On the Linearization Covering Technique

On the Linearization Covering Technique
and its Application to Integrable Nonlinear
Differential Systems

Anatolij K. PRYKARPATSKI †‡

A.K. Prykarpatski

† The Department of Physics, Mathematics and Computer Science,
† Cracow University of Technology, Kraków 30-155, Poland ‡ Ivan Franko State
Pedagogical University of Drohobych, Lviv Region, Ukraine
<EMAIL_ADDRESS>
Received January 22, 2018, in final form February 28, 2018; Published online
March 16, 2018

In this letter I analyze a covering jet manifold scheme, its relation to the
invariant theory of the associated vector fields, and applications to the
Lax–Sato-type integrability of nonlinear dispersionless differential systems.
The related contact geometry linearization covering scheme is also discussed.
The devised techniques are demonstrated for such nonlinear Lax–Sato integrable
equations as Gibbons–Tsarev, ABC, Manakov–Santini and the differential Toda
singular manifold equations.

covering jet manifold; linearization; Hamilton–Jacobi equations; Lax–Sato
representation; ABC equation; Gibbons–Tsarev equation; Manakov–Santini
equation; contact geometry; differential Toda singular manifold equations

17B68; 17B80; 35Q53; 35G25; 35N10; 37K35; 58J70; 58J72; 34A34; 37K05; 37K10

##  1 Introductory notions and examples

Some three years ago I. Krasil’shchik in his work [21] has analyzed the so-
called Gibbons–Tsarev equation

$\displaystyle z_{yy}+z_{t}z_{ty}-z_{y}z_{tt}+1=0,$ (1.1)

and its so-called nonlinear first-order differential covering

$\displaystyle\frac{\partial w}{\partial
t}-\frac{1}{z_{y}+z_{t}w-w^{2}}=0,\qquad\frac{\partial w}{\partial
y}+\frac{z_{t}-w}{u_{y}+z_{t}w-w^{2}}=0,$ (1.2)

which for any solution $z\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$ to
equation (1.1) is compatible for all $(t,y)\in\mathbb{R}^{2}$. He has shown
that the latter makes it possible to determine for any smooth solution
$w\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$ to the equation (1.2) suitable
smooth functions
$\psi\colon\mathbb{R}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ satisfying the
associated linear representation of Lax–Sato type:

$\displaystyle\frac{\partial\psi}{\partial
t}+\frac{1}{z_{y}+z_{t}\lambda-\lambda^{2}}\frac{\partial\psi}{\partial\lambda}=0,\qquad\frac{\partial\psi}{\partial
y}-\frac{z_{t}-\lambda}{z_{y}+z_{t}\lambda-\lambda^{2}}\frac{\partial\psi}{\partial\lambda}=0,$

which for any solution to the equation (1.1) is also compatible for all
$(t,y)\in\mathbb{R}^{2}$ and an arbitrary parameter $\lambda\in\mathbb{R}$.

In his work [21] I. Krasil’shchik also posed an interesting problem of
providing a differential-geometric explanation of the linearization procedure
for a given nonlinear differential-geometric relation
$J^{1}(\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R})|_{\mathcal{E}}$ in the
jet manifold
$J^{1}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}\big{)}$,
$n\in\mathbb{Z}_{+}$, realizing a compatible covering [1] for the associated
nonlinear differential equation $\mathcal{E}[x,\tau;u]=0$, imbedded into some
jet manifold
$J^{k}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}^{m}\big{)}$ for
some $k,m\in\mathbb{Z}_{+}$. His extended version of explanation of this
procedure, presented in [21], appeared abstract enough and held it in a hidden
form with no new example demonstrating its application. From this point of
view, our work seeks to unveil some important points of this linearization
procedure in the framework of the classical inhomogeneous vector field
equations and present its new and important applications. We consider the jet
manifold
$J^{k}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}^{m}\big{)}$ for
some fixed $k,m\in\mathbb{Z}_{+}$ and some differential relation [18] in a
general form $\mathcal{E}[x,\tau;u]=0$, satisfied for all $(x;\tau)\in$
$\mathbb{R}^{n}\times\mathbb{R}^{2}$ and suitable smooth mappings
$u\colon\mathbb{R}^{n}\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{m}$.

As a new and interesting example, we can take $n=1=m$, $k=2$ and choose a
differential relation $\mathcal{E}[x;y,t;u]=0$ of the form

$\displaystyle u_{t}u_{xy}-k_{1}u_{x}u_{ty}-k_{2}u_{y}u_{tx}=0,$ (1.3)

the so-called ABC equation, or the Zakharevich equation, first discussed in
[31] for $k_{1}+k_{2}-1=0$, where
$u\colon\mathbb{R}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ and
$k_{1},k_{2}\in\mathbb{R}$ are arbitrary parameters satisfying the conditions

$\displaystyle k_{1}+k_{2}-1=0\vee k_{1}+k_{2}-1\neq 0.$

The first case $k_{1}+k_{2}-1=0$ was before intensively investigated in [12,
25, 31] and recently in [19], where there was found its Lax–Sato type
linearization and many other its interesting mathematical properties were
revealed. For the second case $k_{1}+k_{2}-1\neq 0$ the following crucial
result was stated in equivalent form by P.A. Burovskiy, E.V. Ferapontov and
S.P. Tsarev in [8] and recently by I. Krasil’shchik, A. Sergyeyev and O.
Morozov in [22].

######  Proposition 1.1.

The covering system
$J^{1}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$
of quasi-linear first-order differential relations

$\displaystyle\frac{\partial w}{\partial
t}+\frac{u_{t}w}{u_{x}k_{1}(k_{1}+k_{2}-1)}\frac{\partial w}{\partial
x}-\frac{w(w+k_{1}+k_{2}-1)u_{tx}}{u_{x}k_{1}}=0,$
$\displaystyle\frac{\partial w}{\partial
y}+\frac{u_{y}w}{u_{x}k_{1}(w+k_{1}+k_{2}-1)}\frac{\partial w}{\partial
x}-\frac{w(k_{1}+k_{2}-1)u_{yx}}{u_{x}k_{1}}=0$ (1.4)

on the jet manifold
$J^{1}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}\big{)}$ is compatible,
that is, it holds for any its smooth solution
$w\colon\mathbb{R}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ at all points
$(x;y,t)\in\mathbb{R}\times\mathbb{R}^{2}$, iff the function
$u\colon\mathbb{R}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ satisfies the
ABC-equation (1.3).

Moreover, this result was recently generalized in [22] to the following
“linearizing” proposition.

######  Proposition 1.2.

A system $J_{\rm
lin}^{1}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$
of linear first-order differential relations

$\displaystyle\frac{\partial\psi}{\partial t}+\frac{\lambda
u_{x}^{\frac{k_{2}-1}{k_{1}}}u_{t}}{k_{1}(k_{1}+k_{2}-1)}\frac{\partial\psi}{\partial
x}-\frac{\lambda^{2}u_{x}^{\frac{k_{2}-k_{1}-1}{k_{1}}}(u_{t}u_{xx}-k_{1}u_{xx}u_{xt})}{k_{1}^{2}}\frac{\partial\psi}{\partial\lambda}=0,$
$\displaystyle\frac{\partial\psi}{\partial y}+\frac{k_{2}\lambda
u_{x}^{\frac{k_{2}-1}{k_{1}}}u_{y}}{k_{1}\big{(}\lambda
u_{x}^{\frac{k_{2}-k_{1}-1}{k_{1}}}+k_{1}+k_{2}-1\big{)}}\frac{\partial\psi}{\partial
x}-\frac{\lambda^{2}k_{2}(k_{1}+k_{2}-1)u_{x}^{\frac{k_{2}-k_{1}-1}{k_{1}}}u_{y}u_{xx}}{k_{1}^{2}\big{(}\lambda
u_{x}^{\frac{k_{2}-k_{1}-1}{k_{1}}}+k_{1}+k_{2}-1\big{)}}\frac{\partial\psi}{\partial\lambda}=0$
(1.5)

on the covering jet manifold
$J^{1}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}$ is
compatible, that is, it holds for any its smooth solution
$\psi\colon\mathbb{R}^{2}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ at all
points $(x,\lambda;y,t)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$, iff the
function $u\colon\mathbb{R}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$
satisfies the ABC-equation (1.3).

