Lax Pair for a Novel Two-Dimensional Lattice

Lax Pair for a Novel Two-Dimensional Lattice

Maria N. KUZNETSOVA

M.N. Kuznetsova

Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of
Sciences,
112 Chernyshevsky Street, Ufa 450008, Russia<EMAIL_ADDRESS>
Received February 09, 2021, in final form September 15, 2021; Published online
September 26, 2021

In paper by I.T. Habibullin and our joint paper the algorithm for
classification of integrable equations with three independent variables was
proposed. This method is based on the requirement of the existence of an
infinite set of Darboux integrable reductions and on the notion of the
characteristic Lie–Rinehart algebras. The method was applied for the
classification of integrable cases of different subclasses of equations
$u_{n,xy}=f(u_{n+1},u_{n},u_{n-1},u_{n,x},u_{n,y})$ of special forms. Under
this approach the novel integrable chain was obtained. In present paper we
construct Lax pair for the novel chain. To construct the Lax pair, we use the
scheme suggested in papers by E.V. Ferapontov. We also study the periodic
reduction of the chain.

Lax pair; two-dimensional lattice; integrable reduction; characteristic
algebra; Lie–Rinehart algebra; Darboux integrable system; higher symmetry;
$x$-integral

37K10; 37K30; 37D99

##  1 Introduction

In a number of recent publications [8, 13, 14, 15, 16, 18] the problem of
integrable classification of two-dimensional lattices

$u_{n,xy}=f(u_{n+1},u_{n},u_{n-1},u_{n,x},u_{n,y}),\qquad-\infty<n<\infty,$
(1.1)

was studied. Here the sought function $u_{n}=u_{n}(x,y)$ depends on the real
variables $x$, $y$ and the integer variable $n$. In these papers we proposed
the method for seeking and classifying integrable equations with three
independent variables based on the requirement of the existence of a set of
Darboux integrable reductions and on the notion of the characteristic
Lie–Rinehart algebras. The method was applied to different subclasses of
equations (1.1) of special forms.

Within this approach we use the following

######  Definition 1.1.

A lattice of the form (1.1) is called integrable if there exist locally
analytic functions $\varphi$ and $\psi$ of two variables such that for any
choice of integers $N_{1}$, $N_{2}$ the hyperbolic type system

$\displaystyle u_{N_{1},xy}=\varphi(u_{N_{1}+1},u_{N_{1}}),$ $\displaystyle
u_{n,xy}=f(u_{n+1},u_{n},u_{n-1},u_{n,x},u_{n,y}),\qquad N_{1}<n<N_{2},$ (1.2)
$\displaystyle u_{N_{2},xy}=\psi(u_{N_{2}},u_{N_{2}-1}),$

obtained from lattice (1.1) by imposing cut-off conditions at $n=N_{1}$ and
$n=N_{2}$, is integrable in the sense of Darboux.

Let us recall what Darboux integrability means.

######  Definition 1.2.

A function $I=I(x,\bar{u},\bar{u}_{x},\bar{u}_{xx},\dots)$ is called an
$y$-integral if it satisfies the equation $D_{y}I=0$ for every solution of
system (1.2). A function $J=J(y,\bar{u},\bar{u}_{y},\bar{u}_{yy},\dots)$ is
called a $x$-integral if it satisfies the equation $D_{x}J=0$. Integrals of
the form $I=I(x)$ and $J=J(y)$ are called trivial.

Here $\bar{u}$ is a vector $\bar{u}=(u_{N_{1}},u_{N_{1}+1},\dots,u_{N_{2}})$,
$\bar{u}_{x}$ is its derivative and so on. The operators $D_{y}$ and $D_{x}$
are operators of the total derivative with respect to the variable $y$ or $x$,
correspondingly, by virtue of system (1.2).

######  Definition 1.3.

A system (1.2) is called Darboux integrable if it possesses $N_{2}-N_{1}+1$
functionally independent nontrivial integrals in both characteristic
directions $x$ and $y$.

Darboux integrable systems are amenable to study by the Lie–Rinehart algebras.
Let $I=I(x,\bar{u},\bar{u}_{x},\bar{u}_{xx},\dots)$ be a nontrivial
$y$-integral for the system (1.2). Then $I$ must satisfy the following system:

$YI=0,\qquad X_{i}I=0,$

where

$X_{i}=\frac{\partial}{\partial u_{i,y}},\qquad
Y=\sum_{i=N_{1}}^{N_{2}}\left(u_{i,y}\frac{\partial}{\partial
u_{i}}+f_{i}\frac{\partial}{\partial
u_{i,x}}+D_{x}(f_{i})\frac{\partial}{\partial u_{i,xx}}+\cdots\right)$

and $f_{i}=f(u_{i+1},u_{i},u_{i-1},u_{i,x},u_{i,y})$. The first equation
follows from the fact that the operator $D_{y}$ acts on functions
$I=I(x,\bar{u},\bar{u}_{x},\bar{u}_{xx},\dots)$ by the rule $D_{y}I=YI$, the
second one arises because $I$ doesn’t depend on variables $u_{i,y}$.

Let us consider the Lie algebra $L_{y}$ generated by the operators $Y$,
$X_{i}$ over the ring $A$ of locally analytic functions of the dynamical
variables $\bar{u}_{y},\bar{u},\bar{u}_{x},\bar{u}_{xx},\dots$. To the
standard multiplication operation $[Z,W]=ZW-WZ$ we add two conditions:
$[Z,aW]=Z(a)W+a[Z,W]$ and $(aZ)b=aZ(b)$ valid for any $Z,W\in L_{y}$ and
$a,b\in A$. These equalities means that for any $Z\in L_{y}$ and any $a\in A$,
the element $aZ\in L_{y}$. In this case the algebra $L_{y}$ is called the
Lie–Rinehart algebra [20, 22].

If there exists a finite basis $Z_{1},Z_{2},\dots,Z_{k}\in L_{y}$ such that an
arbitrary element $Z\in L_{y}$ is represented as a linear combination
$Z=a_{1}Z_{1}+a_{2}Z_{2}+\dots+a_{k}Z_{k}$, where coefficients
$a_{1},a_{2},\dots,a_{k}\in A$; and if the equality $Z=0$ implies that
$a_{1}=a_{2}=\cdots=a_{k}=0$, then algebra $L_{y}$ is of a finite dimension.

The integrability criterion of the hyperbolic type system in the sense of
Darboux is formulated as follows [28, 29]:

######  Theorem 1.4.

System (1.2) admits a complete set of the $y$-integrals $($a complete set of
the $x$-integrals$)$ if and only if its characteristic algebra $L_{y}$
$($respectively, characteristic algebra $L_{x}$$)$ is of finite dimension.

######  Corollary 1.5.

System (1.2) is integrable in the sense of Darboux if both characteristic
algebras $L_{x}$ and $L_{y}$ are of finite dimension.

The above statements play a key role in our classification works. Within the
scope of this paper we need one of our results: paper [16] provides a complete
list of integrable two-dimensional lattices of the form

$\displaystyle
u_{n,xy}=\alpha(u_{n+1},u_{n},u_{n-1})u_{n,x}u_{n,y}+\beta(u_{n+1},u_{n},u_{n-1})u_{n,x}$
$\displaystyle\hphantom{u_{n,xy}=}{}+\gamma(u_{n+1},u_{n},u_{n-1})u_{n,y}+\delta(u_{n+1},u_{n},u_{n-1}),$
(1.3)

with the coefficient $\alpha$ satisfying the conditions
$\frac{\partial\alpha(u_{n+1},u_{n},u_{n-1})}{\partial u_{n\pm 1}}\neq 0$.
This list consists of two equations:

######  Theorem 1.6.

Integrable equation of the form (1.3) can be reduced by a point transformation
to one of the following forms:

$\displaystyle u_{n,xy}=\alpha_{n}u_{n,x}u_{n,y},$ (1.4) $\displaystyle
u_{n,xy}=\alpha_{n}\big{(}u_{n,x}-u^{2}_{n}-1\big{)}\big{(}u_{n,y}-u^{2}_{n}-1\big{)}+2u_{n}\big{(}u_{n,x}+u_{n,y}-u^{2}_{n}-1\big{)},$
(1.5)

where

$\alpha_{n}=\frac{1}{u_{n}-u_{n-1}}-\frac{1}{u_{n+1}-u_{n}}=\frac{u_{n+1}-2u_{n}+u_{n-1}}{(u_{n+1}-u_{n})(u_{n}-u_{n-1})}.$

Equation (1.4) was found before in papers [7, 23] by Ferapontov and Shabat and
Yamilov. Equation (1.5) appeared in [16] as a result of the classification
procedure.

The aim of the paper is to find Lax pair for novel chain (1.5), to explain the
method of finding Lax pairs and to prove that periodic closings of the chain
possesses higher symmetries.

The Lax pair for equation (1.4)

$\psi_{n,x}=\frac{u_{n,x}}{u_{n+1}-u_{n}}(\psi_{n+1}-\psi_{n}),\qquad\psi_{n,y}=\frac{u_{n,y}}{u_{n}-u_{n-1}}(\psi_{n}-\psi_{n-1})$

was found by E.V. Ferapontov. To construct Lax pair for chain (1.5), we use
the scheme suggested in paper [12]. Let us describe the procedure in detailed.
First of all, we represent lattice (1.5) in the equivalent following form:

$u_{xy}=\big{(}u_{x}-u^{2}-1\big{)}\big{(}u_{y}-u^{2}-1\big{)}\frac{\triangle_{z\bar{z}}u}{\triangle_{z}u\triangle_{\bar{z}}u}+2u\big{(}u_{x}+u_{y}-u^{2}-1\big{)}.$
(1.6)

Here $\triangle_{z}=\frac{T_{z}-1}{\epsilon}$,
$\triangle_{\bar{z}}=\frac{1-T_{\bar{z}}}{\epsilon}$ are the forward/backward
discrete derivatives and
$\triangle_{z\bar{z}}=\frac{T_{z}+T_{\bar{z}}-2}{\epsilon^{2}}$ is the
symmetrised second-order discrete derivative; the operators $T_{z}$,
$T_{\bar{z}}$ are the forward and backward $\epsilon$-shifts operators in the
variable $z$.

The method consists of three steps:

  1. 1)

First we construct the dispersionless limit of the equation (obtained as
$\epsilon\rightarrow 0$).

  2. 2)

Secondly, for the equation found at the previous step we find dispersionless
Lax pair. Usually this problem is effectively solved.

  3. 3)

Finally, we reconstruct Lax pair by appropriate “quantization” of
dispersionless Lax pair as proposed in [27].

The paper is organized as follows. In Section 2 Lax pair for chain (1.5) is
constructed. Section 3 is devoted to periodic closings. Namely, we impose the
periodic closure conditions $u_{n+2}=u_{n}$ to infinite chains (1.4), (1.5)
and obtain finite systems. Lax pairs and higher symmetries of the second order
are constructed for obtained finite systems. Conclusion contains a discussion
of the results.

