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			{\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]  %Headings in Russian
			{\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
			{\large Siberian Electronic Mathematical Reports} \\[1mm]
			{\LARGE\tt{http://semr.math.nsc.ru}}\\[0.5mm]
			%     {\small Институт математики им. С.Л.Соболева СО РАН}\\[-1mm]
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		{\flushleft\it Том 17, стр. 1697--1709 (2020) \hspace{60mm}{\rm\small УДК \udcs}} %Volume, pages in Russian,
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		\newline
		{\rm\small DOI~10.33048/semi.2020.17.114}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}{\rm\small MSC\ \ \mscs }
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\title[Some representations of solutions to Blokhintsev
equation]{Some representations of solutions to Blokhintsev
	equation}
\author{{Yu.E. Anikonov, N.B. Ayupova, M.V. Neshchadim}}%
\address{Yurii Evgenievich Anikonov
	\newline\hphantom{iii} Sobolev Institute of Mathematics,
	\newline\hphantom{iii} 4,  Koptyuga ave.,
	\newline\hphantom{iii} Novosibirsk, 630090,  Russia}%
	\email{anikon@math.nsc.ru}%

%\author{{N.B. Ayupova}}%
\address{Natalia Borisovna Ayupova
\newline\hphantom{iii} Sobolev Institute of Mathematics,
	\newline\hphantom{iii} 4,  Koptyuga ave.,
	\newline\hphantom{iii} Novosibirsk, 630090,  Russia%
	\newline\hphantom{iii} Novosibirsk State University,
	\newline\hphantom{iii} 1, Pirogova str.,
	\newline\hphantom{iii} Novosibirsk, 630090,  Russia}%
	\email{ayupova@math.nsc.ru}%






%\author{{}}%
\address{Mikhail Vladimirovich Neshchadim
	\newline\hphantom{iii} Sobolev Institute of Mathematics,
	\newline\hphantom{iii} 4,  Koptyuga ave.,
	\newline\hphantom{iii} Novosibirsk, 630090,  Russia%
	\newline\hphantom{iii} Novosibirsk State University,
	\newline\hphantom{iii} 1, Pirogova str.,
	\newline\hphantom{iii} Novosibirsk, 630090,  Russia}%
\email{ neshch@math.nsc.ru}%



\thanks{\sc  Anikonov, Yu.E.,  Ayupova, N.B.,  Neshchadim, M.V., Some representations of solutions to Blokhintsev
equation}
\thanks{\copyright \ 2020 Anikonov Yu.E.,  Ayupova N.B.,  Neshchadim  M.V}
\thanks{\rm The work was partially supported by by the Program of fundamental scientific
	researches of the SB RAS N I.1.5., project N 0314-2019-0011
	and by RFFR (grant 18-01-00057)}
\thanks{\it Received  May, 8, 2020, published  October, 21,  2020.}%


\semrtop \vspace{1cm}
\maketitle {\small
	\begin{quote}
		\noindent{\sc Abstract. } In the paper, we obtain some representations for solutions and coefficients of Blokhintsev equation under condition the solutions satisfy to supplementary quasi-linear equation. 
		These results may be used in the problems of  
		identification of solutions and coefficients given 
		supplementary initial-boundary information.
		\medskip
		
		\noindent{\bf Keywords:}  Blokhintsev equation, method of differential 
		constrains,\\
		over\-determined systems of partial differential equations.
	\end{quote}
}









\section{Introduction}






In this paper, we study  Blokhintsev equation \cite{1} on the base of method of differential constrains
\cite{2}.



Blokhintsev equation is written by
\begin{equation}\label{eq-01-aan}
D_t^2 w=\nu(x,t)\Delta w
+(\nabla P, \nabla w)+
(\overline{V}(x,t),\nabla\ln \nu(x,t)) D_t w,
\end{equation}
for $x\in D$, $t\ge 0$, $D$ is a domain in Euclidean space
$\mathbb{R}^{n}$.
Here, the function  $w(x,t)$ is a potential of velocity of  acoustic oscillations,
$P$ is a potential of pressure,
$\overline{V}(x,t)$ is vector of stream velocity,
$\nu(x,t)=c^{2}(x,t)$ is sound velocity squared,  $\nabla u$
is gradient of  function $u(x,t)$ with respect to variable $x$,
that is
 \begin{equation*}
 \nabla u=
 \left(
 \dfrac{\partial u}{\partial x_{1}},
 \dfrac{\partial u}{\partial x_{2}}, \ldots,
 \dfrac{\partial u}{\partial x_{n}}
 \right).
 \end{equation*}
If $a=(a_{1}, a_{2},\ldots, a_{n})$,
$b=(b_{1}, b_{2},\ldots, b_{n})$
are two vectors then  $\sum\limits_{i=1}^{n}a_{i}b_{i}=(a,b)$ is a scalar product.