A similar result, when $n=1$, $m,k=2$, we have stated for the Manakov–Santini
equations

$\displaystyle u_{tx}+u_{yy}+(uu_{x})_{x}+v_{x}u_{xy}-v_{y}u_{xx}=0,$
$\displaystyle v_{xt}+v_{yy}+uv_{xx}+v_{x}u_{xy}-v_{y}v_{xx}=0,$ (1.6)

whose Lax–Sato integrability was extensively studied earlier in [6, 11, 13,
19, 23, 26]. The system (1.6) as a jet submanifold
$J^{1}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}\subset
J^{1}(\mathbb{R}\times\mathbb{R}^{2};\mathbb{R})$ allows the following
nonlinear first-order differential Lax-type representation

$\displaystyle\frac{\partial w}{\partial
t}+\big{(}w^{2}-wv_{x}+u-v_{y}\big{)}\frac{\partial w}{\partial
x}+u_{x}w-u_{y}+v_{yy}+v_{x}(v_{y}-u)_{x}=0,$ $\displaystyle\frac{\partial
w}{\partial y}+w\frac{\partial w}{\partial x}-v_{xx}w+(u-v_{y})_{x}=0,$ (1.7)

compatible on solutions to the nonlinear differential relation
$\mathcal{E}[x,\tau;u]=0$ (1.6) on
$J^{2}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}^{2}\big{)}$. The
existence of the compatible representation (1.7) makes it possible to state
the following proposition.

######  Proposition 1.3.

A covering system $J_{\rm
lin}^{1}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$
of linear first-order differential relations

$\displaystyle\frac{\partial\psi}{\partial t}+\big{(}\lambda^{2}+\lambda
v_{x}+u-v_{y}\big{)}\frac{\partial\psi}{\partial x}+(u_{y}-\lambda
u_{x})\frac{\partial\psi}{\partial\lambda}=0,$
$\displaystyle\frac{\partial\psi}{\partial
y}+(v_{x}+\lambda)\frac{\partial\psi}{\partial
x}-u_{x}\frac{\partial\psi}{\partial\lambda}=0$

on the jet manifold
$J^{1}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}$ is
compatible, that is, it holds for any its smooth solution
$\psi\colon\mathbb{R}^{2}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ at all
points $(x,\lambda;y,t)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$, iff the
function $u\colon\mathbb{R}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$
satisfies the Manakov–Santini equation (1.6).

From the point of view, based on the propositions above, the substance of our
present analysis, similarly to that suggested in [21], is to recover the
intrinsic mathematical structure responsible for the existence of the
“linearizing” covering jet manifold mappings

$\displaystyle J_{\rm
lin}^{1}\big{(}\mathbb{R}^{n+1}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}\rightleftharpoons
J^{1}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$
(1.8)

for any dimension $n\in\mathbb{Z}_{+}$, compatible with our differential
relation $\mathcal{E}[x,\tau;u]=0$, as it was presented above in the form
(1.4) and (1.5) for the differential relation (1.3). Thus, for a given
nonlinear differential relation $\mathcal{E}[x,\tau;u]=0$ on the jet manifold
$J^{k}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}^{m}\big{)}$ for
some $k\in\mathbb{Z}_{+}$ one can pose the following problem:

Suppose there holds a compatible system
$J^{1}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}^{m}\big{)}|_{\mathcal{E}}\subset
J^{1}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}^{m}\big{)}$ of
quasi-linear first-order differential relations. How then can one construct a
linearizing first-order differential system $J_{\rm
lin}^{1}\big{(}\mathbb{R}^{(1+n)+2};\mathbb{R}\big{)}|_{\mathcal{E}}\subset
J^{1}\big{(}\mathbb{R}^{(1+n)+2};\mathbb{R}\big{)}$ in the form of equations
involving vector fields on the covering space
$\mathbb{R}^{n+1}\times\mathbb{R}^{2}$, implementing the implications (1.8)?

The latter is interpreted as the associated Lax–Sato representation [32],
recently developed in [3, 4, 5, 13, 14, 29, 30], for the given differential
relation $\mathcal{E}[x,\tau;u]=0$ on the jet manifold
$J^{k}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}^{m}\big{)}$.

As a dual approach to this linearization covering scheme we present also the
so-called contact geometry linearization, proposed recently in [27] and
generalizing from three to four dimensions the well-known, see, e.g., [28] and
reference therein, Hamiltonian linearization covering method. As an example,
we state the following proposition.

######  Proposition 1.4.

The following [8] nonlinear singular manifold Toda differential relation

$\displaystyle u_{xy}\sinh^{2}u_{t}=u_{x}u_{y}u_{tt}$ (1.9)

on the jet manifold
$J^{2}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}$ allows the
Lax–Sato type linearization covering

$\displaystyle\frac{\partial\psi}{\partial
t}+\frac{\big{(}e^{-2u_{t}}-1\big{)}}{2u_{x}}\frac{\partial\psi}{\partial
x}-\left[\lambda\left(\frac{e^{-2u_{t}}-1}{2u_{x}}\right)_{x}+\lambda^{2}\left(\frac{e^{-2u_{t}}-1}{2u_{x}}\right)_{z}\right]\frac{\partial\psi}{\partial\lambda}=0,$
$\displaystyle\frac{\partial\psi}{\partial
y}-\frac{u_{y}e^{-2u_{t}}}{u_{x}}\frac{\partial\psi}{\partial
x}+\left[\lambda\left(\frac{u_{y}e^{-2u_{t}}}{u_{x}}\right)_{x}+\lambda^{2}\left(\frac{u_{y}e^{-2u_{t}}}{u_{x}}\right)_{z}\right]\frac{\partial\psi}{\partial\lambda}=0$

for smooth invariant functions $\psi\in
C^{2}\big{(}\mathbb{R}^{3}\times\mathbb{R}^{2};\mathbb{R}\big{)}$, all
$(x,z,\lambda;\tau)\in\mathbb{R}^{3}\times\mathbb{R}^{2}$ and any smooth
solution $u\colon\mathbb{R}^{2}\times\mathbb{R}^{2}\mathbb{\rightarrow R}$ to
the differential relation (1.9).

##  2 The linearization covering scheme

Implementation of the scheme (1.8) is based on the notion of invariants of
suitably specified vector fields on the extended base space
$\mathbb{R}^{n+1}\times\mathbb{R}^{2}$, whose definition suitable for our
needs is as follows: a smooth mapping
$\psi\colon\mathbb{R}^{n+1}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ is an
invariant of a set of vector fields

$\displaystyle
X^{(k)}:=\frac{\partial}{\partial\tau_{k}}+\sum_{j=1}^{n}a_{j}^{(k)}(x,\lambda;\tau)\frac{\partial}{\partial
x_{j}}+b^{(k)}(x,\lambda;\tau)\frac{\partial}{\partial\lambda}$

on $\mathbb{R}^{n+1}\times\mathbb{R}^{2}$ involving parameters
$\tau\in\mathbb{R}^{2}$ with smooth coefficients
$\big{(}a^{(k)},b^{(k)}\big{)}\colon\mathbb{R}^{n+1}\times\mathbb{R}^{2}\rightarrow\mathbb{E}^{n}\times\mathbb{R}$,
$k=1,2$, if the conditions

$\displaystyle X^{(k)}\psi=0$ (2.1)

hold for $k=1,2$ and all
$(x,\lambda;\tau)\in\mathbb{R}^{n+1}\times\mathbb{R}^{2}$. The system of
linear equations (2.1) is equivalently representable as a jet submanifold
$J_{\rm
lin}^{1}\big{(}\mathbb{R}^{n+1}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}\subset
J^{1}\big{(}\mathbb{R}^{n+1}\times\mathbb{R}^{2};\mathbb{R}\big{)}$. It is
also well known [9] that simultaneously the following vector field flows

$\displaystyle\frac{\partial
x_{j}}{\partial\tau_{k}}=a_{j}^{(k)}(x,\lambda;\tau),\qquad\frac{\partial\lambda}{\partial\tau_{k}}=b^{(k)}(x,\lambda;\tau)$
(2.2)

are compatible for any $j=1,\dots,n$, $k=1,2$ and all
$(x,\lambda;\tau)\in\mathbb{R}^{n+1}\times\mathbb{R}^{2}$ too. Taking now into
account that there is such an invariant function
$\psi\colon\mathbb{R}^{n+1}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$, which
can be represented as $\psi(x,\lambda;\tau)=w(x;\tau)-\lambda:=0$ for some
smooth mapping
$w\colon\mathbb{R}^{n}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$, it gives
upon its substitution into (2.1) the following a priori compatible reduced
system of quasi-linear first-order differential relations

$\displaystyle\frac{\partial
w}{\partial\tau_{k}}+\sum_{j=1}^{n}a_{j}^{(k)}(x,w;\tau)\frac{\partial
w}{\partial x_{j}}-b^{(k)}(x,w;\tau)=0$ (2.3)

for $k=1,2$ on the jet manifold
$J^{0}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}\big{)}$. Moreover,
subject to the system (2.3) one can also observe [9, 17] that, modulo
solutions to the vector field flow evolution equations (2.2), the expression
$w(x;\tau)=\psi(x,\lambda(\tau);\tau)+\lambda(\tau)$ for all
$(x;\tau)\in\mathbb{R}^{n}\times\mathbb{R}^{2}$, where
$\psi\colon\mathbb{R}^{n+1}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ is a
first integral of the vector field flows (2.2). Thus, the reduction scheme
just described above provides the algorithm

$\displaystyle J_{\rm
lin}^{1}\big{(}\mathbb{R}^{n+1}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}\rightarrow
J^{1}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$

from the implications (1.8) formulated above. The corresponding inverse
implication

$\displaystyle J_{\rm
lin}^{1}\big{(}\mathbb{R}^{n+1}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}\leftarrow
J^{1}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$
(2.4)

can be algorithmically described as follows.