##  2 Construction Lax pair for equation (1.5)

The main result of this section is as follows:

######  Theorem 2.1.

Equation (1.5) possesses the Lax pair

$\displaystyle\psi_{n,x}=\frac{u_{n,x}-u^{2}_{n}-1}{u_{n+1}-u_{n}}(\psi_{n+1}-\psi_{n})+u_{n}\psi_{n},$
$\displaystyle\psi_{n,y}=\frac{u_{n,y}-u^{2}_{n}-1}{u_{n}-u_{n-1}}(\psi_{n}-\psi_{n-1})+u_{n}\psi_{n}.$

###### Proof.

The dispersionless limit of the equation (1.6) coincides with equation:

$u_{xy}=\big{(}u_{x}-u^{2}-1\big{)}\big{(}u_{y}-u^{2}-1\big{)}\frac{u_{zz}}{u^{2}_{z}}+2u\big{(}u_{x}+u_{y}-u^{2}-1\big{)}.$
(2.1)

There exists a direct method for finding Lax pairs for equations of this form.
Lax pair is sought in the following form:

$\displaystyle S_{x}=F(u,u_{x},u_{y},u_{z},S_{z}),$ (2.2) $\displaystyle
S_{y}=G(u,u_{x},u_{y},u_{z},S_{z}).$ (2.3)

The compatibility condition $S_{xy}=S_{yx}$ of system (2.2), (2.3) by virtue
of equation (2.1) leads to the overdetermined equation

$\displaystyle
F_{u_{y}}u_{yy}u^{2}_{z}\\!-G_{u_{x}}u_{xx}u^{2}_{z}\\!-(G_{S_{z}}F_{u_{x}}-G_{u_{x}}F_{S_{z}}+G_{u_{z}})u_{zx}u^{2}_{z}\\!-(G_{u_{y}}F_{S_{z}}-G_{S_{z}}F_{u_{x}}-F_{u_{z}})u_{zy}u^{2}_{z}$
$\displaystyle\qquad{}+u_{zz}\big{(}\big{(}u^{2}-u_{y}+1\big{)}\big{(}u^{2}-u_{x}+1\big{)}(F_{u_{x}}-G_{u_{y}})-u^{2}_{z}(G_{S_{z}}F_{u_{z}}-G_{u_{z}}F_{S_{z}})\big{)}$
$\displaystyle\qquad{}-u^{2}_{z}\big{(}2u\big{(}1+u^{2}-u_{x}-u_{y}\big{)}(F_{u_{x}}-G_{u_{y}})+u_{z}(G_{S_{z}}F_{u}-G_{u}F_{S_{z}})+u_{x}G_{u}-u_{y}F_{u}\big{)}=0.$

Because of the fact that variables $u$, $u_{x}$, $u_{y}$, $u_{z}$, $u_{xx}$,
$u_{yy}$, $u_{zx}$, $u_{zy}$, $u_{zz}$ are independent, this equation splits
down into the overdetermined system of equations:

$\displaystyle F_{u_{y}}=0,\qquad G_{u_{x}}=0,$ (2.4) $\displaystyle
G_{S_{z}}F_{u_{x}}-G_{u_{x}}F_{S_{z}}+G_{u_{z}}=0,$ (2.5) $\displaystyle
G_{u_{y}}F_{S_{z}}-G_{S_{z}}F_{u_{x}}-F_{u_{z}}=0,$ (2.6)
$\displaystyle\big{(}u^{2}-u_{y}+1\big{)}\big{(}u^{2}-u_{x}+1\big{)}(F_{u_{x}}-G_{u_{y}})-u^{2}_{z}(G_{S_{z}}F_{u_{z}}-G_{u_{z}}F_{S_{z}})=0,$
(2.7) $\displaystyle
2u\big{(}1+u^{2}-u_{x}-u_{y}\big{)}(F_{u_{x}}-G_{u_{y}})+u_{z}(G_{S_{z}}F_{u}-G_{u}F_{S_{z}})+u_{x}G_{u}-u_{y}F_{u}=0.$
(2.8)

Equations (2.4) mean that $F=F(u,u_{x},u_{z},S_{z})$ and
$G=G(u,u_{y},u_{z},S_{z})$. Substituting $F$ and $G$ into (2.5), (2.6), we
arrive at the equations:

$G_{u_{z}}+G_{S_{z}}F_{u_{x}}=0,\qquad F_{u_{z}}+G_{u_{y}}F_{S_{z}}=0.$ (2.9)

We differentiate the first equation (2.9) by $u_{x}$, the second equation
(2.9) – by $u_{y}$, and obtain that $G_{S_{z}}F_{u_{x}u_{x}}=0$,
$F_{S_{z}}G_{u_{y}u_{y}}=0$. Obviously that the functions $F$ and $G$ take the
following forms:

$\displaystyle
F(u,u_{x},u_{z},S_{z})=F_{2}(u,u_{z},S_{z})u_{x}+F_{3}(u,u_{z},S_{z}),$
$\displaystyle
G(u,u_{y},u_{z},S_{z})=F_{4}(u,u_{z},S_{z})u_{y}+F_{5}(u,u_{z},S_{z}).$

Then we rewrite (2.9) and (2.7), (2.8) using the last formulas. Because of the
fact that the variables $u$, $u_{x}$, $u_{y}$, $u_{z}$ are independent,
obtained equations split down one more time. Thus we arrive at the system for
unknown functions $F_{i}(u,u_{z},S_{z})$, $i=2,3,4,5$:

$\displaystyle F_{2}F_{4,S_{z}}+F_{4,u_{z}}=0,\qquad
F_{4}F_{2,S_{z}}+F_{2,u_{z}}=0,$ (2.10) $\displaystyle
F_{2}-F_{4}+u^{2}_{z}(F_{2,S_{z}}F_{4,u_{z}}-F_{4,S_{z}}F_{2,u_{z}})=0,$
(2.11) $\displaystyle
F_{4,u}-F_{2,u}+u_{z}(F_{2,u}F_{4,S_{z}}-F_{4,u}F_{2,S_{z}})=0,$ (2.12)
$\displaystyle F_{4}F_{3,S_{z}}+F_{3,u_{z}}=0,$ (2.13)
$\displaystyle\big{(}1+u^{2}\big{)}(F_{4}-F_{2})+u^{2}_{z}(F_{3,S_{z}}F_{4,u_{z}}-F_{4,S_{z}}F_{3,u_{z}})=0,$
(2.14) $\displaystyle
2u(F_{4}-F_{2})+u_{z}(F_{3,u}F_{4,S_{z}}-F_{4,u}F_{3,S_{z}})-F_{3,u}=0,$
(2.15) $\displaystyle F_{2}F_{5,S_{z}}+F_{5,u_{z}}=0,$ (2.16) $\displaystyle
2u(F_{4}-F_{2})+u_{z}(F_{2,u}F_{5,S_{z}}-F_{5,u}F_{2,S_{z}})+F_{5,u}=0,$
(2.17)
$\displaystyle\big{(}u^{2}+1\big{)}(F_{4}-F_{2})+u^{2}_{z}(F_{2,S_{z}}F_{5,u_{z}}-F_{5,S_{z}}F_{2,u_{z}})=0,$
(2.18)
$\displaystyle\big{(}u^{2}+1\big{)}^{2}(F_{2}-F_{4})+u^{2}_{z}(F_{3,S_{z}}F_{5,u_{z}}-F_{5,S_{z}}F_{3,u_{z}})=0,$
(2.19) $\displaystyle
2u\big{(}u^{2}+1\big{)}(F_{2}-F_{4})+u_{z}(F_{3,S_{z}}F_{5,u}-F_{5,S_{z}}F_{3,u})=0.$
(2.20)

Now we will work with equations (2.10)–(2.12) to clarify functions $F_{2}$,
$F_{4}$. Let us express $F_{4,u_{z}}$, $F_{2,u_{z}}$ from (2.10) and
substitute them into (2.12). This leads to the condition $F_{4}=F_{2}$ or to
the equation

$\big{(}1-u^{2}_{z}F_{2,S_{z}}F_{4,S_{z}}\big{)}=0.$ (2.21)

Let us consider case (2.21). We look for $F_{2}$, $F_{4}$ in the following
form:

$F_{2}(u,u_{z},S_{z})=\frac{A(u,S_{z})}{u_{z}},\qquad
F_{4}(u,u_{z},S_{z})=\frac{B(u,S_{z})}{u_{z}}.$

Then $A$, $B$ have to satisfy the system obtained using (2.21), (2.10), and
(2.11),

$\displaystyle
1-A_{S_{z}}B_{S_{z}}=0,\qquad-A+BA_{S_{z}}=0,\qquad-B+AB_{S_{z}}=0,$ (2.22)
$\displaystyle B_{u}-A_{u}+B_{S_{z}}A_{u}-A_{S_{z}}B_{u}=0.$ (2.23)

This system has the solution:

$A(u,S_{z})=\frac{{\rm e}^{a_{1}(u)S_{z}+a_{1}(u)a_{2}(u)}-1}{a_{1}(u)}.$

Here $a_{1}$, $a_{2}$ are arbitrary functions. Similarly, we find that

$B(u,S_{z})=\frac{{\rm e}^{a_{4}(u)S_{z}+a_{4}(u)a_{3}(u)}-1}{a_{4}(u)}$

with arbitrary functions $a_{3}$, $a_{4}$. Under obtained $A$ and $B$ the
first equation (2.22) becomes

$1-{\rm e}^{(a_{1}(u)+a_{4}(u))S_{z}+a_{2}(u)a_{1}(u)+a_{3}(u)a_{4}(u)}=0.$

Thus one can derive that $a_{4}=-a_{1}$, $a_{3}=a_{2}$. Finally, equation
(2.23) takes the form

$\displaystyle\big{(}{-}a_{1}(u)a^{\prime}_{1}(u)S_{z}-a^{2}_{1}(u)a^{\prime}_{2}(u)-a_{1}(u)a_{2}(u)a^{\prime}_{1}(u)+2a^{\prime}_{1}(u)\big{)}{\rm
e}^{a_{1}(u)(S_{z}+a_{2}(u))}$
$\displaystyle{}+\big{(}a^{2}_{1}(u)a^{\prime}_{2}(u)+a_{1}(u)a_{2}(u)a^{\prime}_{1}(u)+2a^{\prime}_{1}(u)+a_{1}(u)a^{\prime}_{1}(u)S_{z}\big{)}{\rm
e}^{-a_{1}(u)(S_{z}+a_{2}(u))}-4a^{\prime}_{1}(u)=0.$

We assume essential dependence on $S_{z}$ for functions $F_{2}$, $F_{4}$ and,
therefore, for $A$, $B$, so the functions ${\rm e}^{a_{1}(u)S_{z}}$, ${\rm
e}^{-a_{1}(u)S_{z}}$, ${\rm e}^{a_{1}(u)S_{z}}S_{z}$, ${\rm
e}^{-a_{1}(u)S_{z}}S_{z}$ are independent. Hence we have $a_{1}(u)=c_{1}$,
$a_{2}(u)=c_{2}$, where $c_{1}$, $c_{2}$ are arbitrary constants.