Derivatives $D_t w$, $D_t^{2} w$
are equal to
\begin{equation}\label{eq-02-aan}
D_t w=\dfrac{\partial w}{\partial t}+
(\overline{V}(x,t),\nabla w)\equiv \psi(x,t), \quad D_t^{2} w=\dfrac{\partial \psi}{\partial t}+
(\overline{V}(x,t),\nabla \psi)
\end{equation}
simultaneously.
 Substituting derivatives (\ref{eq-02-aan}) in the equation (\ref{eq-01-aan}), we obtain an equa\-tion
\begin{multline}\label{eq-03-aan}
\dfrac{\partial}{\partial t}
\left[
\dfrac{\partial w}{\partial t}+
(\overline{V}(x,t),\nabla w)
\right]+
\left(
\overline{V}(x,t),
\nabla\left(
\dfrac{\partial w}{\partial t}+
(\overline{V}(x,t),\nabla w)
\right)
\right)=\\
=\nu(x,t)\Delta w
+
(\nabla P, \nabla w)+
\left[
\dfrac{\partial w}{\partial t}+
(\overline{V}(x,t),\nabla w)
\right]
\left(\overline{V}(x,t), \nabla\ln \nu(x,t)\right).
\end{multline}

In \cite{3},
the Blokhintsev equation was represented  as
 non-homogeneous hyperbolic equation
with a source function $R(x,t)$ which depends on
 solution and coefficients
\begin{equation*}
\dfrac{\partial^{2}w}{\partial t^{2}}=
\nu(x,t)\Delta w+
(\nabla P, \nabla w)+R(x,t)
\end{equation*}
where
\begin{multline*}
R(x,t)=\left[
\dfrac{\partial w}{\partial t}+
(\overline{V}(x,t),\nabla w)
\right]\left(\overline{V}(x,t), \nabla\ln \nu(x,t)
\right)-
\dfrac{\partial}{\partial t}
(\overline{V}(x,t),\nabla w)\\
-\left(
\overline{V}(x,t),
\nabla
\left(
\dfrac{\partial w}{\partial t}+
(\overline{V}(x,t),\nabla w)
\right)
\right).
\end{multline*}

    General and explicit representations of solutions to
equation were obtained for cases $w(x,t)=f(u(x,t))$ and
$w(x,t)=\varphi(x,t)F
\left(
\dfrac{\psi(x,t)}{\varphi(x,t)}
\right)$.



We obtain the main results of this paper to on the hypotheses
that a solution $w=w(x,t)$ to the equation (\ref{eq-01-aan}) 
satisfies to supplementary quasi-linear equation
\begin{equation}\label{eq-04-aan}
D_t w=\psi,
\end{equation}
where $\psi=\psi(x,t)$ is a function of variables $(x,t)$.
If the function  $\psi(x,t)$ is arbitrary then system
 (\ref{eq-01-aan}), (\ref{eq-04-aan})
is equivalent to equation (\ref{eq-01-aan}), 
that is  condition (\ref{eq-04-aan}) does  not apply
supplementary restriction to the function $w(x,t)$. In this way, for separation some class of solutions  from
all solutions to equation (\ref{eq-01-aan}), it is necessary to apply
 some supplementary restrictions to the function
$\psi(x,t)$. For example, 
if $\psi=0$,
we obtain the solution $w(x,t)$ is a Lagrange variable of
operator $D_t$.
It is reasonable to require   $\psi(x,t)$ is a function of $w$,
that is $\psi=\Psi(w)$, or does not depend on part of variables
$(x,t)$ and so on.
Supposing  $w$ satisfies to condition (\ref{eq-04-aan}) with some
restrictions right-hand part
$\psi$, we obtain overdetermined system of equations for 
function  $w$.
Reducing of this system to completely  	integrable type 
is a classical problem
 of the theory of overdetermined systems of 
partial differential equations
 \cite{4,5}, and obviously the final 
result should depend on collection of
functional parameters-coefficients $\nu$, $V$, $P$, 
$\psi$.
Hence, problem (\ref{eq-01-aan}), (\ref{eq-04-aan}) 
consists in determination of conditions
for coefficients  $\nu$, $V$, $P$, $\psi$ which lead to 
nontrivial sub-classes of solutions to equation
(\ref{eq-01-aan}).

In the second section of this paper, we consider so called one-parametric solutions  \cite{6} to equation (\ref{eq-01-aan}), such that all considered functions depend on
Lagrange variable of operator $D_t$
and (\ref{eq-04-aan}) is fulfilled in the case
  $\psi=0$.
Explicit formulas for solutions and coefficients are obtained, in general in complex-valued functions.
Note, in some applications, the solutions to classic wave equation
 is given by complex-valued function
\cite{7} --
%8, 
%9,10,11,
%12,13,14,
\cite{15}. Note, the solutions constructed in
theorem~\ref{th-03} are determined by  
function of distance to some surface in $\mathbb{R}^{n}$.

In sections 3, 4, we consider one-dimensional equation (\ref{eq-01-aan}), $n=1$, under conditions
$D_t w=0$ and $D_t w=w$ and by methods of  system of
partial differential equations
these systems are reduced to completely  	integrable type
\cite{4,5,16}. Also,
in section 5, we state a general remark on possibility 
of reducing system (\ref{eq-01-aan}), (\ref{eq-04-aan})
in the case $\psi=0$
to completely  	integrable type. More exactly, 
we obtain a system of two second order partial equations which have
only derivatives with respect to variables
 $x$ of the function $w$. It appears that
reducing of this new system to completely  	integrable type
 is the main problem.

In final part of the paper, we discuss a problem  of 
finding of functional-invariant solutions to Blokhintsev equations.

We suppose all considered functions are sufficiently differentiable.