Consider now a compatible system
$J^{1}\big{(}\mathbb{R}^{n+2};\mathbb{R}\big{)}|_{\mathcal{E}}\subset
J^{1}\big{(}\mathbb{R}^{n+2};\mathbb{R}\big{)}$ of the first-order nonlinear
differential relations

$\displaystyle\frac{\partial
w}{\partial\tau_{k}}+\sum_{j=1}^{n}a_{j}^{(k)}(x,w;\tau)\frac{\partial
w}{\partial x_{j}}-b^{(k)}(x,w;\tau)=0$ (2.5)

with smooth coefficients
$\big{(}a^{(k)},b^{(k)}\big{)}\colon\mathbb{R}^{n+1}\times\mathbb{R}^{2}\rightarrow\mathbb{E}^{n}\times\mathbb{R}$,
$k=1,2$. As the first step it is necessary to check whether the associated
system of vector field flows

$\displaystyle\frac{\partial x_{j}}{\partial\tau_{k}}=a_{j}^{(k)}(x,w;\tau)$
(2.6)

on $\mathbb{R}^{n+1}$ modulo the flows (2.5) for all $j=1,\dots,n$ and $k=1,2$
is compatible too. If the answer is “yes”, then it just means [9] that any
solution to (2.5) as a complex function
$w\colon\mathbb{R}^{n}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ is
representable as $w(x;\tau)-\lambda=\alpha(\psi(x,\lambda;\tau))$ for any
$\lambda\in\mathbb{R}$ and some smooth mapping
$\alpha\colon\mathbb{R}\rightarrow\mathbb{R}$, where the mapping
$\psi\colon\mathbb{R}^{n+1}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ is a
first integral of the vector field equations

$\displaystyle\frac{\partial\psi}{\partial\tau_{k}}+\sum_{j=1}^{n}a_{j}^{(k)}(x,\lambda;\tau)\frac{\partial\psi}{\partial
x_{j}}+b^{(k)}(x,\lambda;\tau)\frac{\partial\psi}{\partial\lambda}=0$ (2.7)

on the extended space $\mathbb{R}^{n+1}\times\mathbb{R}$ for all
$(x,\lambda;\tau)\in\mathbb{R}^{n+1}\times\mathbb{R}^{2}$. Moreover, the value
$w(x(\tau);\tau)=\lambda\in\mathbb{R}$ for all $\tau\in\mathbb{R}^{2}$ is
constant as it follows from the condition $\alpha(\psi(x(\tau),w;\tau))=0$ for
the $\tau\in\mathbb{R}^{2}$. Thus, we derived that the vector field equations
(2.7) realize the covering linear first-order differential relations $J_{\rm
lin}^{1}\big{(}\mathbb{R}^{n+1}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}\subset
J^{1}\big{(}\mathbb{R}^{n+1}\times\mathbb{R}^{2};\mathbb{R}\big{)}$ for
$k=1,2$ and all $(x,\lambda;\tau)\in\mathbb{R}^{n+1}\times\mathbb{R}^{2}$,
linearizing the first-order nonlinear differential relations (2.5) and
interpreting it as the corresponding Lax–Sato representation.

On the contrary, if the associated system of vector field flows (2.6) is not
compatible, it is necessary to recover a hidden isomorphism transformation

$\displaystyle
J^{1}\big{(}\mathbb{R}^{n+1}\times\mathbb{R}^{2};\mathbb{R}\big{)}\ni(x,w;\tau)\rightarrow(x,\tilde{w};\tau)\in
J^{1}\big{(}\mathbb{R}^{n+1}\times\mathbb{R}^{2};\mathbb{R}\big{)},$ (2.8)

for which the resulting a priori compatible first-order nonlinear differential
relations

$\displaystyle\frac{\partial\tilde{w}}{\partial\tau_{k}}+\sum_{j=1}^{n}\tilde{a}_{j}^{(k)}(x,\tilde{w};\tau)\frac{\partial\tilde{w}}{\partial
x_{j}}-\tilde{b}^{(k)}(x,\tilde{w};\tau)=0$ (2.9)

possess already a compatible associated system of the corresponding vector
field flows

$\displaystyle\frac{\partial
x_{j}}{\partial\tau_{k}}=\tilde{a}_{j}^{(k)}(x,\tilde{w};\tau)$ (2.10)

on the space $\mathbb{R}^{n}\times\mathbb{R}$, for which any solution
$\tilde{w}\colon\mathbb{R}^{n+2}\rightarrow\mathbb{R}$ generates a first
integral
$\tilde{\psi}\colon\mathbb{R}^{n+1}\times\mathbb{R}^{2}\mathbb{\rightarrow R}$
of an associated compatible system of the linear vector field equations

$\displaystyle\frac{\partial\tilde{\psi}}{\partial\tau_{k}}+\sum_{j=1}^{n}\tilde{a}_{j}^{(k)}(x,\lambda;\tau)\frac{\partial\tilde{\psi}}{\partial
x_{j}}+\tilde{b}^{(k)}(x,\lambda;\tau)\frac{\partial\tilde{\psi}}{\partial\lambda}=0$
(2.11)

on the space $\mathbb{R}^{n+1}\times\mathbb{R}^{2}\mathbb{\rightarrow R}$ for
$k=1,2$, where
$\tilde{\psi}(x,\lambda;\tau):=\tilde{\alpha}(\tilde{w}(x;\tau)-\lambda)$ for
all $(x,\lambda;\tau)\in\mathbb{R}^{n+1}\times\mathbb{R}$ and some smooth
mapping $\tilde{\alpha}\colon\mathbb{R}\rightarrow\mathbb{R}$. From this one
easily infers, as it was done above, a linearized covering jet submanifold
$J_{\rm
lin}^{1}\big{(}\mathbb{R}^{n+1}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}\subset
J^{1}\big{(}\mathbb{R}^{n+1}\times\mathbb{R}^{2};\mathbb{R}\big{)}$, as a
compatible system of the vector field equations (2.11), generated by the
nonlinear first-order differential system
$J^{1}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}\subset
J^{1}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}\big{)}$ on the space
$\mathbb{R}^{n}\times\mathbb{R}^{2}$. The latter exactly determines the
inverse implication (2.4) as applied to general compatible first-order
nonlinear differential relations (2.7), providing for the nonlinear first-
order differential system
$J^{1}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}\subset
J^{1}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}\big{)}$ its
corresponding Lax–Sato representation.