Thus, we have clarified the right hand sides of Lax pair (2.2), (2.3)

$\displaystyle S_{x}=F(u,u_{x},u_{y},u_{z},S_{z})=\frac{\big{(}{\rm
e}^{c_{1}(S_{z}+c_{2})}-1\big{)}u_{x}}{c_{1}u_{z}}+F_{3}(u,u_{z},S_{z}),$
$\displaystyle S_{y}=G(u,u_{x},u_{y},u_{z},S_{z})=-\frac{\big{(}{\rm
e}^{-c_{1}(S_{z}+c_{2})}-1\big{)}u_{y}}{c_{1}u_{z}}+F_{5}(u,u_{z},S_{z}).$

By the shift transformation $S\rightarrow S-c_{2}z$ and by the scaling
$z\rightarrow c_{1}z$ these equations can be reduced to

$\displaystyle S_{x}=F(u,u_{x},u_{y},u_{z},S_{z})=\frac{\big{(}{\rm
e}^{S_{z}}-1\big{)}u_{x}}{u_{z}}+F_{3}(u,u_{z},S_{z}),$ $\displaystyle
S_{y}=G(u,u_{x},u_{y},u_{z},S_{z})=-\frac{\big{(}{\rm
e}^{-S_{z}}-1\big{)}u_{y}}{u_{z}}+F_{5}(u,u_{z},S_{z}).$

To clarify $F_{3}$, we substitute the above functions into (2.13), (2.14), and
(2.15)

$\displaystyle\big{(}{\rm e}^{-S_{z}}-1\big{)}u_{z}F_{3,u}-2u\big{(}{\rm
e}^{S_{z}}+2\big{)}=0,$ $\displaystyle u_{z}\big{(}{\rm
e}^{-S_{z}}-1-u_{z}{\rm
e}^{-S_{z}}\big{)}F_{3,S_{z}}-\big{(}u^{2}+1\big{)}\big{(}{\rm e}^{S_{z}}+{\rm
e}^{-S_{z}}-2\big{)}=0,$ $\displaystyle-({\rm
e}^{-S_{z}}-1)F_{3,S_{z}}+u_{z}F_{3,u_{z}}=0.$

This system has the solution

$F_{3}(u,u_{z},S_{z})=-\frac{\big{(}{\rm
e}^{S_{z}}-1\big{)}\big{(}u^{2}+1\big{)}}{u_{z}}.$

Now we rewrite equations (2.16)–(2.20) and we obtain the system on the unknown
function $F_{5}$:

$\displaystyle\big{(}{\rm e}^{S_{z}}-1\big{)}F_{5,S_{z}}+u_{z}F_{5,u_{z}}=0,$
$\displaystyle u_{z}\big{(}{\rm e}^{S_{z}}-1\big{)}F_{5,S_{z}}+u^{2}_{z}{\rm
e}^{S_{z}}F_{5,u_{z}}-\big{(}u^{2}+1\big{)}\big{(}{\rm e}^{S_{z}}+{\rm
e}^{-S_{z}}-2\big{)}=0,$ $\displaystyle-u_{z}(-{\rm e}^{-2S_{z}}+3{\rm
e}^{-S_{z}}-3+{\rm e}^{S_{z}})F_{5,S_{z}}-u^{2}_{z}({\rm e}^{S_{z}}+{\rm
e}^{-S_{z}}-2)F_{5,u_{z}}$
$\displaystyle\qquad{}+\big{(}u^{2}+1\big{)}\big{(}{-}4+{\rm e}^{S_{z}}-4{\rm
e}^{-2S_{z}}+6{\rm e}^{-S_{z}}+{\rm e}^{-3S_{z}}\big{)}=0,$ $\displaystyle
u_{z}\big{(}1-{\rm e}^{S_{z}}\big{)}F_{5,u}-2u\big{(}{\rm e}^{S_{z}}+{\rm
e}^{-S_{z}}-2\big{)}=0,$ $\displaystyle 2uu_{z}\big{(}{\rm e}^{-2S_{z}}-3{\rm
e}^{-S_{z}}+3-{\rm
e}^{S_{z}}\big{)}F_{5,S_{z}}+\big{(}u^{2}+1\big{)}u_{z}\big{(}{\rm
e}^{S_{z}}+{\rm e}^{-S_{z}}-2\big{)}F_{5,u}$
$\displaystyle\qquad{}+2u\big{(}u^{2}+1\big{)}\big{(}{\rm e}^{-3S_{z}}+6{\rm
e}^{-S_{z}}-4{\rm e}^{-2S_{z}}+{\rm e}^{S_{z}}-4\big{)}=0.$

This system possesses the solution

$F_{5}(u,u_{z},S_{z})=-\frac{\big{(}1-{\rm
e}^{-S_{z}}\big{)}\big{(}u^{2}+1\big{)}}{u_{z}}.$

Thus we have found the Lax pair

$\displaystyle S_{x}=\frac{u_{x}-u^{2}-1}{u_{z}}\big{(}{\rm
e}^{S_{z}}-1\big{)}+\frac{1}{u_{z}},$ (2.24) $\displaystyle
S_{y}=\frac{u_{y}-u^{2}-1}{u_{z}}\big{(}1-{\rm
e}^{-S_{z}}\big{)}-\frac{1}{u_{z}}$ (2.25)

for equation (2.1).

Now we reconstruct the dispersive Lax pair by an appropriate quantization the
dispersionless Lax pair (2.24), (2.25). First, we “quantise” [27] the terms in
every equation (2.24), (2.25): $u_{z}$ is replaced by $\triangle_{z}u$; ${\rm
e}^{S_{z}}-1$ by $\triangle_{z}\psi$ due to the formal representation ${\rm
e}^{\frac{\partial}{\partial z}}\approx 1+\frac{\partial}{\partial z}+\cdots$,
and, similarly $1-{\rm e}^{-S_{z}}$ by $\triangle_{\bar{z}}\psi$.

In most cases, this procedure provides the necessary Lax pair. But in this
case we do not obtain the Lax pair for (1.6) if we act in the same way. It was
experimentally found that we should fit the second term in the r.h.s. of
equations (2.24), (2.25) by the following way (i.e., we guess some part):

$\displaystyle\psi_{x}=\frac{u_{x}-u^{2}-1}{\triangle_{z}u}\triangle_{z}\psi+P(u)\psi,$
$\displaystyle\psi_{y}=\frac{u_{y}-u^{2}-1}{\triangle_{\bar{z}}u}\triangle_{\bar{z}}\psi+Q(u)\psi.$

The compatibility condition $\psi_{xy}=\psi_{yx}$ is straightforward to solve.
Thus we find that equation (1.6) possesses the Lax pair

$\displaystyle\psi_{x}=\frac{u_{x}-u^{2}-1}{\triangle_{z}u}\triangle_{z}\psi+u\psi,\qquad\psi_{y}=\frac{u_{y}-u^{2}-1}{\triangle_{\bar{z}}u}\triangle_{\bar{z}}\psi+u\psi.$

It finally proved Theorem 2.1. ∎

##  3 Higher symmetries of periodic closings

Let us impose the periodic closure conditions $u_{n+2}=u_{n}$ to infinite
lattice (1.4). Then we obtain the following finite system:

$u_{0,xy}=\frac{2}{u_{0}-u_{1}}u_{0,x}u_{0,y},\qquad
u_{1,xy}=\frac{2}{u_{1}-u_{0}}u_{1,x}u_{1,y}.$ (3.1)

System (3.1) has the $x$-integral and the $y$-integral

$w=\frac{u_{0,y}u_{1,y}}{(u_{0}-u_{1})^{2}},\qquad
W=\frac{u_{0,x}u_{1,x}}{(u_{0}-u_{1})^{2}}.$ (3.2)

Lax pair for (3.1) has the form

$\Psi_{x}=(A\lambda+B)\Psi,\qquad\Psi_{y}=\big{(}\tilde{A}\lambda^{-1}+\tilde{B}\big{)}\Psi,$
(3.3)

where $\Psi=(\psi_{1},\psi_{0})^{\rm T}$ and

$\displaystyle A=\left(\begin{matrix}0&0\\\
\dfrac{u_{1,x}}{u_{0}-u_{1}}&0\end{matrix}\right),\qquad
B=\left(\begin{matrix}\dfrac{u_{0,x}}{u_{0}-u_{1}}&-\dfrac{u_{0,x}}{u_{0}-u_{1}}\vspace{1mm}\\\
0&-\dfrac{u_{1,x}}{u_{0}-u_{1}}\end{matrix}\right),$
$\displaystyle\tilde{A}=\left(\begin{matrix}0&0\\\
0&-\dfrac{u_{0,y}}{u_{0}-u_{1}}\end{matrix}\right),\qquad\tilde{B}=\left(\begin{matrix}-\dfrac{u_{0,y}}{u_{0}-u_{1}}&0\vspace{1mm}\\\
-\dfrac{u_{1,y}}{u_{1}-u_{0}}&-\dfrac{u_{1,y}}{u_{1}-u_{0}}\end{matrix}\right),$

$\lambda$ is a spectral parameter.

The classical symmetry can be found directly from the consistency condition
$(u_{i,xy})_{t_{1}}=(u_{i,t_{1}})_{xy}$:

$\displaystyle u_{0,t_{1}}=u_{0,x}F(W)+c_{1}u^{2}_{0}+c_{2}u_{0}+c_{3},$
$\displaystyle u_{1,t_{1}}=u_{1,x}F(W)+c_{1}u^{2}_{1}+c_{2}u_{1}+c_{3},$

where $F$ is an arbitrary function depending on the $y$-integral $W$ defined
by the second formula of (3.2); $c_{1}$, $c_{2}$, $c_{3}$ are arbitrary
constants. The classical symmetry in the another direction is simply found
because the system is symmetric under the change of variables
$x\leftrightarrow y$:

$\displaystyle
u_{0,t_{2}}=u_{0,y}G(w)+\tilde{c}_{1}u^{2}_{0}+\tilde{c}_{2}u_{0}+\tilde{c}_{3},$
$\displaystyle
u_{1,t_{2}}=u_{1,y}G(w)+\tilde{c}_{1}u^{2}_{1}+\tilde{c}_{2}u_{1}+\tilde{c}_{3}.$

Higher symmetry of the second order is sought in the following form:

$\displaystyle
u_{i,\tau_{1}}=a_{i}(u_{0},u_{1},u_{0,x},u_{1,x})u_{0,xx}+b_{i}(u_{0},u_{1},u_{0,x},u_{1,x})u_{1,xx}+h_{i}(u_{0},u_{1},u_{0,x},u_{1,x}),$

$i=1,2$, where $a_{i}$, $b_{i}$, $h_{i}$ are functions to be found. To find
the higher symmetry we use Lax pair (3.3). Let us consider the linear problem

$\Psi_{\tau_{1}}=\big{(}\alpha\lambda^{2}+\beta\lambda+\gamma\big{)}\Psi,$
(3.4)

where $\alpha=(\alpha_{i,j})$, $\beta=(\beta_{i,j})$, $\gamma=(\gamma_{i,j})$,
$i,j=1,2$ are matrices to be found. It is assumed that elements of the
matrices depend on the variables $u_{0}$, $u_{1}$, $u_{0,x}$, $u_{1,x}$,
$u_{0,xx}$, $u_{1,xx}$. The compatibility condition
$(\Psi_{x})_{\tau_{1}}=(\Psi_{\tau_{1}})_{x}$ for the systems

$\Psi_{x}=(A\lambda+B)\Psi,\qquad\Psi_{\tau_{1}}=\big{(}\alpha\lambda^{2}+\beta\lambda+\gamma\big{)}\Psi,$

results in the system of relations

$\displaystyle A\alpha=\alpha A,\qquad A\beta+B\alpha=\alpha_{x}+\alpha
B+\beta A,$ $\displaystyle A_{\tau_{1}}+A\gamma+B\beta=\beta_{x}+\beta
B+\gamma A,\qquad B_{\tau_{1}}+B\gamma=\gamma_{x}+\gamma B.$

A complete study of these equations leads to the following formulas:

$\displaystyle
u_{0,\tau_{1}}=H(W)u_{0,xx}+\frac{u^{2}_{0,x}}{(u_{0}-u_{1})^{2}}\Phi(W)u_{1,xx}+(u_{0}-u_{1})g(u_{0},u_{1},u_{0,x},u_{1,x})$
$\displaystyle\hphantom{u_{0,\tau_{1}}=}{}+(u_{0}-u_{1})(c_{0}-c_{1}u_{1}-\frac{c_{2}}{2})-(c_{1}u^{2}_{1}+c_{2}u_{1}+c_{3}),$
$\displaystyle
u_{1,\tau_{1}}=\frac{u_{1,x}}{u_{0,x}}H(W)u_{0,xx}+W\Phi(W)u_{1,xx}+\frac{(u_{0}-u_{1})u_{1,x}}{u_{0,x}}g(u_{0},u_{1},u_{0,x},u_{1,x})$
$\displaystyle\hphantom{u_{1,\tau_{1}}=}{}+\frac{(u_{0}-u_{1})u_{1,x}}{u_{0,x}}(c_{0}+c_{1}u_{0}+\frac{c_{2}}{2})-(c_{1}u^{2}_{1}+c_{2}u_{1}+c_{3}),$

where $H$, $\Phi$, $g$ are arbitrary functions; $c_{i}$ are arbitrary
constants. To define precisely obtained formulas we substitute them into the
compatibility condition $(u_{i,xy})_{\tau_{1}}=(u_{i,\tau_{1}})_{xy}$. Thus,
we finally found the higher symmetry of the second order:

$\displaystyle
u_{0,\tau_{1}}=\left(u_{0,xx}+\frac{u_{0,x}}{u_{1,x}}u_{1,xx}-\frac{2u_{0,x}(u_{0,x}-u_{1,x})}{(u_{0}-u_{1})}\right)F(W),$
(3.5) $\displaystyle
u_{1,\tau_{1}}=\left(u_{1,xx}+\frac{u_{1,x}}{u_{0,x}}u_{0,xx}-\frac{2u_{1,x}(u_{0,x}-u_{1,x})}{(u_{0}-u_{1})}\right)F(W),$
(3.6)

where $F$ is an arbitrary function; $W$ is the $y$-integral defined by the
second formula of (3.2). Also we finally found matrices $\alpha$, $\beta$,
$\gamma$ involved in (3.4):

$\displaystyle\alpha=\left(\begin{matrix}\alpha_{11}&0\\\
0&\alpha_{11}\end{matrix}\right),\qquad\beta=\left(\begin{matrix}\beta_{11}&0\\\
\beta_{21}(\overline{u},\overline{u}_{x},\overline{u}_{xx})&\beta_{11}\end{matrix}\right),$
$\displaystyle\gamma=\left(\begin{matrix}\gamma_{11}(\overline{u},\overline{u}_{x},\overline{u}_{xx})&\gamma_{12}(\overline{u},\overline{u}_{x},\overline{u}_{xx})\\\
0&\gamma_{22}(\overline{u},\overline{u}_{x},\overline{u}_{xx})\end{matrix}\right),$

where

$\displaystyle\beta_{21}(\overline{u},\overline{u}_{x},\overline{u}_{xx})=\left(\frac{u_{1,x}}{u_{0,x}(u_{0}-u_{1})}u_{0,xx}+\frac{1}{(u_{0}-u_{1})}u_{1,xx}-\frac{2u_{1,x}(u_{0,x}-u_{1,x})}{(u_{0}-u_{1})^{2}}\right)F(W),$
$\displaystyle\gamma_{11}(\overline{u},\overline{u}_{x},\overline{u}_{xx})=\left(\frac{1}{(u_{0}-u_{1})}u_{0,xx}+\frac{u_{0,x}}{u_{1,x}(u_{0}-u_{1})}u_{1,xx}-\frac{2u_{0,x}(u_{0,x}-u_{1,x})}{(u_{0}-u_{1})^{2}}\right)F(W),$
$\displaystyle\gamma_{12}(\overline{u},\overline{u}_{x},\overline{u}_{xx})=\left(-\frac{1}{(u_{0}-u_{1})}u_{0,xx}-\frac{u_{0,x}}{u_{1,x}(u_{0}-u_{1})}u_{1,xx}+\frac{u_{0,x}(u_{0,x}-u_{1,x})}{(u_{0}-u_{1})^{2}}\right)F(W),$

$\alpha_{11}$, $\beta_{11}$ are arbitrary constants. Thus it is seen that
definitive answer is given by formulas (3.5), (3.6) and

$\Psi_{\tau_{1}}=(\beta\lambda+\gamma)\Psi,\qquad\beta=\left(\begin{matrix}0&0\\\
\beta_{21}&0\end{matrix}\right),\qquad\gamma=\left(\begin{matrix}\gamma_{11}&\gamma_{12}\\\
0&\gamma_{22}\end{matrix}\right),$

where $\beta_{21}$, $\gamma_{ij}$ have been described just above.

######  Remark 3.1.

The symmetry given by (3.5), (3.6) can be written as111I am grateful to the
anonymous referee for this constructive comment.

$u_{0,\tau_{1}}=u_{0,x}F(W)\frac{W_{x}}{W},\qquad
u_{1,\tau_{1}}=u_{1,x}F(W)\frac{W_{x}}{W}.$

Therefore this is actually the classical symmetry in disguise.

Let us consider chain (1.5). We impose the periodic closure conditions
$u_{n+2}=u_{n}$ to infinite chain (1.5) and obtain the following finite
system:

$\displaystyle
u_{0,xy}=\frac{2}{u_{0}-u_{1}}\big{(}u_{0,x}-u^{2}_{0}-1\big{)}\big{(}u_{0,y}-u^{2}_{0}-1\big{)}+2u_{0}\big{(}u_{0,x}+u_{0,y}-u^{2}_{0}-1\big{)},$
$\displaystyle
u_{1,xy}=\frac{2}{u_{1}-u_{0}}\big{(}u_{1,x}-u^{2}_{1}-1\big{)}\big{(}u_{1,y}-u^{2}_{1}-1\big{)}+2u_{1}\big{(}u_{1,x}+u_{1,y}-u^{2}_{1}-1\big{)}.$
(3.7)

This system possesses the $y$-integral and the $x$-integral

$P=\frac{\big{(}u_{0,x}-u^{2}_{0}-1\big{)}\big{(}u_{1,x}-u^{2}_{1}-1\big{)}}{(u_{0}-u_{1})^{2}},\qquad
J=\frac{\big{(}u_{0,y}-u^{2}_{0}-1\big{)}\big{(}u_{1,y}-u^{2}_{1}-1\big{)}}{\big{(}u_{0}-u_{1}\big{)}^{2}}.$
(3.8)

System (3.7) is the compatibility condition for the Lax pair

$\Phi_{x}=(S\lambda+T)\Phi,\qquad\Phi_{y}=\big{(}\tilde{S}\lambda^{-1}+\tilde{T}\big{)}\Phi,$
(3.9)

where $\Phi=(\phi_{0},\phi_{1})^{\rm T}$,

$\displaystyle S=\left(\begin{matrix}0&0\\\
\dfrac{u_{1,x}-u^{2}_{1}-1}{u_{0}-u_{1}}&0\end{matrix}\right),\qquad
T=\left(\begin{matrix}-\dfrac{u_{0,x}-u^{2}_{0}-1}{u_{1}-u_{0}}+u_{0}&\dfrac{u_{0,x}-u^{2}_{0}-1}{u_{1}-u_{0}}\vspace{1mm}\\\
0&-\dfrac{u_{1,x}-u^{2}_{1}-1}{u_{0}-u_{1}}+u_{1}\end{matrix}\right),$
$\displaystyle\tilde{S}=\left(\begin{matrix}0&-\dfrac{u_{0,y}-u^{2}_{0}-1}{u_{0}-u_{1}}\\\
0&0\end{matrix}\right),\qquad\tilde{T}=\left(\begin{matrix}\dfrac{u_{0,y}-u^{2}_{0}-1}{u_{0}-u_{1}}+u_{0}&0\vspace{1mm}\\\
-\dfrac{u_{1,y}-u^{2}_{1}-1}{u_{1}-u_{0}}&\dfrac{u_{1,y}-u^{2}_{1}-1}{u_{1}-u_{0}}+u_{1}\end{matrix}\right).$

To find the higher symmetry it is sufficient (as we have just seen) to
consider the system

$\Phi_{\tau_{2}}=\big{(}\tilde{\beta}\lambda+\tilde{\gamma}\big{)}\Phi,$
(3.10)

compatible with the first equation of (3.9). In this way we obtained the
higher symmetry of system (3.7):

$\displaystyle
u_{0,\tau_{2}}=\left(u_{0,xx}+\frac{u_{0,x}-u^{2}_{0}-1}{u_{1,x}-u^{2}_{1}-1}u_{1,xx}+\frac{2\varphi(u_{0},u_{1},u_{0,x},u_{1,x})}{(u_{1,x}-u^{2}_{1}-1)(u_{0}-u_{1})}\right)F(P),$
(3.11) $\displaystyle
u_{1,\tau_{2}}=\left(\frac{u_{1,x}-u^{2}_{1}-1}{u_{0,x}-u^{2}_{0}-1}u_{0,xx}+u_{1,xx}+\frac{2\varphi(u_{0},u_{1},u_{0,x},u_{1,x})}{(u_{0,x}-u^{2}_{0}-1)(u_{0}-u_{1})}\right)F(P),$
(3.12)

where $P$ is the $y$-integral given by the first formula of (3.8),

$\displaystyle\varphi(u_{0},u_{1},u_{0,x},u_{1,x})=u_{0,x}u_{1,x}(u_{1,x}-u_{0,x})+u^{2}_{0,x}\big{(}1+u^{2}_{1}\big{)}-u^{2}_{1,x}\big{(}1+u^{2}_{0}\big{)}$
$\displaystyle\hphantom{\varphi(u_{0},u_{1},u_{0,x},u_{1,x})=}{}-u_{0,x}\big{(}1+u^{2}_{1}+u_{0}u_{1}+u_{0}u^{3}_{1}\big{)}+u_{1,x}\big{(}1+u^{2}_{0}+u_{0}u_{1}+u^{3}_{0}u_{1}\big{)}.$
(3.13)

Matrices $\tilde{\beta}$, $\tilde{\gamma}$ (see (3.10)) are defined by the
following formulas:

$\displaystyle\tilde{\beta}=\left(\begin{matrix}0&0\\\
\tilde{\beta}_{21}(\bar{u},\bar{u}_{x},\bar{u}_{xx})&0\end{matrix}\right),\qquad\tilde{\gamma}=\left(\begin{matrix}\tilde{\gamma}_{11}(\bar{u},\bar{u}_{x},\bar{u}_{xx})&\tilde{\gamma}_{12}(\bar{u},\bar{u}_{x},\bar{u}_{xx})\\\
0&\tilde{\gamma}_{22}(\bar{u},\bar{u}_{x},\bar{u}_{xx})\end{matrix}\right),$

where

$\displaystyle\tilde{\beta}_{21}(\bar{u},\bar{u}_{x},\bar{u}_{xx})=\left(\frac{u_{1,x}-u^{2}_{1}-1}{(u_{0}-u_{1})\big{(}u_{0,x}-u^{2}_{0}-1\big{)}}u_{0,xx}+\frac{u_{1,xx}}{u_{0}-u_{1}}\right.$
$\displaystyle\left.\hphantom{\tilde{\beta}_{21}(\bar{u},\bar{u}_{x},\bar{u}_{xx})=}{}+\frac{2\varphi(u_{0},u_{1},u_{0,x},u_{1,x})}{\big{(}u_{0,x}-u^{2}_{0}-1\big{)}(u_{0}-u_{1})^{2}}\right)F(P),$
$\displaystyle\tilde{\gamma}_{11}(\bar{u},\bar{u}_{x},\bar{u}_{xx})=\left(\frac{u_{0,xx}}{u_{0}-u_{1}}+\frac{\big{(}u_{0,x}-u^{2}_{0}-1\big{)}u_{1,xx}}{\big{(}u_{1,x}-u^{2}_{1}-1\big{)}(u_{0}-u_{1})}\right.$
$\displaystyle\left.\hphantom{\tilde{\gamma}_{11}(\bar{u},\bar{u}_{x},\bar{u}_{xx})=}{}+\frac{2\varphi(u_{0},u_{1},u_{0,x},u_{1,x})}{(u_{0}-u_{1})^{2}\big{(}u_{1,x}-u^{2}_{1}-1\big{)}}\right)F(P),$
$\displaystyle\tilde{\gamma}_{12}(\bar{u},\bar{u}_{x},\bar{u}_{xx})=\left(-\frac{u_{0,xx}}{u_{0}-u_{1}}-\frac{\big{(}u_{0,x}-u^{2}_{0}-1\big{)}}{\big{(}u_{1,x}-u^{2}_{1}-1\big{)}(u_{0}-u_{1})}u_{1,xx}\right.$
$\displaystyle\left.\hphantom{\tilde{\gamma}_{12}(\bar{u},\bar{u}_{x},\bar{u}_{xx})=}{}-\frac{2\varphi(u_{0},u_{1},u_{0,x},u_{1,x})}{(u_{0}-u_{1})^{2}\big{(}u_{1,x}-u^{2}_{1}-1\big{)}}\right)F(P),$
$\displaystyle\tilde{\gamma}_{22}(\bar{u},\bar{u}_{x},\bar{u}_{xx})=\left(-\frac{\big{(}u_{1,x}-u^{2}_{1}-1\big{)}}{(u_{0}-u_{1})\big{(}u_{0,x}-u^{2}_{0}-1\big{)}}u_{0,xx}-\frac{u_{1,xx}}{u_{0}-u_{1}}\right.$
$\displaystyle\left.\hphantom{\tilde{\gamma}_{22}(\bar{u},\bar{u}_{x},\bar{u}_{xx})=}{}-\frac{2\varphi(u_{0},u_{1},u_{0,x},u_{1,x})}{\big{(}u_{0,x}-u^{2}_{0}-1\big{)}(u_{0}-u_{1})^{2}}\right)F(P),$

$\varphi(u_{0},u_{1},u_{0,x},u_{1,x})$ is defined by (3.13).