\section{One-parametric solutions to Blokhintsev equation}



Let functions $w$, $P$ and a vector-function  $\overline{V}$
depend on a function $u=u(x,t)$
\begin{equation*}%\label{eq-5n}
w=f(u),\quad P=g(u), \quad V_k=V_k(u), \quad k=1,\ldots,n,
%\eqno{(5)}
\end{equation*}
where $f$, $g$, $V_k$, $k=1,\ldots,n$ are  some functions of one variable. Then
$$
\frac{\partial w}{\partial t}=f' \frac{\partial u}{\partial t}, \quad
\frac{\partial w}{\partial x_k}=f' \frac{\partial u}{\partial x_k}, \quad
\nabla w= f' \nabla u, \quad \nabla P= g' \nabla u, \quad
$$
$$
\frac{\partial^2 w}{\partial t^2}=f' \frac{\partial^2 u}{\partial t^2}+
f'' \left(\frac{\partial u}{\partial t}\right)^2,\quad
 \Delta w=f' \Delta u+ f'' |\nabla u|^2.
$$
The equation (\ref{eq-03-aan}) takes the form
\begin{multline}\label{eq-05-aan}
f' u_{tt}+f'' u_{t}^2=\nu \left( f' \Delta u+ f'' |\nabla u|^2 \right)+ f' g' |\nabla u|^2+
f' \left( u_t + \left\langle \overline{V}, \nabla u\right\rangle \right)
\left\langle \overline{V}, \nabla \ln \nu\right\rangle\\
-\frac{\partial }{\partial t}\left( f'\left\langle \overline{V}, \nabla u\right\rangle\right)-
\left\langle \overline{V}, \nabla \left( f' \left( u_t + \left\langle \overline{V}, \nabla u\right\rangle \right)   \right) \right\rangle.
\end{multline}

In addition suppose, the function 
 $u(x,t)$ is a solution to a quasi-linear equation
\begin{equation}\label{eq-06-aan}
u_t + \left( \overline{V}, \nabla u\right) =0,
\end{equation}
that is $D_t w=0$. Then we rewrite the equation (\ref{eq-05-aan}) in the form
\begin{equation}\label{eq-07-aan}
\nu \left( f' \Delta u+ f'' |\nabla u|^2 \right)+ f' g' |\nabla u|^2=0.
\end{equation}
It is possible to write a general solution to the equation  (\ref{eq-06-aan})  implicitly
\begin{equation}\label{eq-08-aan}
u=F(y_1,\ldots,y_n),
%\eqno{(9)}
\end{equation}
where $y_k=x_k-t V_k(u)$, $k=1,\ldots,n$
and $F(y)$ is an arbitrary function of variables $y=(y_1,\ldots,y_n)$.

We deduce from (\ref{eq-08-aan}) that
$$
\frac{\partial u}{\partial x_i}=
\frac{\partial F}{\partial y_i}-
t\frac{\partial u}{\partial x_i}\sum\limits_{k=1}^{n} \frac{\partial F}{\partial y_k}V_k',
$$
and
\begin{equation}\label{eq-09-aan}
\frac{\partial u}{\partial x_i}=\frac{1}{m}\frac{\partial F}{\partial y_i} ,
\end{equation}
where $m= 1+t \sum\limits_{k=1}^{n} V_k' \dfrac{\partial F}{\partial y_k} $.
Therefore,
\begin{equation}\label{eq-10-aan}
|\nabla u|^2=\frac{1}{m^2} |\nabla F|^2.
\end{equation}
Next, it follows from  (\ref{eq-09-aan})

\begin{multline*}
\frac{\partial^2 u}{\partial x_i^2}=
\frac{1}{m}\frac{\partial }{\partial x_i}\left( \frac{\partial F}{\partial y_i} \right)-
\frac{1}{m^2}\frac{\partial F}{\partial y_i}\frac{\partial m}{\partial x_i}\\
=\frac{1}{m}\frac{\partial }{\partial x_i}\left( \frac{\partial F}{\partial y_i} \right)-
\frac{t}{m^2}\frac{\partial F}{\partial y_i}
\sum\limits_{k=1}^{n}
\left(
V_k''\frac{\partial u}{\partial x_i} \frac{\partial F}{\partial y_k}+
V_k'\frac{\partial }{\partial x_i}\left( \frac{\partial F}{\partial y_k}\right)
\right)\\
=\frac{1}{m}\frac{\partial }{\partial x_i}\left( \frac{\partial F}{\partial y_i} \right)-
\frac{t}{m^2}\frac{\partial F}{\partial y_i}
\sum\limits_{k=1}^{n}V_k'\frac{\partial }{\partial x_i}\left( \frac{\partial F}{\partial y_k}\right)
-
\frac{t}{m^3}\left(\frac{\partial F}{\partial y_i}\right)^2
\sum\limits_{k=1}^{n}V_k'' \frac{\partial F}{\partial y_k}.
\end{multline*}
Then
\begin{multline*}
\frac{\partial }{\partial x_i}\left( \frac{\partial F}{\partial y_k}\right)=
\sum\limits_{j=1}^{n}\frac{\partial^2 F}{\partial y_k \partial y_j}\frac{\partial y_j}{\partial x_i}=
\sum\limits_{j=1}^{n}\frac{\partial^2 F}{\partial y_k \partial y_j}
\left( -tV_j' \frac{\partial u}{\partial x_i}\right) +\frac{\partial^2 F}{\partial y_k \partial y_i}\\
=- \frac{t}{m}\sum\limits_{j=1}^{n}\frac{\partial^2 F}{\partial y_k \partial y_j}
V_j' \frac{\partial F}{\partial y_i} +\frac{\partial^2 F}{\partial y_k \partial y_i}.
\end{multline*}
Thus,
\begin{multline*}
\frac{\partial^2 u}{\partial x_i^2}=
-\frac{t}{m^2}\frac{\partial F}{\partial y_i}
\sum\limits_{j=1}^{n}\frac{\partial^2 F}{\partial y_i \partial y_j}V_j' +
\frac{1}{m}\frac{\partial^2 F}{\partial  y_i^2}\\
-\frac{t}{m^2}\frac{\partial F}{\partial y_i}
\sum\limits_{k=1}^{n}V_k'
\left( - \frac{t}{m}\sum\limits_{j=1}^{n}\frac{\partial^2 F}{\partial y_k \partial y_j}
V_j' \frac{\partial F}{\partial y_i} +\frac{\partial^2 F}{\partial y_k \partial y_i}  \right)
-
\frac{t}{m^3}\left(\frac{\partial F}{\partial y_i}\right)^2
\sum\limits_{k=1}^{n}V_k'' \frac{\partial F}{\partial y_k}\\
=
-\frac{2t}{m^2}\frac{\partial F}{\partial y_i}
\sum\limits_{k=1}^{n}\frac{\partial^2 F}{\partial y_i \partial y_k}V_k' +
\frac{1}{m}\frac{\partial^2 F}{\partial  y_i^2}+
\frac{t^2}{m^3}\left(\frac{\partial F}{\partial y_i}\right)^2
\sum\limits_{k,j=1}^{n}V_k'V_j'\frac{\partial^2 F}{\partial y_k \partial y_j}\\
-\frac{t}{m^3}\left(\frac{\partial F}{\partial y_i}\right)^2
\sum\limits_{k=1}^{n}V_k'' \frac{\partial F}{\partial y_k}.
\end{multline*}
In this way,
\begin{multline}\label{eq-11-aan}
\Delta u=
-\frac{t}{m^2}\sum\limits_{k=1}^{n}V_k' \frac{\partial}{\partial y_k} \left( |\nabla F|^2\right)+
\frac{1}{m}\Delta F\\
+
\frac{t^2}{m^3}|\nabla F|^2
\sum\limits_{k,j=1}^{n}V_k'V_j'\frac{\partial^2 F}{\partial y_k \partial y_j}
-\frac{t}{m^3}|\nabla F|^2
\sum\limits_{k=1}^{n}V_k'' \frac{\partial F}{\partial y_k}.
%\eqno{(12)}
\end{multline}