######  Remark 2.1.

The existence of the mapping (2.8) can be inferred from the following
reasoning. Assume that the mapping (2.8) exists and is equivalent to the
relation

$\displaystyle w(x;\tau):=\rho(x,\tilde{w}(x;\tau);\tau)$ (2.12)

for some smooth function
$\rho\colon\mathbb{R}^{n+1}\times\mathbb{R}^{2}\mathbb{\rightarrow R}$ and all
$(x;\tau)\in\mathbb{R}^{n}\times\mathbb{R}^{2}$, where, by definition, the
corresponding mapping
$\tilde{w}\colon\mathbb{R}^{n}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$
satisfies the following system of first-order differential relations:

$\displaystyle\frac{\partial\tilde{w}}{\partial\tau_{k}}+\sum_{j=1}^{n}a_{j}^{(k)}(x,\rho(x,\tilde{w};\tau);\tau)\frac{\partial\tilde{w}}{\partial
x_{j}}=\tilde{b}^{(k)}(x,\tilde{w};\tau),$ (2.13)

compatible for all $\tau_{k}\in\mathbb{R}$, $k=1,2$, and $x\in\mathbb{R}^{n}$.
Here the functions
$\tilde{b}^{(k)}\colon\mathbb{R}^{n+1}\times\mathbb{R}^{2}\mathbb{\rightarrow
R}$, $k=1,2$, defined as

$\displaystyle\tilde{b}^{(k)}(x,\tilde{w};\tau):=\left[b^{(k)}-\sum_{j=1}^{n}\left(\frac{\partial\rho}{\partial\tau_{k}}+a_{j}^{(k)}\frac{\partial\rho}{\partial
x_{j}}\right)\right]\left.\left(\frac{\partial\rho}{\partial
x_{j}}\right)^{-1}\right|_{w=\rho(x,\tilde{w};\tau)},$

should depend on the mapping (2.12) in such a way that the vector fields

$\displaystyle\frac{\partial
x_{j}}{\partial\tau_{k}}=a_{j}^{(k)}(x,\rho(x,\tilde{w};\tau);\tau):=\tilde{a}_{j}^{(k)}(x,\tilde{w};\tau)$
(2.14)

be also compatible for all $j=1,\dots,n$ and $k=1,2$ modulo the flows (2.13).
The latter means that the equation (2.13) can be equivalently represented as a
compatible system of the following vector field equations

$\displaystyle\frac{\partial\tilde{\psi}}{\partial\tau_{k}}+\sum_{j=1}^{n}\tilde{a}_{j}^{(k)}(x,\lambda;\tau)\frac{\partial\tilde{\psi}}{\partial
x_{j}}+\tilde{b}^{(k)}(x,\rho(x,\lambda;\tau);\tau)\frac{\partial\tilde{\psi}}{\partial\lambda}=0$
(2.15)

on its first integral
$\tilde{\psi}\colon\mathbb{R}^{n+1}\times\mathbb{R}^{2}\mathbb{\rightarrow
R}$, where $\tilde{\psi}(x,\lambda;\tau)=\alpha(\tilde{w}(x;\tau)-\lambda)$
for an arbitrarily chosen smooth mapping $\alpha\colon\mathbb{R\rightarrow
R}$, any parameter $\lambda\in\mathbb{R}$ and all
$(x;\tau)\in\mathbb{R}^{n}\times\mathbb{R}^{2}$. The resulting system (2.15)
means exactly a suitable Lax–Sato type linearization of the compatible quasi-
linear first-order differential relations (2.3). Regarding the mapping (2.12)
and the functions
$\tilde{b}^{(k)}\colon\mathbb{R}^{n+1}\times\mathbb{R}^{2}\mathbb{\rightarrow
R}$, $k=1,2$, that depend on it, one can easily observe that the compatibility
condition for the vector fields (2.14) reduces to the a priori compatible
first-order quasi-linear differential relations

$\displaystyle\frac{\partial
a_{j}^{(k)}}{\partial\rho}\frac{\partial\rho}{\partial\tau_{s}}-\frac{\partial
a_{j}^{(s)}}{\partial\rho}\frac{\partial\rho}{\partial\tau_{k}}+\left(\frac{\partial
a_{j}^{(k)}}{\partial\rho}\tilde{b}^{(s)}-\frac{\partial
a_{j}^{(s)}}{\partial\rho}\tilde{b}^{(k)}\right)\frac{\partial\rho}{\partial\tilde{w}}$
$\displaystyle\qquad{}+\sum_{m=1}^{n}\left(\frac{\partial
a_{j}^{(k)}}{\partial\rho}a_{m}^{(s)}-\frac{\partial
a_{j}^{(s)}}{\partial\rho}a_{m}^{(k)}\right)\frac{\partial\rho}{\partial
x_{m}}=0,$ (2.16)

where $j=1,\dots,n$ and $k\neq s=1,2$, and whose solution is exactly the
searched-for mapping (2.12). Taking here into account that we have only two
functional parameters
$b^{(s)}\colon\mathbb{R}^{n+1}\times\mathbb{R}^{2}\mathbb{\rightarrow R}$,
$s=1,2$, the system of $2n$ differential relations (2.16) can be, in general,
compatible only for the case $n=1$. For all other cases $n\geq 2$ the
compatibility condition for (2.16) should be checked separately by direct
calculations.

###  2.1 Example: the Gibbons–Tsarev equation

As a first degenerate case of the above scheme (2.4) we consider a compatible
nonlinear first-order system
$J^{1}\big{(}\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$ $\subset
J^{1}\big{(}\mathbb{R}^{2};\mathbb{R}\big{)}$ for $n=0$, discussed before in
[21]:

$\displaystyle\frac{\partial w}{\partial
t}-\frac{1}{z_{y}+z_{t}w-w^{2}}=0,\qquad\frac{\partial w}{\partial
y}+\frac{z_{t}-w}{u_{y}+z_{t}w-w^{2}}=0,$ (2.17)

where $(t,y;w)\in\mathbb{R}^{2}\times\mathbb{R}$ and a mapping
$u\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$ satisfies the Gibbons–Tsarev
equation $\mathcal{E}[y,t;u]=0$ in the form

$\displaystyle z_{yy}+z_{t}z_{ty}-z_{y}z_{tt}+1=0,$ (2.18)

first derived in [15]. As the nonlinear system (2.17) is compatible and the
associated system of vector field flows (2.10) is empty, one easily infers
[26] that any solution $w\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$ to (2.17)
generates a first integral
$\psi\colon\mathbb{R}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ of a system of
the vector field equations

$\displaystyle\frac{\partial\psi}{\partial
t}+\frac{1}{z_{y}+z_{t}\lambda-\lambda^{2}}\frac{\partial\psi}{\partial\lambda}=0,\qquad\frac{\partial\psi}{\partial
y}-\frac{z_{t}-\lambda}{z_{y}+z_{t}\lambda-\lambda^{2}}\frac{\partial\psi}{\partial\lambda}=0,$
(2.19)

where, by definition, $\psi(\lambda;y,t):=\alpha(w(t,y)-\lambda)$ for all
$(\lambda;t,y)\in\mathbb{R}\times\mathbb{R}^{2}$ and some smooth mapping
$\alpha\colon\mathbb{R}\rightarrow\mathbb{R}$. The compatible system (2.19)
considered as the jet submanifold $J_{\rm
lin}^{1}\big{(}\mathbb{R}^{1}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}\subset
J^{1}\big{(}\mathbb{R}^{1}\times\mathbb{R}^{2};\mathbb{R}\big{)}$ solves the
problem of constructing the linearizing implication (2.4).

As it was demonstrated in [16, 24], the substitution

$\displaystyle
u:=\frac{1}{2}\left(-z_{t}+\sqrt{z_{t}^{2}+4z_{y}}\right),\qquad
v:=\frac{1}{2}\left(-z_{t}-\sqrt{z_{t}^{2}+4z_{y}}\right)$

gives rise to the next equivalent dynamical system

$\displaystyle u_{y}=vu_{t}-(u-v)^{-1},\qquad v_{y}=uv_{t}+(u-v)^{-1}$ (2.20)

on a functional space $M\subset
C^{\infty}\big{(}\mathbb{R};\mathbb{R}^{2}\big{)}$ subject to the evolution
parameter $y\in\mathbb{R}$ modulo evolution with respect to the joint
evolution parameter $t\in\mathbb{R}$. Taking into account the Lax–Sato
representation (2.19), one easily infers [26] the corresponding linearizing
Lax–Sato representation for the dynamical system (2.20):

$\displaystyle\frac{\partial\psi}{\partial
t}-\frac{1}{(\lambda+u)(\lambda+v)}\frac{\partial\psi}{\partial\lambda}=0,\qquad\frac{\partial\psi}{\partial
y}-\frac{u+v+\lambda}{(\lambda+u)(\lambda+v)}\frac{\partial\psi}{\partial\lambda}=0.$
(2.21)

The above Lax–Sato representation can be now reanalyzed more deeply within the
Lie-algebraic scheme devised recently in [19]. Namely, we define the complex
torus diffeomorphism Lie group
$\tilde{G}:=\operatorname{Diff}\big{(}\mathbb{T}_{\mathbb{C}}^{1}\big{)}$,
holomorphically extended to the interior $\mathbb{D}_{+}^{1}\subset\mathbb{C}$
and to the exterior $\mathbb{D}_{-}^{1}\subset\mathbb{C}$ regions of the unit
disc $\mathbb{D}^{1}\subset\mathbb{C}^{1}$, such that for any
$g(\lambda)\in\tilde{G}|_{\mathbb{D}_{-}^{1}}$,
$\lambda\in\mathbb{D}_{-}^{1}$,
$g(\infty)=1\in\operatorname{Diff}\big{(}\mathbb{T}^{1}\big{)}$, and study its
specially chosen coadjoint orbits related to the compatible system of linear
vector field equations (2.21).