######  Remark 3.2.

The symmetry given by (3.11), (3.12) can be written as

$u_{0,\tau_{2}}=\big{(}u_{0,x}-u^{2}_{0}-1\big{)}F(P)\frac{P_{x}}{P},\qquad
u_{1,\tau_{2}}=\big{(}u_{1,x}-u^{2}_{1}-1\big{)}F(P)\frac{P_{x}}{P}.$

Therefore this is actually the classical symmetry in disguise.

Note, that periodic closing obtained by the conditions $u_{n+3}=u_{n}$
imposing on infinite chain (1.5) leads to the system

$\displaystyle
u_{0,xy}=\left(\frac{1}{u_{0}-u_{2}}-\frac{1}{u_{1}-u_{0}}\right)\big{(}u_{0,x}-u^{2}_{0}-1\big{)}\big{(}u_{0,y}-u^{2}_{0}-1\big{)}+2u_{0}\big{(}u_{0,x}+u_{0,y}-u^{2}_{0}-1\big{)},$
$\displaystyle
u_{1,xy}=\left(\frac{1}{u_{1}-u_{0}}-\frac{1}{u_{2}-u_{1}}\right)\big{(}u_{1,x}-u^{2}_{1}-1\big{)}\big{(}u_{1,y}-u^{2}_{1}-1\big{)}+2u_{1}\big{(}u_{1,x}+u_{1,y}-u^{2}_{1}-1\big{)},$
$\displaystyle
u_{2,xy}=\left(\frac{1}{u_{2}-u_{1}}-\frac{1}{u_{0}-u_{2}}\right)\big{(}u_{2,x}-u^{2}_{2}-1\big{)}\big{(}u_{2,y}-u^{2}_{2}-1\big{)}+2u_{2}\big{(}u_{2,x}+u_{2,y}-u^{2}_{2}-1\big{)}.$

This system has $y$-integral and $x$-integral

$\displaystyle
W=\frac{\big{(}u_{0,x}-u^{2}_{0}-1\big{)}\big{(}u_{1,x}-u^{2}_{1}-1\big{)}\big{(}u_{2,x}-u^{2}_{2}-1\big{)}}{(u_{2}-u_{1})(u_{0}-u_{1})(u_{0}-u_{2})},$
$\displaystyle
w=\frac{\big{(}u_{0,y}-u^{2}_{0}-1\big{)}\big{(}u_{1,y}-u^{2}_{1}-1\big{)}\big{(}u_{2,y}-u^{2}_{2}-1\big{)}}{(u_{2}-u_{1})(u_{0}-u_{1})(u_{0}-u_{2})}.$

Lax pair has the following form:

$\Psi_{x}=(A\lambda+B)\Psi,\qquad\Psi_{y}=\big{(}\tilde{A}\lambda^{-1}+\tilde{B}\big{)}\Psi,$

where $\Psi=(\psi_{0},\psi_{1},\psi_{2})^{\rm T}$,

$\displaystyle A=\left(\begin{matrix}0&0&0\\\ 0&0&0\\\
\dfrac{u_{2,x}-u^{2}_{0}-1}{u_{1}-u_{0}}&0&0\end{matrix}\right),$
$\displaystyle
B=\left(\begin{matrix}-\dfrac{u_{0,x}-u^{2}_{0}-1}{u_{1}-u_{0}}+u_{0}&\dfrac{u_{0,x}-u^{2}_{0}-1}{u_{1}-u_{0}}&0\vspace{1mm}\\\
0&-\dfrac{u_{1,x}-u^{2}_{1}-1}{u_{2}-u_{1}}+u_{1}&\dfrac{u_{1,x}-u^{2}_{1}-1}{u_{2}-u_{1}}\vspace{1mm}\\\
0&0&-\dfrac{u_{2,x}-u^{2}_{2}-1}{u_{0}-u_{2}}+u_{2}\end{matrix}\right),$
$\displaystyle\tilde{A}=\left(\begin{matrix}0&0&-\dfrac{u_{0,y}-u^{2}_{0}-1}{u_{0}-u_{2}}\\\
0&0&0\\\ 0&0&0\end{matrix}\right),$
$\displaystyle\tilde{B}=\left(\begin{matrix}\dfrac{u_{0,y}-u^{2}_{0}-1}{u_{0}-u_{2}}+u_{0}&0&0\vspace{1mm}\\\
-\dfrac{u_{1,y}-u^{2}_{1}-1}{u_{1}-u_{0}}&\dfrac{u_{1,y}-u^{2}_{1}-1}{u_{1}-u_{0}}+u_{1}&0\vspace{1mm}\\\
0&-\dfrac{u_{2,y}-u^{2}_{2}-1}{u_{2}-u_{1}}&\dfrac{u_{2,y}-u^{2}_{2}-1}{u_{2}-u_{1}}+u_{2}\end{matrix}\right).$

##  4 Conclusion

The problem of classification multidimensional equations is actively studied
by many authors, using different algebraic and geometry approaches [1, 2, 3,
5, 6, 9, 10, 11, 21]. We note that the classification algorithm for integrable
two-dimensional lattices proposed in our previous papers does not provide any
algorithm for constructing the Lax pair.

It is known that finite systems obtained from infinite integrable chains by
degenerate boundary conditions imposing at the two points of the form
$u_{n+k}=c_{1}$, $u_{n+s}=c_{2}$ (where $c_{1}$, $c_{2}$ are constants) are
integrable in the sense of Darboux (they have complete set of integrals in
both characteristic directions, i.e., the number of independent integrals is
equal to the order of the system). We study finite systems obtained from
infinite chains (1.4), (1.5) by periodic closure conditions. It is interesting
fact that each of these systems also has one $x$-integral and one
$y$-integral. We obtained that symmetries of these systems depend on
integrals. It is known that Darboux integrable systems possesses symmetries
which depend on integrals [24, 30]. Symmetries of systems with incomplete sets
of integrals might depend on these integrals [17, 19]. In a discrete version,
this fact is discussed in paper [26]. In papers [4, 25] an algorithm is
proposed which allows one to construct higher symmetries of arbitrary order
for some special classes of hyperbolic systems possessing the integrals.

### Acknowledgements

The author gratefully thanks I.T. Habibulin for assignment the problem and
useful discussions, E.V. Ferapontov for explaining the method of the
construction of Lax pairs and S.Ya. Startsev for valuable comments. The author
gratefully thanks anonymous referees for a contribution to improve the paper.

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# Dataset of Solution-based Inorganic Materials Synthesis Recipes Extracted
from the Scientific Literature

Zheren Wang1,2, Olga Kononova1,2, Kevin Cruse1,2 Tanjin He1,2,
Haoyan Huo1,2, Yuxing Fei1,2, Yan Zeng2, Yingzhi Sun1,2,
Zijian Cai1,2, Wenhao Sun3, Gerbrand Ceder1,2*

###### Abstract

The development of a materials synthesis route is usually based on heuristics
and experience. A possible new approach would be to apply data-driven
approaches to learn the patterns of synthesis from past experience and use
them to predict the syntheses of novel materials. However, this route is
impeded by the lack of a large-scale database of synthesis formulations. In
this work, we applied advanced machine learning and natural language
processing techniques to construct a dataset of 35,675 solution-based
synthesis “recipes” extracted from the scientific literature. Each recipe
contains essential synthesis information including the precursors and target
materials, their quantities, and the synthesis actions and corresponding
attributes. Every recipe is also augmented with the reaction formula. Through
this work, we are making freely available the first large dataset of solution-
based inorganic materials synthesis recipes.

1Department of Materials Science and Engineering, University of California,
Berkeley, CA 94720, USA;
2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley,
CA 94720, USA;
3Department of Materials Science and Engineering, University of Michigan, Ann
Arbor, MI, USA
*Corresponding author: Gerbrand Ceder<EMAIL_ADDRESS>

##  1 Background & Summary

Big-data-driven approaches have helped to establish a new paradigm of
scientific research[1, 2, 3]. In materials science specifically, the Materials
Genome Initiative (MGI) effort has significantly facilitated and accelerated
materials discovery and design by deploying large-scale ab initio computation
and building computed databases of structure–property relationships [4, 5, 6].
Unlike computational data, the experimentally determined properties and
structures of inorganic materials are mainly available in manually curated
databases[7, 8, 9, 10, 11]. Such well-curated experimental databases have led
to early machine learning models that address difficult problems in materials
research, such as structure prediction[3]. The ability to efficiently design
and predict the structure of novel advanced materials with the assistance of
computed and experimental databases has shifted the materials innovation
challenge toward understanding and determining the synthesis routes for novel
materials[12]. In principle, this challenge could also be mastered using data-
driven approaches, since very few guiding principles are available for
materials synthesis. [13, 14] Indeed, in organic chemistry, AI-guided
synthesis planning [15, 16] has already been successfully implemented in
certain cases, such as in predicting retrosynthesis [17] and in complex
natural product synthesis design [18]. Although datasets for the synthesis of
organic materials are widely available [19, 20], there is not yet a large-
scale database of inorganic synthesis routes, which is needed to train
advanced deep-learning models to enable a breakthrough in AI-assisted design
and optimization of inorganic materials synthesis.

Scientific publications represent the largest repository of knowledge about
material synthesis and can be used as a reliable source of data. However,
human-written descriptions of syntheses require additional levels of
interpretation for conversion into a codified, machine-operable format.
Additionally, manual extraction of synthesis information is laborious, even
for a very limited number of papers[21, 22, 23, 24]. Given these obstacles, an
automated information extraction pipeline can accelerate data collection and
assist in building structured synthesis “recipes” from scientific text.
Natural language processing (NLP) approaches have been widely developed in the
past decade, and various advanced tools for high-quality information
extraction from unstructured text are available to researchers.

In materials science, NLP has been used to extract and analyze materials
properties [25, 26, 27], applications [28, 29], and synthesis conditions for
some limited cases [30]. Various NLP tools, including ChemDataExtractor [31],
OSCAR[32], ChemicalTagger[33], and others [34, 35, 21], have been developed to
extract information from chemical text. Recently, advanced models based on
deep convolutional and recurrent neural networks [36, 37, 38, 39] have been
proposed to improve the accuracy of chemical data extraction.