If $\Delta F=|\nabla F|^2=0$  then the relation (\ref{eq-07-aan}) is fulfilled identically by  (\ref{eq-10-aan})
and (\ref{eq-11-aan}). Thus, a following theorem takes place

\begin{theorem}
Suppose

1) $F(y_1,\ldots,y_n)$ be a function of complex variables $y_1,\ldots,y_n$
such that
$$
\Delta F=|\nabla F|^2=0;
$$

2) $f(z)$, $g(z)$, $V_k(z)$, $k=1,\ldots,n$ be arbitrary functions of one variable $z$;

3) function $u=u(x,t)$ be an implicit solution to equation
$$
u=F(x_1-tV_1(u),\ldots,x_n-tV_n(u)).
$$

Then functions  $w=f(u)$, $P=g(u)$, $\overline{V}=\overline{V}(u)$
for arbitrary $\nu=\nu(x,t)$ reduce \textup{(\ref{eq-03-aan})} to identity.
\end{theorem}

{\bf Example.}
Let $F=\varphi(y_1+\mathrm{i}y_2)$. Then
$\Delta F=|\nabla F|^2=0$. We can find $u(x_1, x_2, t)$
from an equation
$$
u=\varphi(x_1+\mathrm{i}x_2-t(V_1(u)+\mathrm{i}V_2(u))).
$$

By taking $\varphi(z)=z$, $V_1(u)=a_1 u$, $V_2(u)=a_2 u$, $a_1, a_2 \in \mathbb{C}$,
we find
$$
u=\frac{x_1+\mathrm{i}x_2}{1+t(a_1+\mathrm{i}a_2)}.
$$


By taking $\varphi(z)=z$, $V_1(u)=a_1 u^2$, 
$V_2(u)=a_2 u^2$, $a_1, a_2 \in \mathbb{C}$,
we find multivalued solution
$$
u=\frac{1+
 \sqrt{1+4t(a_1+\mathrm{i}a_2)(x_1+\mathrm{i}x_2)}}
 {2t(a_1+\mathrm{i}a_2)}.
$$

By taking $V_1(u)=a_1 \varphi^{-1}(u)$, $V_2(u)=a_2 \varphi^{-1}(u)$, $a_1, a_2 \in \mathbb{C}$,
we find
$$
u=\varphi \left(\frac{x_1+\mathrm{i}x_2}{1+t(a_1+\mathrm{i}a_2)}\right),
$$
where $\varphi(z)$ is arbitrary function and $\varphi^{-1}(z)$
is inverse function.



