1})$ as the dual forms to this basis: $\alpha_{i}(\psi_{j})=\delta_{ij}$, $\alpha_{i}(\phi_{k}(t^{n}))=0$, $\theta_{k,n}(\psi_{i})=0$, $\theta_{k,n}(\phi_{l}(t^{m}))=\delta_{kl}\delta_{nm}$. Put $\Theta=\sum\limits_{k=0}^{3}\sum\limits_{m=0}^{\infty}\frac{h_{1}^{m}}{m!}\,h_{2}^{k}\,\theta_{k,m}$, where $h_{1}$ and $h_{2}$ are (formal) parameters such that $dh_{i}=0$, and denote $\nabla_{i}=\partial_{h_{i}}$. Then the commutator table (8) gives the Maurer–Cartan structure equations

[TABLE]

of $\mathfrak{sym}_{0}(\mathcal{E}_{1})$.

3.2 Second exotic cohomology group and non-central extension

From the structure equations (9) it follows that $H^{1}(\mathfrak{sym}_{0}(\mathcal{E}_{1}))=\mathbb{R}\alpha_{0}$ and

[TABLE]

Moreover, we have $H^{2}_{\alpha_{0}}(\mathfrak{b}_{\diamond})\subseteq H^{2}_{\alpha_{0}}(\mathfrak{sym}_{0}(\mathcal{E}_{1}))$. Hence the nontrivial 2-cocycle $\alpha_{0}\wedge\alpha_{1}$ of the differential $d_{\alpha_{0}}$ defines a non-central extension $\widehat{\mathfrak{b}}_{\diamond}$ of the Lie algebra $\mathfrak{b}_{\diamond}$ and thus a non-central extension $\mathfrak{b}_{\infty}\rtimes\widehat{\mathfrak{b}}_{\diamond}$ of the Lie algbera $\mathfrak{sym}_{0}(\mathcal{E}_{1})$. The additional Maurer–Cartan form $\sigma$ for the extended Lie algebra is a solution to $d_{\alpha_{0}}\sigma=\alpha_{0}\wedge\alpha_{1}$, that is, to equation

[TABLE]

This equation is automatically compatible with the structure equations (9) of the Lie algebra $\mathfrak{sym}_{0}(\mathcal{E}_{1})$.

3.3 Maurer–Cartan forms and Lax representation

For the purposes of the present paper we need explicit expressions for the Maurer–Cartan forms $\alpha_{i}$, $\theta_{k,0}$, and $\sigma$. We can compute them via two approaches. The first one is to integrate equations (9), (10) step by step. Each integration gives certain number of new coordinates (the ‘integration constants’) to express the new form, while it is not clear how these coordinates relate to the coordinates of $\mathcal{E}_{1}$. For example, integrating the first two equations from system (9) and then equation (10) we obtain

[TABLE]

where $q$, $s$, and $v$ are free parameters and $q>0$ (we put $\alpha_{0}=dq/q$ instead of the natural choice $\alpha_{0}=dq$ to simplify the further computations). The second approach to computing the Maurer–Cartan forms is to use Cartan’s method of equivalence, [5, 6, 7, 8, 38, 16], see details and examples of applying the method to symmetry pseudo-groups to pdes in [30, 31, 32]. In case of the structure equations (9) this technique shows that

1. (i)

$\theta_{0,0}$ is a multiple of $dt$, $\theta_{1,0}$ is a linear combination of $dy$, $dt$, $\theta_{2,0}$ is a linear combination of $dx$, $dy$, $dt$, 2. (ii)

$\theta_{3,0}$ is a multiple of the contact form $du-u_{t}\,dt-u_{x}\,dx-u_{y}\,dy$.

Using (i) we have $\theta_{0,0}=a_{0}\,dt$, $\theta_{1,0}=a_{0}\,q\,(dy+a_{1}\,dt)$, $\theta_{2,0}=a_{0}\,q^{2}\,(dx+(a_{1}+s)\,dy+a_{2}\,dt)$, with new parameters $a_{0}\neq 0$, $a_{1}$, $a_{2}$, while (ii) then gives $a_{1}=-u_{x}-2\,s$, $a_{2}=-u_{y}+s\,u_{x}+s^{2}$, and $\theta_{3,0}=a_{0}\,q^{3}\,(du-u_{t}\,dt-u_{x}\,dx-u_{y}\,dy)$.