Text-mining approaches in materials science have also been used to construct
automated pipelines for collecting information about materials synthesis from
publications and to build large-scale publicly available datasets from such
collected data, including datasets of synthesis formulations for metal
oxides[40, 30, 41], germanium-containing zeolites[42], and perovskites[43]. In
recent work, our group has developed a text-mining pipeline to construct the
first large-scale dataset of solid-state ceramics synthesis “recipes”, which
includes not only the starting materials and final products but also the
synthesis actions, their attributes, and balanced chemical-reaction equations
[44].

In the current work, we built a more advanced extraction pipeline (Figure. 1)
which uses various advanced machine learning and natural language processing
techniques to extract precise recipe data for solution-based inorganic
materials synthesis procedures from the scientific literature. Solution
recipes are considerably more complex than solid state synthesis and require
the precise extraction of not only the chemicals involved but also their
respective amounts (since they determine concentration in solution). In
addition, more complex organic and mixed organic-inorganic compounds are used
to solubilize ions or to control solution conditions. By applying the
extraction pipeline, we codified 35,675 solution-based inorganic materials
synthesis “recipes” from over 4 million papers. Extracted information includes
target material and precursors, their quantities, and the synthesis operations
and their attributes. Information about the targets and precursors is then
used to build a reaction formula for every synthesis procedure. This dataset
is the first large-scale dataset of solution-based synthesis recipes, and
should pave the way for future data-driven approaches to inorganic materials
synthesis and synthesizability, and to design optimized synthesis procedures
in automated experimentation.

##  2 Methods

###  2.1 Content acquisition

The journal articles used in this work were downloaded with publisher consent
from Wiley, Elsevier, the Royal Society of Chemistry, the Electrochemical
Society, the American Chemical Society, the American Physical Society, the
American Institute of Physics, and Nature Publishing Group. A customized web-
scraper, Borges (see Codes Availability section below), was used to
automatically download a broad selection of materials-relevant papers
published after the year 2000 from publishers’ websites in HTML/XML format. We
selected 2000 as the cutoff year as parsing of materials science papers stored
as image PDFs (as for most papers published before 2000) introduces a
significant number of errors due to the limitations of currently available
optical character recognition models on chemistry containing text[45, 46].

To convert the articles from HTML/XML into raw-text files, we developed the
LimeSoup toolkit (see Codes Availability section below), which takes into
account the specific format standards of various publishers and journals. The
full-text and metadata of the articles such as the journal name, article
title, abstract, author names, etc., are stored in a MongoDB (www.mongodb.com)
database collection. To date, we have accumulated 4.06 million articles, which
are used for further processing down the pipeline (Figure 1).

###  2.2 Paragraph classification

Paragraphs containing information about solution synthesis (referred to as
“synthesis paragraphs” throughout this paper) were identified using a
Bidirectional Encoder Representations from Transformers (BERT) model [47]. The
model was pre-trained on full-text paragraphs of 2 million papers randomly
drawn from our database in a self-supervised way, i.e., by predicting masked
words based on their surrounding context. After training the BERT model, we
fine-tuned the paragraph classifier using 7,292 paragraphs labeled as either
“solid-state synthesis”, “sol-gel precursor synthesis”, “hydrothermal
synthesis”, “precipitation synthesis”, or “none of the above”. The resulting
F1 score of the paragraph classification is 99.5%, an improvement over the F1
score of 94.6% in our previous work [48], when evaluated using the same
labeled training dataset.

###  2.3 Synthesis recipe extraction

A solution-based synthesis recipe includes the precursors and target
materials, their quantities, and the synthesis actions and their attributes,
properly sequenced. This is the minimum essential information required to
complete a synthesis route. A schematic representation of the recipe is shown
in the bottom panel in Figure 1. In the sections below, we provide a brief
overview of the methods used for each step of the recipe extraction.

####  2.3.1 Materials entity recognition (MER)

Materials entities in synthesis paragraphs are identified and classified as
_target_ , _precursor_ , or _other_ via a two-step sequence-to-sequence model
as introduced in our previous work [37]. In the current work, we replaced the
original Word2Vec embedding model used previously [49] with a BERT model
trained on papers from the materials science domain (see Section 2.2 above).
First, each word token was transformed into a digitized BERT embedding vector.
A bi-directional long-short-term memory neural network with a conditional
random-field top layer (BiLSTM-CRF) was used to determine whether the token
was a materials entity or a regular word, and each materials entity was
replaced with the keyword <MAT> before being classified as either a _target_ ,
_precursor_ , or _other_ material using a second BERT-based BiLSTM-CRF
network. In addition to the 834 annotated solid-state synthesis paragraphs
from 750 papers used in our previous work[37], we manually annotated 447
solution-based synthesis paragraphs from 405 papers by labeling each word
token as _material_ , _target_ , _precursor_ , or _outside_. The annotated
dataset was split into training, validation, and test sets with a paper-wise
ratio of 700:150:305 to train the aforementioned two neural networks.

####  2.3.2 Extraction of synthesis actions and attributes

We implemented an algorithm which combines a neural network and sentence
dependency tree analysis to identify synthesis actions in the text. First, the
Word2Vec model from the Gensim library [50] was re-trained on $\sim$400,000
synthesis paragraphs of four synthesis types (see Section 2.2 above). These
word embeddings were used as the input for a recurrent neural network that
takes a sentence word-by-word and assigns labels to the verb tokens: _not-
operation_ , _mixing_ , _heating_ , _cooling_ , _shaping_ , _drying_ , or
_purifying_. For each obtained synthesis action, we parsed a dependency sub-
tree using the SpaCy library [51] to obtain information about the
corresponding temperature, time, and environment. To extract the corresponding
values of these attributes, we used a rule-based regular expression approach
[52].

####  2.3.3 Extraction of material quantities

To extract the numerical values of material quantities and assign them to the
corresponding materials obtained using the MER model (see Section 2.3.1
above), we applied a rule-based approach to search along the syntax tree [52].
The NLTK library [53] was used to build the syntax trees for each sentence in
a paragraph. The words in given sentences are leaf nodes of syntax trees. We
then applied an algorithm to cut the syntax tree of each sentence into the
largest sub-trees for every material, with each sub-tree having only one
material entity: 1. we first identified the materials on leaf nodes; 2\.
starting from each material, we identified the largest sub-trees, i.e., we
traversed the syntax tree upwards until there was more than one material leaf
node descending from the same node; 3\. the largest sub-tree for a given
material was defined as the sub-tree formed by the node and its descendants
identified in step 2. Next, we searched for the quantities in each sub-tree
given as molarity, concentration, or volume. Finally, we assigned the
quantities found to the unique material entity in the sub-tree.

####  2.3.4 Building reaction formulas

For every synthesis procedure described in a paragraph, we built a chemical
formula. Every material entity was converted from a text-string representation
into a chemical-data structure using an in-house material parser toolkit (see
Codes Availability section below). The data structure included information
about the material formula, composition, and ions. We then paired the target
with precursors containing at least one element in the target except for
hydrogen and oxygen and defined those precursors as “precursor candidates”.
Next, we computed the oxidation state change of elements from each “precursor
candidate” to the target and determined whether the precursor was oxidized or
reduced. If precursors were reduced or oxidized, we also included the
corresponding redox agents in the reaction formula. The agents can either be
another “precursor candidate” or a commonly used oxidizing or reducing agent
from the remaining material entities marked as _other_ or _precursor_ by the
MER algorithm (see Section 2.3.1 above).

###  2.4 Dataset generation

The dataset generation followed the protocol displayed in Figure 1. We
downloaded a total of 4,061,814 papers using web scraping and identified the
experimental sections by keyword matching in section headings, with keywords
including “experiment”, “synthesis”, “preparation”, and their morphological
derivations. ChemDataExtractor [31] was used to split the plain-text
paragraphs into sentences and words. After classification (see Section 2.2
above), 364,076 paragraphs describing solid-state, hydrothermal, sol-gel, and
precipitation syntheses were obtained. Among them, 189,553 paragraphs
described hydrothermal or precipitation syntheses, which we categorize as
solution-based synthesis methods. These paragraphs were further processed to
extract the precursors, targets, quantities, and operations with corresponding
conditions and to build the reaction formula (Figure 1).

##  3 Data Records

The solution-based synthesis dataset is provided as a single JSON file,
available at
https://doi.org/10.6084/m9.figshare.16583387.v1. There are 20,037 hydrothermal
synthesis reactions and 15,638 precipitation synthesis reactions. Each record
corresponds to a synthesis recipe extracted from a paragraph and is
represented as an individual JSON object. If a paragraph reported the
synthesis of several materials, the corresponding reactions were split into
separate data records. In addition to the chemical formula, the metadata for
each reaction returns the data structure used in our previous work[44], which
includes: DOI of the paper, a snippet of the corresponding synthesis paragraph
(50 first and 50 last characters to facilitate its lookup), chemical
information about the target and precursor materials used in the reaction, and
operations with their corresponding attributes. We also included the materials
with their corresponding quantities in the metadata. The details of the data
format are given in Table 1

The chemical formula for the reaction is stored as a string (reaction_string)
as well as in a dictionary containing lists of precursors (left_side) and
target materials (right_side) in reaction.

The metadata for target materials and precursors used to construct the
chemical formula are represented by the following data structure:

  * •

material_string: string of material as given in the original paragraph before
being parsed into a chemical composition.

  * •

material_formula: chemical formula associated with the material (given
originally or constructed empirically by parser).

  * •

composition: chemical composition of the material derived from its formula.
Aside from single-compound materials, we found that a large portion of the
materials (predominantly target materials) are composites, mixtures, solid
solutions, or alloys written as a sequence of compound-fraction pairs.
Therefore, a chemical-composition entity is represented by a list of
dictionary entries, where each item is associated with a compound found in the
materials formula. The fraction of each compound in the material is given in
amount, and its chemical composition (i.e., the elements and stoichiometry) is
given in elements. If a material is one compound, the list has only one item
and amount=1.0. If a material is a hydrate, water is added to the composition
list with amount corresponding to the amount of water molecules (if
specified).

  * •

additives: list of additive elements (i.e., elements used for doping,
stabilization, or substitution) resolved from the material string.

  * •

elements_vars: lists all variable elements and their corresponding values
found in the materials.

  * •

amounts_vars: lists all variable element ratios and their corresponding values
found in the material formula. The values of each variable are given as a
structure with values listing the values of each specific variable and
max_value/min_value values if a range is given in the paragraph.

  * •

oxygen_deficiency: yes/no attribute that reflects whether a material was
synthesized with unspecified oxygen stoichiometry.

  * •

mp_id: ID of the lowest-energy polymorph entry in the Materials Project
database (materialsproject.org) if the material is found there.

To facilitate querying of the dataset, the targets_string field contains the
target material formulas, and the solvents field contains all solvent(s) from
matching material entities marked as _other_ by the MER model with a table of
common solvents adopted from Common Solvents Used in Organic Chemistry
(organicchemistrydata.org/solvents/).

##  4 Technical Validation

###  4.1 Extraction completeness and accuracy

To ensure high accuracy of the dataset, we included only those data that
produced complete reaction formulas at the final step of the pipeline. This
strategy reduced potential errors in the dataset that may have been caused by
composition-parsing failure, incomplete extraction, or incomplete information
provided by the text. We applied the extraction pipeline to 189,553 solution-
based synthesis paragraphs, 28,749 of which generated a reaction formula,
giving an extraction yield of $\sim$15%. To evaluate the source of the loss,
we randomly selected and manually checked 100 solution-based synthesis
paragraphs that did not produce any reactions. Among those 100 paragraphs, 36
were written with an incomplete list of precursors or targets in the text,
such that human experts would not be able to reconstruct the reaction based
solely on the information provided in the paragraph. For the remaining 64
paragraphs, the loss was due to: 1. the use of organic precursors with complex
groups or complicated notation (e.g., acronyms) that could not be parsed into
a chemical composition by our parser or 2. MER misidentification resulting in
an incomplete or incorrect list of precursors and (or) target entities such
that the reactions could not be built.