We can now return to (\ref{eq-07-aan}) and rewrite it in the form
\begin{equation*}%\label{eq-13-new}
\nu =- \frac{f' g' |\nabla u|^2}{\Delta f(u)},
%\eqno{(13)}
\end{equation*}
allowing denominator is not zero.
Like this the equality assumes formula for function $\nu$.
\begin{theorem}
Let

1) $F(y_1,\ldots,y_n)$ be an arbitrary function of variables $y_1,\ldots,y_n$;

2) $f(z)$, $g(z)$, $V_k(z)$, $k=1,\ldots,n$ be an arbitrary function of one variable   $z$;

3) a function $u=u(x,t)$ be an implicit solution to equation
$$
u=F(x_1-tV_1(u),\ldots,x_n-tV_n(u)).
$$

Then functions  $w=f(u)$, $P=g(u)$, $\overline{V}=\overline{V}(u)$
and a function  $\nu=\nu(x,t)$ which are defined by formula
\begin{equation*}%\label{eq-13n}
\nu =- \frac{f' g' |\nabla u|^2}{\Delta f(u)},
\end{equation*}
reduce \textup{(\ref{eq-03-aan})} to an identity. Here
$|\nabla u|^2$ and $\Delta u$ are defined by formulas
\textup{(\ref{eq-10-aan})} and \textup{(\ref{eq-11-aan})}.
\end{theorem}


Rewrite the relation (\ref{eq-07-aan}) in the form
$$
\nu  f' \Delta u+ \left(\nu f'' + f' g' \right)|\nabla u|^2=0.
$$
Substitute $|\nabla u|^2$ and $\Delta u$ from (\ref{eq-10-aan}) and (\ref{eq-11-aan}):
\begin{multline*}
\nu  f' 
\left(
\frac{1}{m}\Delta F-
\frac{t}{m^2}\sum\limits_{k=1}^{n}V_k' \frac{\partial}{\partial y_k} \left( |\nabla F|^2\right)+
\frac{t^2}{m^3}|\nabla F|^2
\sum\limits_{k,j=1}^{n}V_k'V_j'\frac{\partial^2 F}{\partial y_k \partial y_j}
\right.\\
\left.
-\frac{t}{m^3}|\nabla F|^2
\sum\limits_{k=1}^{n}V_k'' \frac{\partial F}{\partial y_k}
\right)+
\left(\nu f'' + f' g' \right)\frac{|\nabla F|^2}{m^2} =0.
\end{multline*}
The following 
statement immediate from this equality:

\begin{theorem}\label{th-03}
Let 

1) $F(y_1,\ldots,y_n)$ be a function of variables $y_1,\ldots,y_n$
which satisfies to an equation
$$
|\nabla F|^2=\kappa^2=const;
$$

2) $f(z)$, $g(z)$, $V_k(z)$, $k=1,\ldots,n$ be arbitrary 
functions of one variable $z$;

3) a function $u=u(x,t)$ be implicit solution to an equation
$$
u=F(x_1-tV_1(u),\ldots,x_n-tV_n(u)).
$$

Then functions $w=f(u)$, $P=g(u)$, $\overline{V}=\overline{V}(u)$
and function  $\nu=\nu(x,t)$ which is defined by formula
\begin{equation*}
\nu =\frac{\kappa^2}{m}\cdot
\frac{t\sum\limits_{k=1}^{n}V_k'' \dfrac{\partial F}{\partial y_k}-
t^2\sum\limits_{k,j=1}^{n}V_k'V_j'\dfrac{\partial^2 F}{\partial y_k \partial y_j}-mf' g'}{
m f' + \kappa^2 f''},
\end{equation*}
where $m= 1+t \sum\limits_{k=1}^{n} V_k' \dfrac{\partial F}{\partial y_k} $,
reduce \textup{(\ref{eq-03-aan})} to identity.
\end{theorem}

{\bf Remark.}
If $\Omega=\left\{\, y\in \mathbb{R}^n \, | \, \omega(y)=0  \,  \right\}$
is some surface in $\mathbb{R}^n(y)$, then function
$F(y)=\rho(y, \Omega)$ of distance from point $y$ to the 
surface $\Omega$
satisfies to the equation $|\nabla F|^2=1$.


\section{Solution to one-dimensional Blokhintsev equation 
under condition\\ $D_t w=0$}



Consider one-dimensional Blokhintsev equation
\begin{equation}\label{eq-12-aan}
D_t^2 w=\nu w_{xx}+P_x w_x+ V D_t w (\ln \nu)_x,
\end{equation}
where $D_t=\partial_t +V\partial_x$,
under supplementary condition
\begin{equation}\label{eq-13-aan}
D_t w\equiv \partial_t w +V\partial_x w =0.
\end{equation}


By (\ref{eq-13-aan}), the equation (\ref{eq-12-aan}) 
takes form
\begin{equation*}
\nu w_{xx}+P_x w_x=0.
\end{equation*}
Now then, consider a system
\begin{equation}\label{eq-14-aan}
w_{xx}=-\frac{P_x}{\nu } w_x,
\end{equation}
\begin{equation}\label{eq-15-aan}
 w_t =-V w_x
\end{equation}
and test it for consistency.