As a first step for solving this problem one needs to consider the
corresponding Lie algebra
$\mathcal{\tilde{G}}:=\operatorname{Diff}\big{(}\mathbb{T}_{\mathbb{C}}^{1}\big{)}$
and its decomposition into the direct sum of subalgebras

$\displaystyle\mathcal{\tilde{G}}=\mathcal{\tilde{G}}_{+}\oplus\mathcal{\tilde{G}}_{-}$

of Laurent series with positive as $|\lambda|\rightarrow 0$ and strongly
negative as $\lambda|\rightarrow\infty$ degrees, respectively. Then, owing to
the classical Adler–Kostant–Symes theory, for any element
$\tilde{l}\in\mathcal{\tilde{G}}^{\ast}\simeq\Lambda^{1}\big{(}\mathbb{T}_{\mathbb{C}}^{1}\big{)}$
the following formally constructed flows

$\displaystyle{\rm d}\tilde{l}/{\rm d}y=-\operatorname{ad}_{\nabla
h^{(y)}(\tilde{l})}^{\ast}\tilde{l},\qquad{\rm d}\tilde{l}/{\rm
d}t=-\operatorname{ad}_{\nabla h^{(t)}(\tilde{l})}^{\ast}\tilde{l}$ (2.22)

with the evolution parameters $y,t\in\mathbb{R}^{2}$ are always compatible, if
$h^{(p_{y})}$ and $h^{(p_{t})}\in I\big{(}\mathcal{\tilde{G}}^{\ast}\big{)}$
are arbitrarily chosen functionally independent Casimir functionals on the
adjoint space $\mathcal{\tilde{G}}^{\ast}$, and by definition, $\nabla
h^{(y)}(\tilde{l}):=\nabla h^{(p_{y})}(\tilde{l})_{-}$, $\nabla
h^{(t)}(\tilde{l}):=\nabla h^{(p_{t})}(\tilde{l})_{-}$ are their gradients,
suitably projected onto the subalgebra $\mathcal{\tilde{G}}_{-}$. Bearing in
mind the above classical result, consider the Casimir functional $h^{(p_{y})}$
on $\mathcal{\tilde{G}}^{\ast}$, whose gradient $\nabla
h^{(p_{y})}(\tilde{l}):=\nabla
h^{(p_{y})}(l)\partial/\partial\lambda\mathcal{\in\tilde{G}}$ as
$|\lambda|\rightarrow\infty$ is taken, for simplicity, in the asymptotic form

$\displaystyle\nabla
h^{(p_{y})}(\tilde{l})\sim\left(\frac{\lambda+u+v}{(\lambda+u)(\lambda+v)}+\alpha_{0}+\alpha_{1}\lambda\right)\frac{\partial}{\partial\lambda},$

where $\lambda\in\mathbb{T}_{\mathbb{C}}^{1}$, $|\lambda|\rightarrow\infty$,
and coefficients $\alpha_{j}\in
C^{\infty}\big{(}\mathbb{R}^{2};\mathbb{R}\big{)}$, $j=0,1$, are arbitrarily
chosen nontrivial functional parameters, giving rise to the gradient
projection

$\displaystyle\nabla h^{(y)}(\tilde{l}):=\nabla
h^{(p_{y})}(\tilde{l})|_{-}=\frac{\lambda+u+v}{(\lambda+u)(\lambda+v)},$
(2.23)

generating the first flow of (2.22). As the differential 1-form
$\tilde{l}=l(\lambda;y,t){\rm
d}\lambda\in\Lambda^{1}\big{(}\mathbb{T}_{\mathbb{C}}^{1}\big{)}\simeq\mathcal{\tilde{G}}$
satisfies, by definition, the condition

$\displaystyle\operatorname{ad}_{\nabla
h^{(p_{y})}(\tilde{l})}^{\ast}\tilde{l}=0,$ (2.24)

equivalent to the differential equation

$\displaystyle\frac{{\rm d}}{{\rm d}\lambda}\big{[}l(\lambda;y,t)\big{(}\nabla
h^{(p_{y})}(l)\big{)}^{2}\big{]}=0,$ (2.25)

one easily infers from (2.23) and (2.25) that the coefficient

$\displaystyle l(\lambda;y,t)=\big{(}\nabla
h^{(p_{y})}(l)\big{)}^{-2}=\frac{(\lambda+u)^{2}(\lambda+v)^{2}}{[\lambda+u+v+(\alpha_{0}+\alpha_{1}\lambda)(\lambda+u)(\lambda+v)]^{2}},$

satisfies the relation $l(\lambda;y,t)\big{(}\nabla
h^{(p_{y})}(l)\big{)}^{2}=1$ for all $(t,y)\in\mathbb{R}^{2}$.

Now we formulate the following useful observation.

######  Lemma 2.2.

The set $\mathrm{I}(\mathcal{\tilde{G}}^{\ast})$ of the functionally
independent Casimir invariants is one-dimensional.

###### Proof.

Any asymptotic solution to the determining equation (2.26) is invariant under
the multiplication $\nabla h^{(p_{t})}(\tilde{l})\rightarrow\sigma(t,y)\nabla
h^{(p_{t})}(\tilde{l})$ by an arbitrary smooth function
$\sigma\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$. This proves the lemma. ∎

Consider now the gradient $\nabla
h^{(p_{t})}(\tilde{l})\in\mathcal{\tilde{G}}$ of the Casimir functional
$h^{(p_{t})}\in\mathrm{D}(\mathcal{\tilde{G}}^{\ast}\mathcal{)}$, that
satisfies the condition, which is identical to (2.24), that is,

$\displaystyle\operatorname{ad}_{\nabla
h^{(p_{t})}(\tilde{l})}^{\ast}\tilde{l}=0,$

which is equivalent to the following linear differential equation

$\displaystyle 2l(\lambda;y,t)\frac{\partial}{\partial\lambda}\nabla
h^{(p_{t})}(\tilde{l})+\nabla
h^{(p_{t})}(\tilde{l})\frac{\partial}{\partial\lambda}l(\lambda;y,t)=0,$
(2.26)

and whose solution can be naturally represented as the asymptotic series

$\displaystyle\nabla
h^{(p_{t})}(\tilde{l})\sim\frac{1}{(\lambda+u)(\lambda+v)}+\sum_{j\in\mathbb{Z}_{+}}\beta_{j}\lambda^{j}$
(2.27)

for some nontrivial coefficients $\beta_{j}\in
C^{\infty}\big{(}\mathbb{R}^{2};\mathbb{R}\big{)}$ successively determined
from the equation (2.26).

As follows from Lemma 2.2, the asymptotic solution (2.27) to the determining
equation (2.26) exists exclusively owing to its natural invariance under the
multiplication

$\displaystyle\nabla h^{(p_{t})}(\tilde{l})\rightarrow\sigma(t,y)\nabla
h^{(p_{t})}(\tilde{l})$

by arbitrary smooth function
$\sigma\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$. In particular, this means
that asymptotically as $\lambda\rightarrow\infty$ the product
$l(\lambda;y,t)(\nabla h^{(p_{t})}(l))^{2}\sim 0$, for otherwise, if
$l(\lambda;y,t)(\nabla h^{(p_{t})}(l))^{2}\nrightarrow 0$, this product
becomes, owing to (2.25), a nonzero constant involving the parameter
$\lambda\in\mathbb{C}$. The latter, evidently, means that $\nabla
h^{(p_{t})}(l)=\nabla h^{(p_{y})}(\tilde{l})\sigma(t,y)$ for any smooth
arbitrary function $\sigma\in
C^{\infty}\big{(}\mathbb{R}^{2};\mathbb{R}\big{)}$, producing no new flow with
respect to the evolution parameter $t\in\mathbb{R}$.