To evaluate the quality of the dataset, we had a human expert test 100
randomly pulled entries. The human expert manually extracted the information
presented in the recipe, and the results were compared with those extracted by
the pipeline. Table 2 presents the accuracy statistics, which include the
precision, recall, and F1 scores calculated from the tested entries. For the
fields that included reaction, targets, precursors, operations, operation
temperatures, time, and atmosphere, the F1 scores were over 90%. The
relatively low recall, and hence F1 score, for the extraction of materials
quantities can be mainly explained by the MER algorithm missing the
corresponding material entity and, thus, the quantities not being assigned.
The accuracy of the obtained dataset is comparable to that in our previous
work[44] and of other text-mined datasets[40].

###  4.2 Data exploration analysis

To test the diversity of the dataset and its coverage of the materials space,
we analyzed unique materials (targets and precursors) and reactions. The
dataset contains 11,603 unique reactions that include 2,870 unique precursors
and 5,416 unique targets. The ten most frequent targets in the dataset and
their corresponding precursors are listed in Table 3. The target list captures
materials that have drawn substantial attention in the past two decades:
catalysts (ZnO, Fe2O3, TiO2, Fe3O4, SnO2, ZrO2, CuO), adsorbents (SiO2),
various materials for sensors (ZnO, Fe2O3, WO3), quantum dots (CdS), and
semiconductors (ZnO, TiO2, SnO2, CdS). Unsurprisingly, these most frequent
target materials usually appear in multiple applications, as they possess
desirable physical and chemical properties in many scientific and engineering
fields.

We use the periodic table representation (Figure 2) to visualize the chemical
space covered by the dataset. For each element, the fraction of synthesis
recipes containing this element in the target formula is shown with the
yellow-to-navy blue gradient framed at the top of each element box. The most
data-rich elements are transition metals in the third period, such as Zn, Fe,
Ti, Ni, and Co, in accordance with the compounds listed in Table 3. The next-
most prevalent targets are materials with Bi, Sn, Al, W, Mo, Cu, Zr, or Li.
The least common elements are rare elements such as Ru, Rh, Hf, Ta, Re, and
Ir. The elements Fr, Ra, Tc, and Pm are not present in the target materials of
the dataset. Additionally, we calculated the frequency of co-occurrence of
chemical elements and common ions in precursor materials to understand how
different ions are brought into solution. In Figure 2, the frequencies for
each ion are displayed as colored bars. The length of the bar is the fraction
of one specific ion paired with the element normalized over all precursors for
this element.

The results displayed in the periodic table (Figure 2) agree well with several
rules-of-thumb for the selection of precursors usually applied by scientists
to synthesize materials [54]. First, the commonly used precursors are mainly
those that are widely available from companies such as Sigma-Aldrich and
Fisher Scientific. For example, Li2CO3 or LiOH for Li and sulfate or chloride
for Fe. Second, solubility also affects the choice of precursors. For
instance, inorganic salts (such as nitrates, sulfates, and chlorides) are
commonly used because of their high solubility [54]. Third, the organization
of the periodic table reflects relationships between the various element
properties. We observed that precursors with neighboring elements in the
periodic table tend to have similar ions paired. For instance, nitrates,
sulfates, and chlorides are commonly used anions for 3rd-period transition
metals, whereas the precursors for lanthanides are mostly oxides and nitrates.

We used information about the extracted materials and sequences of synthesis
actions to classify the solution-based synthesis recipes into four categories
of synthesis protocols (table in Figure 3) according to the following
definitions:

  * •

solution-mixing with firing step has a final firing step after the precipitate
is obtained from the solution;

  * •

aqueous solution synthesis has no final firing step after precipitating the
compound from the solution and the solvent is water;

  * •

non-aqueous solution synthesis has no final firing step after precipitating
the compound from solution and the solvent(s) is (are) organic;

  * •

aqueous–non-aqueous mixed solution synthesis has no final firing step after
precipitating the compound from solution and the solvents are a mixture of
water and organic solvent(s).

The resulting distributions of synthesis protocols over the aforementioned
categories are shown in the two pie charts in the top-right corner of Figure
3. Note that as solution-based synthesis includes both hydrothermal and
precipitation synthesis according to our definition (see Section 2.2), we
analyzed these synthesis types separately. As observed in the pie charts, only
20% of the recipes in the hydrothermal synthesis subset have a firing step
after solution mixing. Among those that do not have firing step, 63% use only
water as a solvent, 8% use only organic solvents, and 9% use both water and
organic solvents. In contrast, the fractions in the precipitation synthesis
subset are 43%, 46%, 5%, and 6%, respectively.

These results are as expected. A firing step after solution mixing plays
various roles in synthesis. For example, it can dehydrate the targets,
decompose the intermediates to produce the final products, change the
oxidation state, change the morphology, or improve crystallization. [55, 56,
57]. Thus, the decision of whether to use firing after solution steps strongly
depends on the chemical properties of the target. To explore this in more
detail, we split the targets according to their anion type (oxide, sulfide,
etc.) and different oxidation states of several data-rich transition-metal
elements. We then computed the distribution of synthesis categories for each
of the split subsets. Figure 3 presents the results for the most prevalent
subsets of oxides, sulfides, and elements Fe2+, Fe3+, Co3+, Ni2+, Cu2+, and
Zn2+. The fraction of recipes with a firing step in precipitation synthesis is
larger than that in hydrothermal synthesis. This observation holds for all
targets, all oxides, all sulfides, and individual oxides and sulfides with
queried oxidation states of transition metals. This finding can be also
interpreted as hydrothermal synthesis often being used to obtain final
products in a “one-shot” process, without subsequent firing after solution
mixing, likely because many compounds can be crystallized as anhydrous powders
with controlled size and morphology directly from hydrothermal synthesis.
According to the standard hydrothermal synthesis procedure, the reaction is
performed under high pressure and at a temperature higher than the boiling
point of the solvent under normal pressure in an autoclave. In contrast,
precipitation synthesis is performed under normal pressure. The higher
temperature and high pressure applied during hydrothermal synthesis, relative
to the ambient conditions typically used in precipitation methods, are
associated with enhanced kinetics in chemical transport, nucleation, and
crystal growth and thus with a more effective dissolution–recrystallization
process. Furthermore, the physico-chemical properties, such as the viscosity
and dielectric constant of water or other solvents, change pronouncedly under
conditions of hydrothermal synthesis, affecting the solubility and mobility of
species in the solution and eventually facilitating crystallization [58].
Therefore, hydrothermal synthesis does not need a firing step as often as
precipitation synthesis.

The hydrothermal synthesis data subset has more records for sulfides than the
precipitation subset. One possible reason is that the desired morphology of
sulfides can be achieved directly from hydrothermal synthesis, as we discussed
before[58]. Another possible reason might be that most sulfides are not
thermally stable and may decompose under firing temperature. The thermal
instability also explains why the fraction of recipes with a firing step for
sulfides is significantly lower than that for oxides. Furthermore, sulfide
synthesis often requires temperatures higher than ambient conditions.
Temperatures in precipitation synthesis are not high enough to trigger
reactions to produce sulfides [59, 60].

Solution-based synthesis is an important area of materials synthesis [48]. It
provides the ability to precisely control the morphology of the synthesized
specimen[54] or synthesize metastable phases[61]. In order to extract
solution-based synthesis recipes, we have built NLP/ML approaches wrapped into
an automated text-mining pipeline: 1. we annotated new solution-based
synthesis experimental data; 2. we incorporated a pre-trained BERT model to
predict the type of synthesis described in paragraphs as well as to recognize
material entities; 3. we implemented a recurrent neural network to extract
synthesis actions; 4. we built chemical formulas from material entities based
on chemistry knowledge; and 5. we developed a new rule-based method to extract
the quantities of the materials.

Nevertheless, challenges remain in the mining of scientific literature and
construction of robust and accurate large-scale datasets. First, the organic
precursors with complex radicals commonly used in solution-based synthesis
pose a challenge for parsing and extracting chemical information. Constructing
reaction formulas becomes problematic when the precursor information is lost.
Therefore, these entries are mostly dropped out later in the pipeline. To
address this issue, a universal parser that can parse chemical tokens needs to
be developed.

Second, our data were extracted from the experimental section from the main
body of each paper and do not include any information about the actual
synthesis results, e.g., whether the material was synthesized using the
reported procedure or which structure was obtained. This problem can be
overcome by introducing a model that can parse characterization data (e.g.,
X-ray diffraction patterns or electron microscopy images) and relate them to
the corresponding synthesis conditions. To the best of our knowledge, this
task has not been performed to date and is the subject of future efforts. Even
though the actual results of a synthesis can be extracted from a paper, there
remains the challenge of data interpretation and usage, as the authors usually
report only successful and “cherry-picked” experimental results. This
introduces significant anthropogenic bias toward “positive” data with little
“negative” content in the dataset, thus limiting the tasks for future machine-
learning applications [62, 63]. A promising approach to solve this issue is to
incorporate results obtained by autonomous robotic synthesis platforms that
can provide a vast amount of “negative” data in a reasonable time frame [64,
65].

Finally, solution-based synthesis is advantageous when the control of specimen
morphology is required, e.g., when synthesizing noble-metal nanoparticles.
However, this dataset does not provide information about the morphology of the
synthesized materials, though such information is often contained in
characterization or results paragraphs instead of the experimental section.
The extraction of morphology and other solution synthesis outcomes is another
text-mining challenge in materials science research that requires the
development of advanced algorithms and models [21], which is beyond the scope
of the current study.

##  5 Usage Notes

The dataset is provided in JSON format as a single file. All major programming
languages, such as Python, Matlab, R, and Wolfram Mathematica, can be used to
read it. No particular dependency is required.

Because the dataset contains detailed information about chemical formulas as
well as the compositions of the target materials and precursors for each
recipe, it can be easily used to conduct a literature review by querying
desired precursors and (or) targets in different chemical spaces. For example,
selecting all TiO2 synthesized from TiCl4 allows an exploration of how other
synthesis formulations, such as synthesis actions, attributes, and quantities,
affect the results. Furthermore, the materials entries in the dataset are
supplied with the Materials Project [5] identifiers, thus facilitating the
integration of the recipes with the thermochemical data available in the
Materials Project[66, 61].

In addition, this solution-based synthesis dataset keeps the same data
structure as that in the solid-state dataset generated in our previous work
[44]. Therefore, it is easy to analyze the recipes from the two datasets.

Despite the dataset being provided as a static snapshot, we intend to update
it on a regular basis.

##  6 Code availability

The scripts used to classify paragraphs and extract recipes as well as to
perform the data analysis are home-written codes which are publicly available
at the github repository https://github.com/CederGroupHub/text-mined-solution-
synthesis_public with acknowledgement of the current paper.

The underlying libraries used in this project are all open-source:

_Tensorflow_ (www.tensorflow.org)

_Keras_ (keras.io)

_SpaCy_ (spacy.io) [51]

_NLTK_ (https://www.nltk.org/) [53]

_gensim_ (radimrehurek.com) [50]

_scikit-learn_ (scikit-learn.org) [67]

_ChemDataExtractor_ (chemdataextractor.org) [31]

_Material Parser_ (github.com/CederGroupHub/MaterialParser)

_Borges_ (github.com/CederGroupHub/Borges)

LimeSoup (github.com/CederGroupHub/LimeSoup).

##  7 Acknowledgements

This material is based upon work supported by the National Science Foundation
under Grant No. DMR-1922372. We also thank Chris Bartel, and Amalie Trewartha
for valuable discussions.