From(\ref{eq-15-aan}) by virtue (\ref{eq-14-aan}), we get
\begin{equation}\label{eq-16-aan}
 w_{tx} =-V w_{xx}-V_xw_x=V \frac{P_x}{\nu } w_x-V_x w_x=w_x\left(V \frac{P_x}{\nu }- V_x\right),
\end{equation}
\begin{multline}\label{eq-17-aan}
 w_{txx} =w_{xx}\left(V \frac{P_x}{\nu }- V_x\right)+w_x\left(V \frac{P_x}{\nu }- V_x\right)_x\\
=w_{x}\left(  \left(V \frac{P_x}{\nu }- V_x\right)_x-
\frac{P_x}{\nu } \left(V \frac{P_x}{\nu }- V_x\right) \right).
\end{multline}
From (\ref{eq-14-aan}) by virtue (\ref{eq-16-aan}), we obtain
\begin{equation}\label{eq-18-aan}
w_{xxt}=-\frac{P_x}{\nu } w_{xt}-\left(\frac{P_x}{\nu }\right)_t w_x=
w_x \left(  -\left(\frac{P_x}{\nu }\right)_t-
\frac{P_x}{\nu } \left(V \frac{P_x}{\nu }- V_x\right) \right).
\end{equation}
Generate consistency condition $w_{xxt}=w_{txx}$ by virtue (\ref{eq-17-aan}) and (\ref{eq-18-aan})
$$
w_x \left( \left(V \frac{P_x}{\nu }- V_x\right)_x + \left(\frac{P_x}{\nu }\right)_t \right)=0.
$$
As far as  $w_x=0$ and $w_t=0$, that is $w$ is a constant function, we can assume
$w_x\neq 0$ and, it follows that,
\begin{equation}\label{eq-19-aan}
 \left(V \frac{P_x}{\nu }- V_x\right)_x + \left(\frac{P_x}{\nu }\right)_t =0.
\end{equation}
Thus, we proved the following theorem.
\begin{theorem}
Blokhintsev equation  \textup{(\ref{eq-12-aan})} under
supplementary condition \textup{(\ref{eq-13-aan})}
has nontrivial solution if and only if, 
coefficients $V(x,t)$, $P(x,t)$, $\nu(x,t)$ satisfy to the equation \textup{(\ref{eq-19-aan})}.
If \textup{(\ref{eq-19-aan})} is fulfilled, then solution   $w(x,t)$ to the system \textup{(\ref{eq-12-aan})}, \textup{(\ref{eq-13-aan})} is
determined from totally-integrable system \textup{(\ref{eq-14-aan})}, \textup{(\ref{eq-15-aan})}
by initial data
$$
w|_{t=x=0}=w_0,\quad w_{x}|_{t=x=0}=w_1,
$$
where $w_0$, $w_1$ are constants.
\end{theorem}
In the case of arbitrary dimension $n\ge 1$, 
suggestion the solution  $w=w(x,t)$ to equation  
(\ref{eq-01-aan}) satisfies to supplementary condition
$D_t w=0$ leads to system of equations
\begin{equation}\label{eq-20-aan}
\nu \Delta w+ \left\langle \nabla P, \nabla w\right\rangle=0,
\end{equation}
\begin{equation}\label{eq-21-aan}
D_t w=0.
\end{equation}
Denote
\begin{equation*}%\label{eq-06_5}
L =\displaystyle\sum\limits_{k=1}^{n}V_k\frac{\partial}{\partial x_k},\quad
L_i =\displaystyle\sum\limits_{k=1}^{n}\frac{\partial V_k}{\partial x_i}\frac{\partial}{\partial x_k}, \quad
L_{ii} =\displaystyle\sum\limits_{k=1}^{n}\frac{\partial^2 V_k}{\partial x_i^2}\frac{\partial}{\partial x_k},
\end{equation*}
%\eqno{(6)}
$i=1,\ldots,n$. Then the operator $D_t $ is written in the form
\begin{equation*}%\label{eq-07_5}
D_t =\frac{\partial }{\partial t} +L.
\end{equation*}
%\eqno{(7)}
Applying operator $D_t $ to the equation (\ref{eq-20-aan}), we get
\begin{equation}\label{eq-22-aan}
D_t(\nu) \Delta w+ \nu D_t(\Delta w)+ \left\langle D_t(\nabla P), \nabla w\right\rangle+
\left\langle \nabla P, D_t(\nabla w)\right\rangle=0.
\end{equation}
As far as
$$
D_t \left(\frac{\partial w}{\partial x_i}\right)=\frac{\partial }{\partial x_i}(D_t w)-L_i w,
$$
then
\begin{equation}\label{eq-23-aan}
D_t \left(\nabla w\right)=\nabla(D_t w)-\mathcal{L} w,
\quad
D_t \left(\Delta w\right)=\Delta(D_t w)-2 \mathrm{div} (\mathcal{L} w) +\Delta(L) w,
\end{equation}
where $\mathcal{L} w=(L_1 w, \ldots, L_n w)$.

Owing to (\ref{eq-23-aan}) and (\ref{eq-21-aan}), the relation (\ref{eq-22-aan}) takes form
\begin{equation}\label{eq-24-aan}
D_t(\nu) \Delta w+
\nu \left( \Delta(L) w -  2 \mathrm{div} (\mathcal{L} w) \right)+
\left\langle D_t(\nabla P), \nabla w\right\rangle -
\left\langle \nabla P, \mathcal{L} w \right\rangle=0.
\end{equation}
%\eqno{(10)}
Owing to (\ref{eq-20-aan}), it is possible to exclude $\Delta w$ from 
(\ref{eq-24-aan}).
Now then, finally we obtain system of two equations to function  $w$:
$$
 \Delta w=- \frac{1}{\nu} \left\langle \nabla P, \nabla w\right\rangle=0,
$$
$$
2 \mathrm{div} (\mathcal{L} w)=
\Delta(L) w-\frac{1}{\nu^2}D_t(\nu)\left\langle \nabla P, \nabla w\right\rangle+
\frac{1}{\nu}\left\langle D_t(\nabla P), \nabla w\right\rangle-
 \frac{1}{\nu}\left\langle \nabla P, \mathcal{L} w \right\rangle.
$$
There are only  derivatives of  function 
$w$ with respect to variables $x$  in this system.