Consider the linear combination

[TABLE]

and assume that $u$ and $v$ are functions of $t$, $x$, $y$. Then $\sigma-\theta_{2,0}=0$ implies $q=\exp(-v_{s})$, $a_{0}=v_{x}\,\exp(v_{s})$. After this change of notation we obtain the Wahlquist–Estabrook form

[TABLE]

of the covering

[TABLE]

This system differs from (2) by notation.

Remark 2. The non-removable parameter $s$ in the above system has appeared during computation of form $\alpha_{1}$, which is dual to the unliftable symmetry $\psi_{1}$, cf. Remark 1. $\diamond$

4 The four-dimensional universal hierarchy equation

4.1 Contact symmetries

The generators of the Lie algebra $\mathfrak{sym}_{0}(\mathcal{E}_{2})$ of the contact symmetries of equation (3) are

[TABLE]

where $A=A(t,y)$ and $B=B(t,y)$ below are arbitrary functions of $t$ and $y$. The commutators of the generators are given by equations

[TABLE]

[TABLE]

[TABLE]

while $\{\psi_{i},\psi_{j}\}=0$ for all the other pairs $i<j$. The commutator table shows that the contact symmetry algebra of equation (3) is a semi-direct sum $\mathfrak{sym}(\mathcal{E}_{2})=\mathfrak{c}_{\infty}\rtimes\mathfrak{c}_{\diamond}$ of the six-dimensional Lie algebra $\mathfrak{c}_{\diamond}=\langle\psi_{i}\,\,|\,\,i\in\{1,\dots,6\}\rangle$ and the infinite-dimensional ideal $\mathfrak{c}_{\infty}=\mathcal{D}^{3}(\mathfrak{sym}(\mathcal{E}_{2}))=\langle\phi_{k}(A)\,\,|\,\,k\in\{0,1\}\rangle$, which, in its turn, is isomorphic to the tensor product $\mathbb{R}_{2}[h]\otimes\mathfrak{q}$ of the algebra of truncated polynomials $\mathbb{R}_{2}[h]=\mathbb{R}[h]/\langle h^{2}=0\rangle$ and the Lie algebra $\mathfrak{q}$ of the vector fields of the form $A(t,y)\,\partial_{y}$.

Remark 3. The spectral parameter $\lambda$ in the covering (4) is non-removable. Indeed, the symmetry $\psi_{4}$ of equation (3) is unliftable to a symmetry of the covering (4). For the associated vector field $V(\psi_{4})=\frac{1}{2}\,z\,\partial_{x}+t\,\partial_{z}$ the action of $\mathrm{exp}(2\,\lambda\,V(\psi_{4}))$ to the Wahlquist–Estabrook form $\omega=dv-u_{z}\,v_{y}\,dt-v_{x}\,dx-v_{y}\,dy-u_{x}\,v_{y}\,dz$ that defines covering (4) with $\lambda=0$ gives $e^{2\,\lambda V(\psi_{4})}\omega=dv-\left(\lambda^{2}\,v_{x}+(\lambda\,u_{x}+u_{z})\,v_{x}\right)\,dt-v_{x}\,dx-v_{y}\,dy-(\lambda\,v_{x}+u_{x}\,v_{y})\,dz$. This is the Wahlquist–Estabrook form of covering (4) with $\lambda\in\mathbb{R}$. $\diamond$

Define the Maurer–Cartan forms $\beta_{i}$, $\theta_{k,m,n}$, $i\in\{1,\dots,6\}$, $k\in\{0,1\}$, $m,n\in\mathbb{Z}_{+}$, of the Lie algebra of $\mathfrak{sym}(\mathcal{E}_{2})$ as dual forms to its basis $\psi_{i}$, $\phi_{k}(t^{m}y^{n})$, that is, put $\beta_{i}(\psi_{j})=\delta_{ij}$, $\beta_{i}(\phi_{k}(t^{m}y^{n}))=0$, $\theta_{k,m,n}(\psi_{i})=0$, $\theta_{k,m,n}(\phi_{p}(t^{q}y^{r}))=\delta_{kp}\delta_{mq}\delta_{nr}$. Denote $\Theta_{k}=\sum\limits_{m=0}^{\infty}\sum\limits_{n=0}^{\infty}\frac{h_{1}^{m}}{m!}\frac{h_{2}^{n}}{n!}\theta_{k,m,n}$ for the formal parameters $h_{1}$, $h_{2}$ such that $dh_{i}=0$. Then the Maurer-Cartan structure equations of $\mathfrak{sym}(\mathcal{E}_{2})$ read