##  8 Author contributions

Z.W. developed material quantities extraction algorithm, building reaction
formula algorithm, Materials Parser, operations and conditions extraction
algorithms, analyzed the data, and wrote the manuscript. O.K. developed the
Materials Parser, operations and conditions extraction algorithms, analyzed
the data, and wrote the manuscript. K.C. developed operations and conditions
extraction algorithms, manually inspected the dataset, and wrote the
manuscript. T.H. developed materials entity recognition algorithm, and
manually inspected the dataset. H.H. curated paragraphs databases, and
developed the paragraphs classifier. Y.F. developed building reaction formula
algorithm and manually inspected the dataset. Y.Z. analyzed the data and
manually inspected the dataset. Y.S. analyzed the data and manually inspected
the dataset. Z.C. analyzed the data and manually inspected the dataset. W.S.
analyzed the data and developed material quantities extraction algorithm. G.C.
developed approach, supervised the project, and wrote the manuscript. All
authors discussed the results and contributed to the final manuscript.

##  9 Competing interests

The authors declare no competing interests.

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## Tables

[b]  Data description Data Key Label Data Type DOI of the original paper doi
_string_ Snippet of the raw text paragraph_string _string_ Chemical formula
reaction Object (_dict_): \- left_side: _list_ of _strings_ \- right_side:
_list_ of _strings_ Chemical formula in string format reaction_string _string_
Target material data target Object (_dict_): \- material_string: _string_ , \-
material_formula: _string_ , \- composition: _list_ of Objects2, \- additives:
_list_ of _strings_ \- elements_vars: {var: _list_ of _strings_} \-
amounts_vars: {var: _list_ of Objects3} \- oxygen_deficiency: _boolean_ \-
mp_id: _string_ List of target formulas obtained after variables substitution
targets_string _list_ of _strings_ Precursor materials data precursors _list_
of Objects (See target) List of solvent formulas solvents_string _list_ of
_strings_ Sequence of synthesis steps and corresponding conditions operations
_list_ of Objects (_dict_): \- token: _string_ , \- type: _string_ \-
conditions: Object \- - temperature: _list_ of Objects4 \- - time: _list_ of
Objects4, \- - atmosphere: _list_ of _strings_ \- - mixing_device: _list_ of
_strings_ \- - mixing_media: _list_ of _strings_ Materials with corresponding
quantities quantities _list_ of Objects (_dict_): \- material: _string_ , \-
quantity: _list_ of Objects5 Synthesis type type _string_

  * 1

{amount: _float_ , material: _string_}

  * 2

{formula: _string_ , elements: {element: amount of element}, amount: _string_}

  * 3

{max_value: _float_ , min_value: _float_ , values: _list_ of _floats_}

  * 4

{max_value: _float_ , min_value: _float_ , values: _list_ of _floats_ , units:
_string_ }

  * 5

{number: _float_ , unit: _string_ }

Table 1:  Format of each data record: description, key label, data type.

Data attribute | Precision | Recall | F1 score
---|---|---|---
Balanced reactions | 0.94 | / | /
\- targets | 0.97 | / | /
\- precursors | 0.98 | 0.99 | 0.98
Operations | 0.96 | 0.85 | 0.90
Conditions |  |  |
\- temperature | 0.97 | 0.92 | 0.94
\- time | 0.98 | 0.89 | 0.93
\- atmosphere | 0.97 | 0.92 | 0.94
Quantities | 0.90 | 0.85 | 0.87
Table 2:  Performance of data extraction for dataset entries.  Targets | Common Precursors
---|---
ZnO | Zn(NO3)2, Zn(Ac)2, ZnCl2
TiO2 | Ti(OCH(CH3))4, Ti(OC4H9)4, TiCl4
Fe3O4 | FeCl3, FeCl2
Fe2O3 | FeCl3, Fe(NO3)3
SnO2 | SnCl4
ZrO2 | ZrOCl2, ZrO(NO3)2
CuO | Cu(NO3)2, Cu(Ac)2, CuCl2, CuSO4
SiO2 | Si(OC2H5)4
WO3 | Na2WO4, WCl6, H2WO4
CdS | Na2S, CH4N2S, CdCl2, Cd(NO3)2
Table 3:  Ten most common targets in the dataset with their corresponding
precursors.

## Figures

Figure 1:  Extraction pipeline and example. Top panel: Schematic
representation of the standard text mining pipeline: (i) scrape papers in
markup format from the major publishers; (ii) identify and classify synthesis
sections; (iii) extract key information including materials, amounts,
sequenced operations, and conditions; (iv) store synthesis recipes into the
database for future data mining. Bottom panel: Example of a codified recipe
extracted from a synthesis paragraph.  Figure 2:  The chemical space covered
by the dataset. For each element, the box containing the element name is
colored in a yellow-to-navy blue gradient representing the total amount of
reactions that produce a target compound containing the element. The bar graph
below each element shows the list of ions paired with the element in precursor
compounds. The fractions of the precursors (i.e. element+ion) used are shown
by the length of the bars. Boxes with no bar graph represent elements
occurring in five and fewer targets. “Ac” stands for acetate radical CH3COO–
in the compound formula.  Figure 3:  Correspondence between choice of
synthesis route and selected types of targets. The top table gives an example
of the four synthesis categories defined: with firing step, aqueous, non-
aqueous, and mixed. The two pie-charts on the top-right show the fractions of
synthesis routes in the hydrothermal and precipitation datasets separately.
The four rows of pie charts in the lower half of the figure represent the
fractions of the four synthesis routes (given in the table) for all oxides,
all sulfides, and individual oxides and sulfides with different oxidation
states of data-rich transition metals separately. The first and second rows
are results from the hydrothermal dataset. The third and fourth rows are
results from the precipitation dataset. Each blank space means that there is
not enough data to form a statistic for the corresponding type of target.


# A short proof of Seymour’s 6-flow theorem

Matt DeVos  Email<EMAIL_ADDRESS>Supported by an NSERC Discovery Grant
(Canada) Kathryn Nurse  Email<EMAIL_ADDRESS>
###### Abstract

We give a compact variation of Seymour’s proof that every $2$-edge-connected
graph has a nowhere-zero $\mathbb{Z}_{2}\times\mathbb{Z}_{3}$-flow.

All graphs are finite; loops and multiple edges are allowed. For notation not
defined here we use [1]. Let $G=(V,E)$ be a directed graph, $A$ an additively-
written abelian group, and $f:E\to A$ a function. We say $f$ is an $A$-flow
whenever $\sum_{e\in\delta^{+}(v)}f(e)=\sum_{e\in\delta^{-}(v)}f(e)$ holds for
every $v\in V$, where $\delta^{+}(v)$ ($\delta^{-}(v)$) is the set of edges
whose initial (terminal) vertex is $v$. If $0\not\in f(E)$, then we say $f$ is
_nowhere-zero_. If $f$ is a $\mathbb{Z}$-flow with $f(E)\subseteq\\{0,\pm
1,\pm 2,\dots,\pm(k-1)\\}$, then we say $f$ is a _$k$ -flow_. Note that
reversing an edge $e$ and replacing $f(e)$ with its negation preserves all of
the aforementioned properties; accordingly the existence of a nowhere-zero
$A$-flow or $k$-flow depends only on the underlying graph. A famous conjecture
of Tutte [3] asserts that every $2$-edge-connected graph has a nowhere-zero
$5$-flow. This conjecture remains open with the best result due to Seymour [2]
who proved that such graphs have nowhere-zero $6$-flows. His argument involves
a standard reduction due to Tutte equating the existence of a nowhere-zero
$k$-flow and a nowhere-zero $A$-flow whenever $|A|=k$, together with the
following central result.

######  Theorem 1 (Seymour).

Every $2$-edge-connected digraph has a nowhere-zero
$\mathbb{Z}_{2}\times\mathbb{Z}_{3}$-flow.

We give a compact version of Seymour’s proof. We prove a slightly stronger
statement by induction, using simple contraction-based arguments (and in
particular, we don’t require any special reductions). Our proof of Theorem 2
relies on contracting a set of edges $S$, finding a flow $f$ in the smaller
graph, then uncontracting $S$ and extending the domain of $f$ to include $S$.
Observe that 1. it is always possible to extend the domain while maintaining
that $f$ is a flow, and 2. if $S$ is a set of at least two parallel edges, and
the abelian group has size at least 3, then it is possible to extend the
domain so that additionally $0\not\in f(S)$. In the following we use $G/S$ to
denote the graph obtained from $G$ by contracting the set of edges $S\subseteq
E$, and $\delta(u)$ is the set of edges incident to vertex $u$.

######  Theorem 2.

If $G=(V,E)$ is a $2$-edge-connected digraph, and $u\in V$, then $G$ has a
nowhere-zero flow $f_{2}\times f_{3}:E\to\mathbb{Z}_{2}\times\mathbb{Z}_{3}$
so that $\delta(u)\cap\operatorname{supp}(f_{2})=\emptyset$.

###### Proof.

We proceed by induction on $|V|$, with the base case $|V|=1$ holding
trivially. First, suppose $G-u$ has a 1-edge-cut $E(V_{1},V_{2})=\\{e\\}$.
Choose a partition $\\{E_{1},E_{2}\\}$ of $E\setminus\\{e\\}$ so that for
$i=1,2$ the edges in $E_{i}$ have ends in $V_{i}\cup\\{u\\}$. Let
$G_{i}=G/E_{i}$. By induction, $G_{i}$ has a nowhere-zero flow
$f^{i}_{2}\times f^{i}_{3}:E(G_{i})\to\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ so
the edges incident to the contracted vertex are not in the support of
$f_{2}^{i}$. By possibly replacing $f^{1}_{3}$ with its negation, we may
assume that $f^{1}_{3}(e)=f^{2}_{3}(e)$ and then the
$\mathbb{Z}_{2}\times\mathbb{Z}_{3}$-flows in each $G_{i}$ combine to give the
desired flow in $G$. Thus we may assume $G-u$ has no cut-edge.

Choose distinct edges $ux$ and $ux^{\prime}$ so that $x$ and $x^{\prime}$ are
in the same component of $G-u$ (possibly $x=x^{\prime}$). By our assumptions,
we may choose two edge-disjoint paths $P_{1},P_{2}\subseteq G-u$ from $x$ to
$x^{\prime}$. Set $H=P\cup P^{\prime}$, $S=E(u,V(H))$, $G_{1}=G/E(H)$ with
contracted vertex $u_{1}$, and $G_{2}=G_{1}/S$ with contracted vertex $u_{2}$.
By induction, $G_{2}$ has a flow $f_{2}\times
f_{3}:E(G_{2})\to\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ so that
$\delta(u_{2})\cap\operatorname{supp}(f_{2})=\emptyset$. Because $S$ is a set
of at least two parallel edges, we may extend $f_{3}$ to $E(G_{1})$ so that it
remains a flow and $f_{3}(e)\neq 0$ for all $e\in S$. Because
$\delta(u_{2})\cap\operatorname{supp}(f_{2})=\emptyset$, setting $f_{2}(e)=0$
for all $e\in S$ extends $f_{2}$ to $E(G_{1})$ keeping it a flow. Note that
$\delta_{G_{1}}(u)\cap\operatorname{supp}(f_{2})=\emptyset=\delta_{G_{1}}(u_{1})\cap\operatorname{supp}(f_{2})$.
Now, further extend $f_{3}$ to $E(G)$ so that it remains a flow. Because
$\delta(u_{1})\cap\operatorname{supp}(f_{2})=\emptyset$, and every vertex of
$H$ has even degree, we may extend $f_{2}$ to $E(G)$ by setting $f_{2}(e)=1$
for all $e\in E(H)$, keeping it a flow. Now
$S\subseteq\operatorname{supp}(f_{3}),E(H)\subseteq\operatorname{supp}(f_{2})$
and so $f_{2}\times f_{3}$ is as desired. ∎

## References

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