\section{Solution to one-dimensional Blokhintsev equation 
under condition \\$D_t w=w$}




Consider one-dimensional Blokhintsev equation
\begin{equation}\label{eq-25-aan}
D_t^2 w=\nu w_{xx}+P_x w_x+ D_t w V (\ln \nu)_x,
\end{equation}
%\eqno{(1)}
where $D_t=\partial_t +V\partial_x$,
under supplementary condition
\begin{equation}\label{eq-26-aan}
D_t w\equiv \partial_t w +V\partial_x w =w.
\end{equation}
Owing to (\ref{eq-26-aan}), the equation (\ref{eq-25-aan}) takes form
\begin{equation}\label{eq-27-aan}
w=\nu w_{xx}+P_x w_x+ w V (\ln \nu)_x.
\end{equation}
Denote
\begin{equation*}%\label{eq-04nn}
 \nu= e^{A}.
\end{equation*}
%\eqno{(4)}
Then we can rewrite  (\ref{eq-27-aan}) as
$$
w=e^{A} w_{xx}+P_x w_x+ w V A_x
$$
or
\begin{equation}\label{eq-28-aan}
w_{xx}=-e^{-A}P_x w_x +e^{-A}(1-V A_x)  w.
\end{equation}
Denote
\begin{equation}\label{eq-29-aan}
B=-e^{-A}P_x, \quad C =e^{-A}(1-V A_x).
\end{equation}

\begin{lemma}\label{l-01nn}
If $B(x,t)$, $C(x,t)$, $V(x,t)$ are given functions
then, by virtue \textup{(\ref{eq-29-aan})}, functions $A$ and $P$  are defined by formulas
\begin{equation*}%\label{eq-06-aann}
A=a-\ln \left( \int \frac{C e^{a}}{V} d\,x \right),
\end{equation*}
%\eqno{(7)}
\begin{equation*}
P=-\int B e^{a}  \left( \int \frac{C e^{a}}{V} d\,x \right)^{-1} d\,x,
\end{equation*}
where $a=\displaystyle\int \frac{d\,x}{V}$.
\end{lemma}
\begin{proof}
Consider the second relation of (\ref{eq-29-aan})
\begin{equation}\label{eq-30-aan}
VA_x =1-C e^{A}.
\end{equation}
Make the change of variables
$$
A=A_0+a,\,\,\, \mbox{where} \,\,\, a=\displaystyle\int \frac{d\,x}{V}.
$$
Then (\ref{eq-30-aan}) takes form
$$
V \left( {A_0}_x + \frac{1}{V}\right)=1-C e^{A_0 +a},
$$
$$
V {A_0}_x =-C e^{A_0} e^{a},
$$
$$
 -e^{-A_0}{A_0}_x =\frac{C  e^{a}}{V}.
$$
Therefore
$$
e^{-A_0}=\int \frac{C e^{a}}{V} d\,x,
$$
$$
A_0=-\ln \left( \int \frac{C e^{a}}{V} d\,x \right).
$$
Now then,
$$
A=a-\ln \left( \int \frac{C e^{a}}{V} d\,x \right).
$$
Now, from the first relation of (\ref{eq-29-aan})
$$
P_x=-Be^{A},
$$
we get
$$
P=-\int B e^{a}  \left( \int \frac{C e^{a}}{V} d\,x \right)^{-1} d\,x.
$$
The lemma is proved.
\end{proof}
\vspace{1cm}

Owing to notations (\ref{eq-29-aan}), the equation (\ref{eq-28-aan})
takes form
$$
w_{xx}=B w_x +C w.
$$
Now, consider a problem of  reduction of system
\begin{equation}\label{eq-31-aan}
w_{xx}=B w_x +C w,
\end{equation}
\begin{equation}\label{eq-32-aan}
w_{t}=-V w_x + w,
\end{equation}
in involution.

Owing to (\ref{eq-32-aan}) and (\ref{eq-31-aan}),
$$
w_{tx}=-V w_{xx} -V_x w_{x}+ w_{x}
=-V(B w_x +C w)+w_{x}(1-V_x)=
(1-V_x-VB)w_{x}-CVw.
$$
Now then,
\begin{equation}\label{eq-33-aan}
w_{tx}=(1-V_x-VB)w_{x}-CVw.
\end{equation}
Owing to (\ref{eq-33-aan}) and (\ref{eq-31-aan}),
\begin{multline*}
w_{txx}=(1-V_x-BV)w_{xx}
-(V_{xx}+B_xV+BV_x)w_x-CVw_x-(C_xV+CV_x)w\\
=(1-V_x-BV)(B w_x +C w)-(V_{xx}+B_xV+BV_x+CV)w_x-(C_xV+CV_x)w\\
=(B-2BV_x-B^2V-V_{xx}-B_xV-CV) w_x + (C-2CV_x-BCV-C_xV)w.
\end{multline*}
Now then,
\begin{equation}\label{eq-34-aan}
w_{txx}=(B-2BV_x-B^2V-V_{xx}-B_xV-CV) w_x + (C-2CV_x-BCV-C_xV)w.
\end{equation}
Owing to (\ref{eq-31-aan}), (\ref{eq-32-aan}) and (\ref{eq-33-aan}),
\begin{multline*}
w_{xxt}=B w_{xt} +B_tw_x+Cw_t+C_t w\\
=B \left((1-V_x-VB)w_{x}-CVw\right) +B_tw_x+C(-V w_x + w)+C_t w\\
=(B-BV_x-B^2V+B_t-CV)w_{x} +(C+C_t-BCV)w.
\end{multline*}