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

4.2 Non-central extensions, Maurer–Cartan forms and Lax representation

From the structure equations (11) – (18) it follows that $H^{1}(\mathfrak{sym}(\mathcal{E}_{2}))=\langle\beta_{1},\beta_{3}\rangle$ and

[TABLE]

Moreover, all the nontrivial exotic 2-cocycles of $\mathfrak{c}_{\diamond}$ are nontrivial exotic 2-cocycles of $\mathfrak{sym}_{0}(\mathcal{E}_{2})$ as well. Therefore they define a non-central extension $\widehat{\mathfrak{c}}_{\diamond}$ of the Lie algebra $\mathfrak{c}_{\diamond}$ and hence a non-central extension $\mathfrak{c}_{\infty}\rtimes\widehat{\mathfrak{c}}_{\diamond}$ of the Lie algebra $\mathfrak{sym}_{0}(\mathcal{E}_{2})$. The additional Maurer–Cartan forms $\beta_{7}$, … , $\beta_{12}$ for the extended Lie algebra are solutions to the system

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

This system is automatically compatible with equations (11) – (18).

Combining direct integration of the structure equations with Cartan’s method of equivalence we get the explicit expressions for the Maurer–Cartan forms

[TABLE]

[TABLE]

[TABLE]

as well as the requirement for the linear combination

[TABLE]

to be a multiple of the contact form $du-u_{t}\,dt-u_{x}\,dx-u_{y}\,dy-u_{z}\,dz$. This gives the change of notation $r=a_{0}\,u_{x}$, $a_{1}=u_{z}-s\,u_{x}$, $a_{2}=u_{y}$, $a_{3}=u_{t}-s^{2}\,u_{x}$, which yields

[TABLE]

Our attempts to find a linear combination of the Maurer–Cartan forms $\beta_{1}$, … , $\beta_{12}$, $\theta_{k,m,n}$ have not given a Wahlquist–Estabrook form of any covering over equation (3). Therefore we have extended the Lie algebra $\widehat{\mathfrak{c}}_{\diamond}$ with the structure equations (11) – (24) via the same procedure, that is, by finding nontrivial cocycles from $H^{2}_{c_{1}\beta_{1}+c_{2}\beta_{3}}(\widehat{\mathfrak{c}}_{\diamond})$. The additional Maurer–Cartan forms $\beta_{13}$, … , $\beta_{30}$ of the resulting 18-dimensional non-central extension of $\widehat{\mathfrak{c}}_{\diamond}$ and of $\mathfrak{c}_{\infty}\rtimes\widehat{\mathfrak{c}}_{\diamond}$ are solutions to the structure equations

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

These equations are compatible with equations (11) – (24). Integration of equation (25) gives $\beta_{13}=a_{0}^{2}\,u_{x}^{2}q^{-2}\,(dv-a_{4}\,ds)$. Then the linear combination

[TABLE]

[TABLE]

after the change of notation $q=v_{x}$, $a_{0}=v_{x}^{2}u_{x}^{-2}v_{y}^{-1}$, $a_{4}=v_{s}$ gives

[TABLE]

[TABLE]

which is the Wahlquist–Estabrook form of the covering

[TABLE]

This system differs from (4) by the notation.

Remark 4. The non-removable parameter $s$ in the above system has appeared during computation of form $\beta_{4}$, which is dual to the unliftable symmetry $\psi_{4}$, cf. Remark 3. $\diamond$

5 Conclusion

In the present paper we have shown that the method of [34, 35] is applicable to Lax representations with non-removable parameters, in particular, the Lax representations for equations (1) and (3) can be derived from the non-central extensions of contact symmetry algebras of these equations. In both examples the symmetry algebras have the specific structure of the semi-direct product of an infinite-dimensional ideal and a non-Abelian finite-dimensional Lie subalgebra. The cohomological properties of the finite-dimensional subalgebras appear to be sufficient to reveal non-central extensions that define the Lax representations.

It is natural to ask whether the method can produce the known as well as new Lax representations for other equations. Also, we expect that the proposed technique will be helpful in describing multi-component integrable generalizations of integrable pdes, [14, 25, 3, 33, 23]. We intend to address these issues in the further study.

Acknowledgments

This work was partially supported by the Faculty of Applied Mathematics of AGH UST statutory tasks within subsidy of Ministry of Science and Higher Education.

I am very grateful to I.S. Krasil*′*shchik for useful discussions. I thank L.V. Bogdanov and P. Zusmanovich for important remarks.