Construct now the gradient projection

$\displaystyle\nabla h^{(t)}(\tilde{l}):=\nabla
h^{(p_{t})}(\tilde{l})|_{-}=\frac{1}{(\lambda+u)(\lambda+v)}$

generating the second flow of (2.22). As a consequence of the above results we
can easily derive the following compatibility condition for the flows (2.22):

$\displaystyle\left[\frac{\partial}{\partial y}-\nabla
h^{(y)}(\tilde{l}),\frac{\partial}{\partial y}-\nabla
h^{(ty)}(\tilde{l})\right]=0,$

which is equivalent modulo the dynamical system (2.20) to the following system
of two a priori compatible linear vector field equations:

$\displaystyle\frac{\partial\psi}{\partial
y}-\frac{\lambda+u+v}{(\lambda+u)(\lambda+v)}\frac{\partial\psi}{\partial\lambda}=0,\qquad\frac{\partial\psi}{\partial
t}-\frac{1}{(\lambda+u)(\lambda+v)}\frac{\partial\psi}{\partial\lambda}=0$
(2.28)

for $\psi\in C^{2}(\mathbb{R};\mathbb{R})$ and all
$(\lambda;t,y)\in\mathbb{C\times R}^{2}$. Thus, we can formulate the results,
obtained above, as the following proposition.

######  Proposition 2.3.

A system $J_{\rm
lin}^{1}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$
of the linear first-order differential relations (2.28) on the covering jet
manifold $J^{1}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}\big{)}$ is
compatible, that is, it holds for any its smooth solution
$\psi\colon\mathbb{R}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ at all points
$(\lambda;y,t)\in\mathbb{R}\times\mathbb{R}^{2}$ iff the function
$u\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$ satisfies the Gibbons–Tsarev
equation (2.18).

Moreover, taking into account that the flows (2.22) are, by construction,
Hamiltonian systems on the coadjoint space $\mathcal{\tilde{G}}^{\ast}$, their
reduction on the space of functional variables $(u,v)\in
C^{\infty}\big{(}\mathbb{R}^{2};\mathbb{R}^{2}\big{)}$ will be, respectively,
Hamiltonian too. This reduction scheme is now under study and is expected to
be presented elsewhere.

###  2.2 Example: the ABC equation

For the second example we consider a compatible system
$J^{1}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$
of the nonlinear first-order differential relations (1.4) on the jet manifold
$J^{1}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}\big{)}$. It is easy to
check that the associated system of vector field flows

$\displaystyle\frac{\partial x}{\partial
t}=\frac{u_{t}w}{u_{x}k_{1}(k_{1}+k_{2}-1)},\qquad\frac{\partial x}{\partial
y}=\frac{u_{y}w}{u_{x}k_{1}(w+k_{1}+k_{2}-1)},$

modulo differential relations (1.4), is not compatible for all
$(w;t,y)\in\mathbb{R}\times\mathbb{R}^{2}$. Thus, we need to construct such a
mapping (2.8) that the resulting system of nonlinear differential relations
(2.9) will possess an associated system of vector field flows already
compatible for all $(\tilde{w};t,y)\in\mathbb{R}\times\mathbb{R}^{2}$. To do
this let us get rid from the very beginning in the equations (1.4) of the
strictly linear part, giving rise to the representation its solution as
$w(x;t,y)=(u_{x}(x;t,y))^{\alpha}\tilde{w}(x;t,y)$ for
$\alpha=(k_{1}+k_{2}-1)/k_{1}$ and all
$(x;t,y)\in\mathbb{R}\times\mathbb{R}^{2}$. Substituting this into (1.4)
yields the following a priori compatible system
$J^{1}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$
of the first-order differential relations, cf. [22],

$\displaystyle\frac{\partial\tilde{w}}{\partial
t}+\frac{\tilde{w}u_{x}^{\alpha-1}u_{t}}{k_{1}\alpha}\frac{\partial\tilde{w}}{\partial
x}+\frac{\tilde{w}^{2}u_{x}^{\alpha}(u_{t}u_{xx}-k_{1}u_{xx}u_{xt})}{k_{1}^{2}}=0,$
$\displaystyle\frac{\partial\tilde{w}}{\partial
y}+\frac{k_{2}\tilde{w}u_{x}^{\alpha-1}u_{y}}{k_{1}(\tilde{w}u_{x}^{\alpha}+\alpha)}\frac{\partial\tilde{w}}{\partial
x}+\frac{k_{2}\alpha\tilde{w}^{2}u_{x}^{\alpha}u_{y}u_{xx}}{k_{1}^{2}(\tilde{w}u_{x}^{\alpha}+\alpha)}=0$
(2.29)

on the jet manifold
$J^{1}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}\big{)}$. We can now
easily check that the above expression
$\tilde{w}(x;t,y)=(u_{x}(x;t,y))^{-\alpha}w(x;t,y)$ for all
$(x;t,y)\in\mathbb{R}\times\mathbb{R}^{2}$ determining the mapping (2.8), is
exactly the one we searched for, inasmuch as the associated system of vector
field flows

$\displaystyle\frac{\partial x}{\partial
t}=\frac{\tilde{w}u_{x}^{\alpha-1}u_{t}}{k_{1}\alpha},\qquad\frac{\partial
x}{\partial
y}=\frac{k_{2}\tilde{w}u_{x}^{\alpha-1}u_{y}}{k_{1}(\tilde{w}u_{x}^{\alpha}+\alpha)}$

on $\mathbb{R}\times\mathbb{R}^{2}$ proves to be compatible modulo the system
$J^{1}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$
of a priori compatible differential relations (2.29) on
$J^{1}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}\big{)}$. Based on this
compatibility result one can easily construct the corresponding linearizing
system $J_{\rm
lin}^{1}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$
on the covering jet manifold
$J^{1}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}$ by
implemening the inverse implication (2.4) as the Lax–Sato representation

$\displaystyle\frac{\partial\psi}{\partial t}+\frac{\lambda
u_{x}^{\alpha-1}u_{t}}{k_{1}\alpha}\frac{\partial\psi}{\partial
x}-\frac{\lambda^{2}u_{x}^{\alpha}(u_{t}u_{xx}-k_{1}u_{xx}u_{xt})}{k_{1}^{2}}\frac{\partial\psi}{\partial\lambda}=0,$
$\displaystyle\frac{\partial\psi}{\partial y}+\frac{k_{2}\lambda
u_{x}^{\alpha-1}u_{y}}{k_{1}(\lambda
u_{x}^{\alpha}+\alpha)}\frac{\partial\psi}{\partial
x}-\frac{\lambda^{2}k_{2}\alpha u_{x}^{\alpha-1}u_{y}u_{xx}}{k_{1}^{2}(\lambda
u_{x}^{\alpha}+\alpha)}\frac{\partial\psi}{\partial\lambda}=0,$ (2.30)

exactly coinciding with (1.5), where, by definition,
$\psi(x,\lambda;t,y):=\alpha(\tilde{w}(x;t,y)-\lambda)$ for all
$(x,\lambda;t,y)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$ and any smooth mapping
$\alpha\colon\mathbb{R}\rightarrow\mathbb{R}$. Thus, the linear differential
system (2.30) solves the problem, which was posed above, of constructing the
inverse implication (2.4) for the compatible nonlinear differential system
$J^{1}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$
(1.4), thereby proving Proposition 1.3.

###  2.3 Example: the Manakov–Santini equations

The Manakov–Santini equations

$\displaystyle u_{tx}+u_{yy}+(uu_{x})_{x}+v_{x}u_{xy}-v_{y}u_{xx}=0,$
$\displaystyle v_{xt}+v_{yy}+uv_{xx}+v_{x}v_{xy}-v_{y}v_{xx}=0,$ (2.31)

where $(u,v)\in
C^{\infty}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}^{2}\big{)}$, as is
well known [23], are obtained as some generalization of the dispersionless
limit for the Kadomtsev–Petviashvili equation. Equations (2.31) possess the
following compatible nonlinear first-order differential covering
$J^{1}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}\subset
J^{1}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}\big{)}$:

$\displaystyle\frac{\partial w}{\partial
t}+\big{(}w^{2}-wv_{x}+u-v_{y}\big{)}\frac{\partial w}{\partial
x}+(w-v_{x})(u_{x}-wv_{xx})-u_{y}+v_{yy}+v_{x}v_{xy}=0,$
$\displaystyle\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial
x}-v_{xx}w+(u-v_{y})_{x}=0,$ (2.32)

giving rise for all $(x;t,y)\in\mathbb{R}\times\mathbb{R}^{2}$ to the
Manakov–Santini differential relation $\mathcal{E}[x;y,t;u,v]=0$ (2.31) as a
submanifold on the associated jet manifold
$J^{2}\big{(}\mathbb{R}\times\mathbb{R}^{2};\mathbb{R}^{2}\big{)}$. It is now
easy to check that the system of vector field flows

$\displaystyle\frac{\partial x}{\partial
t}=w^{2}-wv_{x}+u-v_{y},\qquad\frac{\partial x}{\partial y}=w,$

naturally related to (2.32), is for all $(t,y)\in\mathbb{R}^{2}$ not
compatible modulo these differential relations (2.32). Thus, one needs to
construct such an isomorphic transformation

$\displaystyle
J^{1}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}\ni(x,w;t,y)\rightarrow(x,\tilde{w};t,y)\in
J^{1}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}$

that the expression $w(x;y,t)=\beta(x,\tilde{w};y,t)$ for some, in general
nonlinear, smooth mapping
$\beta\colon\mathbb{R}^{2}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$
transforms the first-order differential covering (2.32) into an equivalent
compatible first-order differential covering (2.9), for which the
corresponding vector field flows (2.10) already become compatible. This
problem is solved easily enough giving rise to the simple mapping

$\displaystyle w=\tilde{w}+v_{x}$

using which one obtains the following compatible first-order differential
covering:

$\displaystyle\frac{\partial\tilde{w}}{\partial
t}+\big{(}\tilde{w}^{2}+\tilde{w}v_{x}+u-v_{y}\big{)}\frac{\partial\tilde{w}}{\partial
x}-u_{y}+\tilde{w}u_{x}=0,$ $\displaystyle\frac{\partial\tilde{w}}{\partial
y}+(v_{x}+\tilde{w})\frac{\partial\tilde{w}}{\partial x}+u_{x}=0.$ (2.33)

It is easy to check that the system of vector field flows

$\displaystyle\frac{\partial x}{\partial
t}=\tilde{w}^{2}+\tilde{w}v_{x}+u-v_{y},\qquad\frac{\partial x}{\partial
y}=v_{x}+\tilde{w},$

naturally related to (2.33), is already compatible for all
$(t,y)\in\mathbb{R}^{2}$ modulo the differential relations (2.33). Based on
this compatibility result stated above, one can easily construct the
associated linearizing first-order differential system $J_{\rm
lin}^{1}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$
on the covering jet manifold
$J^{1}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}$, realizing
the inverse implication (2.4) as the Lax–Sato representation

$\displaystyle\frac{\partial\psi}{\partial t}+\big{(}\lambda^{2}+\lambda
v_{x}+u-v_{y}\big{)}\frac{\partial\psi}{\partial x}+(u_{y}-\lambda
u_{x})\frac{\partial\psi}{\partial\lambda}=0,$
$\displaystyle\frac{\partial\psi}{\partial
y}+(v_{x}+\lambda)\frac{\partial\psi}{\partial
x}-u_{x}\frac{\partial\psi}{\partial\lambda}=0,$

thus proving Proposition 1.3.

##  3 The contact geometry linearization covering scheme

###  3.1 The setup

We consider two Hamilton–Jacobi-type compatible for all
$(x;\tau):=(x;t,y)\in\mathbb{R}\times\mathbb{R}^{2}$ in general nonlinear
first-order differential relations:

$\displaystyle\frac{\partial z}{\partial t}+\tilde{H}^{(t)}(x,z,\partial
z/\partial x;t,y)=0,\qquad\frac{\partial z}{\partial
y}+\tilde{H}^{(y)}(x,z,\partial z/\partial x;t,y)=0,$ (3.1)

where $z:\mathbb{R}^{3}\mathbb{\rightarrow R}$ is the so-called “action
function” and
$\tilde{H}^{(t)},\tilde{H}^{(y)}\colon\mathbb{R}^{3}\times\mathbb{R}^{2}\mathbb{\rightarrow
R}$ are some smooth generalized Hamiltonian functions. The relations (3.1)
follow from the contact geometry [7, 10] interpretation of some mechanical
systems generated by some vector fields. Namely, a differential one-form
$\alpha^{(1)}$ $\in\Lambda^{1}\big{(}\mathbb{R}^{3}\times\mathbb{R}\big{)}$,
defined by the expression

$\displaystyle\alpha^{(1)}:={\rm d}z-\lambda{\rm d}x,$

is contact, as $\alpha^{(1)}\wedge d\alpha^{(1)}$ defines a volume form on
$\mathbb{R}^{3}$, and vector fields
$X_{H^{(t)}},X_{H^{(y)}}\in\Gamma\big{(}T\big{(}\mathbb{R}^{3}\times\mathbb{R}\big{)}\big{)}$
are called, respectively, contact vector fields (with respect to
$\alpha^{(1)}$), if there exist some functions
$\mu^{(t)},\mu^{(y)}\colon\mathbb{R}^{3}\times\mathbb{R}\rightarrow\mathbb{R}$
such that for all $(x,z;\tau)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$ the
following equalities

$\displaystyle-i_{X_{H^{(t)}}}\alpha^{(1)}=H^{(t)}:=\tilde{H}^{(t)}|_{\partial
z/\partial x=\lambda},\qquad-
i_{X_{H^{(y)}}}\alpha^{(1)}=H^{(y)}:=\tilde{H}^{(y)}|_{\partial z/\partial
x=\lambda},$
$\displaystyle\mathcal{L}_{H^{(t)}}\alpha^{(t)}=\mu^{(t)}\alpha^{(t)},\qquad\mathcal{L}_{H^{(y)}}\alpha^{(y)}=\mu^{(y)}\alpha^{(y)}$
(3.2)

hold, where $\mathcal{L}_{H^{(t)}}$, $\mathcal{L}_{H^{(y)}}$ are the Lie
derivatives [9, 10, 17] with respect to the vector fields
$X_{H^{(t)}},X_{H^{(y)}}\in\Gamma\big{(}T\big{(}\mathbb{R}^{3}\times\mathbb{R}\big{)}\big{)}$.
From the conditions (3.2) one easily finds [2, 20] that

$\displaystyle X_{H^{(t)}}=\frac{\partial
H^{(t)}}{\partial\lambda}\frac{\partial}{\partial x}-\left(\frac{\partial
H^{(t)}}{\partial x}+\lambda\frac{\partial H^{(t)}}{\partial
z}\right)\frac{\partial}{\partial\lambda}+\left(-H^{(t)}+\lambda\frac{\partial
H^{(t)}}{\partial\lambda}\right)\frac{\partial}{\partial z},$ $\displaystyle
X_{H^{(y)}}=\frac{\partial H^{(y)}}{\partial\lambda}\frac{\partial}{\partial
x}-\left(\frac{\partial H^{(y)}}{\partial x}+\lambda\frac{\partial
H^{(y)}}{\partial
z}\right)\frac{\partial}{\partial\lambda}+\left(-H^{(y)}+\lambda\frac{\partial
H^{(y)}}{\partial\lambda}\right)\frac{\partial}{\partial z},$ (3.3)

where we have put $H^{(t)}:=\tilde{H}^{(t)}|_{\partial z/\partial x=\lambda}$,
$H^{(y)}:=\tilde{H}^{(y)}|_{\partial z/\partial x=\lambda}$, and compatibility
of the nonlinear relations (3.1) is equivalent to the commutativity condition
for the following vector fields:

$\displaystyle\left[\frac{\partial}{\partial
t}+X_{H^{(t)}},\frac{\partial}{\partial y}+X_{H^{(y)}}\right]=0$

for all $(x,z,\lambda;\tau)\in\mathbb{R}^{3}\times\mathbb{R}^{2}$, depending
parametrically on $\lambda\in\mathbb{R}$. The latter condition can be
rewritten as a compatible Lax–Sato representation for the vector field
equations

$\displaystyle\frac{\partial\psi}{\partial
t}+X_{H^{(t)}}\psi=0,\qquad\frac{\partial\psi}{\partial y}+X_{H^{(y)}}\psi=0$

for smooth invariant functions $\psi\in
C^{2}\big{(}\mathbb{R}^{3}\times\mathbb{R}^{2};\mathbb{R}\big{)}$ and all
$(x,z,\lambda;\tau)\in\mathbb{R}^{3}\times\mathbb{R}^{2}$.

######  Remark 3.1.

It is worth mentioning here that in the case when the Hamiltonian functions in
(3.1) do not depend on the “action function”
$z\colon\mathbb{R}^{3}\mathbb{\rightarrow R}$, the contact vector fields
naturally reduce to the classical Hamiltonian ones:

$\displaystyle X_{H^{(t)}}=\frac{\partial
H^{(t)}}{\partial\lambda}\frac{\partial}{\partial x}-\frac{\partial
H^{(t)}}{\partial x}\frac{\partial}{\partial\lambda},\qquad
X_{H^{(y)}}=\frac{\partial H^{(y)}}{\partial\lambda}\frac{\partial}{\partial
x}-\frac{\partial H^{(y)}}{\partial x}\frac{\partial}{\partial\lambda},$

well known [10] from the symplectic geometry.

###  3.2 Example: the differential Toda singular manifold equation

The above general contact geometry linearization scheme is a modification of
the one originally discovered and applied to integrable 4D dispersionless
equations by A. Sergyeyev in [27]. Below we will apply this scheme to a
degenerate case when the system (3.1) is given by the following linear first-
order differential relations:

$\displaystyle\frac{\partial z}{\partial
t}+\frac{\big{(}e^{-2u_{t}}-1\big{)}}{2u_{x}}\frac{\partial z}{\partial
x}=0,\qquad\frac{\partial z}{\partial
y}-u_{y}u_{x}^{-1}e^{-2u_{t}}\frac{\partial z}{\partial x}=0$ (3.4)

for a smooth mapping $z\colon\mathbb{R}\times\mathbb{R}^{2}\mathbb{\rightarrow
R}$, whose compatibility condition is the interesting [8] differential Toda
singular manifold equation (1.9) for a smooth function
$u\colon\mathbb{R}\times\mathbb{R}^{2}\mathbb{\rightarrow R}$ for all
$(x;y,t)\in\mathbb{R}\times\mathbb{R}^{2}$, defining a differential relation
$J^{2}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}\subset
J^{2}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}$ on the jet
manifold $J^{2}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}$

$\displaystyle u_{xy}\sinh^{2}u_{t}=u_{x}u_{y}u_{tt}.$ (3.5)

Although the differential relations (3.4) are linear, they contain no
“spectral” parameter $\lambda\in\mathbb{R}$ using which one can construct the
conservation laws for (2.3) and apply the modified inverse scattering
transform for constructing its particular exact solutions.

However, the above contact geometry linearization scheme makes it possible to
represent the system in question as a set of the compatible Hamilton–Jacobi
equations

$\displaystyle\frac{\partial z}{\partial
t}+\frac{\big{(}e^{-2u_{t}}-1\big{)}}{2u_{x}}\lambda=0,\qquad\frac{\partial
z}{\partial y}-\frac{u_{y}e^{-2u_{t}}}{u_{x}}\lambda=0$

with the contact Hamiltonians

$\displaystyle
H^{(t)}:=\frac{\big{(}e^{-2u_{t}}-1\big{)}}{2u_{x}}\lambda,\qquad
H^{(y)}:=-\frac{u_{y}e^{-2u_{t}}}{u_{x}}\lambda,$

where the function
$u\colon\mathbb{R}^{2}\times\mathbb{R}^{2}\mathbb{\rightarrow R}$ depends here
on the additional variable $z\in\mathbb{R}$. Taking into account (3.3), one
can construct the corresponding extended contact vector fields

$\displaystyle\tilde{X}_{H^{(t)}}:=\frac{\partial}{\partial
t}+X_{H^{(t)}}=\frac{\partial}{\partial
t}+\frac{(e^{-2u_{t}}-1)}{2u_{x}}\frac{\partial}{\partial
x}-\left[\lambda\left(\frac{e^{-2u_{t}}-1}{2u_{x}}\right)_{x}+\lambda^{2}\left(\frac{e^{-2u_{t}}-1}{2u_{x}}\right)_{z}\right]\frac{\partial}{\partial\lambda},$
$\displaystyle\tilde{X}_{H^{(y)}}:=\frac{\partial}{\partial
y}+X_{H^{(y)}}=\frac{\partial}{\partial
y}-\frac{u_{y}e^{-2u_{t}}}{u_{x}}\frac{\partial}{\partial
x}+\left[\lambda\left(\frac{u_{y}e^{-2u_{t}}}{u_{x}}\right)_{x}+\lambda^{2}\left(\frac{u_{y}e^{-2u_{t}}}{u_{x}}\right)_{z}\right]\frac{\partial}{\partial\lambda},$

compatible on the solution set to the nonlinear differential relation (3.5) on
the jet manifold
$J^{2}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}$. The latter
makes it possible to state our final proposition on the contact geometry
linearization covering of the system (3.5).

######  Proposition 3.2.

The linear first-order differential relations (3.4) on the jet manifold
$J^{2}\big{(}\mathbb{R}^{2}\times\mathbb{R}^{2};\mathbb{R}\big{)}$ allow the
following quadratic in the parameter $\lambda\in\mathbb{R}$ Lax–Sato type
linearization covering

$\displaystyle\frac{\partial\psi}{\partial
t}+\frac{\big{(}e^{-2u_{t}}-1\big{)}}{2u_{x}}\frac{\partial\psi}{\partial
x}-\left[\lambda\left(\frac{e^{-2u_{t}}-1}{2u_{x}}\right)_{x}+\lambda^{2}\left(\frac{e^{-2u_{t}}-1}{2u_{x}}\right)_{z}\right]\frac{\partial\psi}{\partial\lambda}=0,$
$\displaystyle\frac{\partial\psi}{\partial
y}-\frac{u_{y}e^{-2u_{t}}}{u_{x}}\frac{\partial\psi}{\partial
x}+\left[\lambda\left(\frac{u_{y}e^{-2u_{t}}}{u_{x}}\right)_{x}+\lambda^{2}\left(\frac{u_{y}e^{-2u_{t}}}{u_{x}}\right)_{z}\right]\frac{\partial\psi}{\partial\lambda}=0$

for smooth invariant functions $\psi\in
C^{2}\big{(}\mathbb{R}^{3}\times\mathbb{R}^{2};\mathbb{R}\big{)}$, all
$(x,z,\lambda;\tau)\in\mathbb{R}^{3}\times\mathbb{R}^{2}$ and any smooth
solution $u\colon\mathbb{R}^{2}\times\mathbb{R}^{2}\mathbb{\rightarrow R}$ to
the differential relation (3.5).

##  4 Conclusions

We have studied an interesting problem posed by I. Krasil’shchik in [21]
concerning a differential-geometric explanation of the linearization procedure
for a given nonlinear differential-geometric relation
$J^{1}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}\big{)}|_{\mathcal{E}}$
in the jet manifold
$J^{1}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}\big{)}$,
$n\in\mathbb{Z}_{+}$, realizing a compatible covering [1, 26] for the
corresponding nonlinear differential equation $\mathcal{E}[x,\tau;u]=0$,
imbedded into some associated jet manifold
$J^{k}\big{(}\mathbb{R}^{n}\times\mathbb{R}^{2};\mathbb{R}^{m}\big{)}$ for
some $k,m\in\mathbb{Z}_{+}$. We have analyzed this covering jet manifold
scheme, its relation to the invariant theory of the quasi-linear associated
vector fields and applications to the Lax–Sato type integrability of nonlinear
dispersionless differential systems. In particular, we discussed the
compatible Jacobi type nonlinear relations, the related contact geometry and
its application to the linearization covering scheme. The devised techniques
were applied to such nonlinear Lax–Sato integrable equations as
Gibbons–Tsarev, ABC, Manakov–Santini and the differential Toda singular
manifold equations.

### Acknowledgements

The author cordially thanks Professors M. Błaszak, B. Szablikowski and J.
Cieślinski for useful discussions of the results during the International
Conference in Functional Analysis dedicated to the 125th anniversary of Stefan
Banach held on 18–23 September, 2017 in Lviv, Ukraine. He is also greatly
indebted to Professors V.E. Zakharov (University of Arizona, Tucson) and J.
Szmigelski (University of Saskatchewan, Saskatoon) for their interest in the
work and instructive discussions during the XXXV Workshop on Geometric Methods
in Physics, held in Białowieża, Poland. The author is grateful to Professor B.
Kruglikov (University of Tromsø, Norway) for interest in the work and
mentioning some misprints and important references, which were very helpful in
preparing the manuscript. He is also indebted to the referees for their
remarks and instrumental suggestions. Last but not least, thanks go to the
Department of Mathematical Sciences of the NJIT (Newark NJ, USA) for the
invitation to visit the NJIT during the Summer Semester of 2017, where an
essential part of this paper was completed. Local support from the Institute
of Mathematics at the Cracow University of Technology is also much
appreciated.

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