Now then,
\begin{equation}\label{eq-35-aan}
w_{xxt}=(B-BV_x-B^2V+B_t-CV)w_{x} +(C+C_t-BCV)w.
\end{equation}

By (\ref{eq-34-aan}) and (\ref{eq-35-aan}), we obtain  
\begin{multline*}
(B-2BV_x-B^2V-V_{xx}-B_xV-CV) w_x + (C-2CV_x-BCV-C_xV)w\\
=(B-BV_x-B^2V+B_t-CV)w_{x} +(C+C_t-BCV)w,
\end{multline*}
$$
-(BV_x+V_{xx}+B_xV) w_x - (2CV_x+C_xV)w=
B_tw_{x} +C_tw.
$$
It follows that,
\begin{equation}\label{eq-36-aan}
(B_t+(BV)_x+V_{xx}) w_x + (C_t+2CV_x+C_xV)w=0.
\end{equation}

If
\begin{equation*}%\label{eq-16nn}
B_t+(BV)_x+V_{xx}=0\,\,\, \mbox{and} \,\,\, C_t+2CV_x+C_xV=0
\end{equation*}
%\eqno{(16)}
then (\ref{eq-36-aan}) is  fulfilled  and the system (\ref{eq-31-aan}), (\ref{eq-32-aan})
is in involution. Taking account of lemma~\ref{l-01nn}, we obtain the following statement

\begin{theorem}\label{th-01n}
Let

1) $V(x,t)\neq 0$ be a given function;

2) functions $B(x,t)$, $C(x,t)$  be solutions of Cauchy problem
$$
B_t=-(BV)_x-V_{xx},
$$
$$
C_t=-2CV_x-C_xV,
$$
$$
B|_{t=0}=B_0(x),\quad C|_{t=0}=C_0(x)
$$
for  $B_0(x)$, $C_0(x)$;

3) functions $P(x,t)$, $\nu(x,t)$ be defined by formulas
\begin{equation*} 
P=-\int B e^{a}  \left( \int \frac{C e^{a}}{V} d\,x \right)^{-1} d\,x,
\end{equation*}
\begin{equation*}
\nu=e^{a} \left( \int \frac{C e^{a}}{V} d\,x \right)^{-1},
\end{equation*}
where $a=\displaystyle\int \frac{d\,x}{V}$.

Then the function $w(x,t)$ is a solution to involutive system
$$
w_{xx}=B w_x +C w,\quad w_{t}=-V w_x + w,
$$
$$
w|_{t=x=0}=w_0,\quad w_{x}|_{t=x=0}=w_1,
$$
where $w_{0}$, $w_{1}$ are constants, satisfy to the Blokhintsev equation  \textup{(\ref{eq-25-aan})}, where
coeffi\-cients $P(x,t)$, $\nu(x,t)$ are defined by the formulas above.
\end{theorem}


Now, if
\begin{equation*}
B_t+(BV)_x+V_{xx}\neq 0,
\end{equation*}
then from  (\ref{eq-36-aan}), we get
\begin{equation*} %\label{eq-18nn}
w_x=D w,
\end{equation*}
where
\begin{equation}\label{eq-37-aan}
D=- \frac{C_t+(CV)_x+CV_{x}}{B_t+(BV)_x+V_{xx}}.
\end{equation}
In this case, the system (\ref{eq-31-aan}), (\ref{eq-32-aan}) is equivalent to the system 
\begin{equation}\label{eq-38-aan}
w_t=(1-DV)w, \quad w_x=D w.
\end{equation}
Give consistency condition $w_{xt}=w_{tx}$:
$$
(1-DV)w_x-(DV)_xw=D w_t+D_tw.
$$
Owing to (\ref{eq-38-aan}),
$$
(1-DV)D w-(DV)_xw=D (1-DV)w+D_tw.
$$
It follows that,
\begin{equation}\label{eq-39-aan}
D_t+(DV)_x=0.
\end{equation}
The relation (\ref{eq-39-aan}) is a unique equation for functions $B$, $C$.
Owing to (\ref{eq-37-aan}), we can give system of Cauchy type:
\begin{equation*}
C_t=-(CV)_x-CV_x-D(B_t+(BV)_x+V_{xx}),\quad
D_t=-(DV)_x
\end{equation*}
for functions  $C$, $D$. In the process the function $B$ leaves over arbitrary.

Taking account of lemma~\ref{l-01nn}, we get the following result.
\begin{theorem}\label{th-02nn}
Let

1) $V(x,t)$, $B(x,t)$  be arbitrary functions such that
$$
B_t+(BV)_x+V_{xx}\neq 0;
$$

2) functions $C(x,t)$, $D(x,t)$  be a solution of Cauchy problem
$$
C_t=-(CV)_x-CV_x-D(B_t+(BV)_x+V_{xx}),\quad
D_t=-(DV)_x,
$$
$$
C|_{t=0}=C_0(x),\quad D|_{t=0}=D_0(x)
$$
for some analytical functions $C_0(x)$, $D_0(x)$;

3) functions  $P(x,t)$, $\nu(x,t)$ be defined by formulas
$$
P=-\int B e^{a}  \left( \int \frac{C e^{a}}{V} d\,x \right)^{-1} d\,x,
$$
$$
\nu=e^{a} \left( \int \frac{C e^{a}}{V} d\,x \right)^{-1},
$$
where $a=\displaystyle\int \frac{d\,x}{V}$.

Then the function $w(x,t)$ which is a solution to an involutive system
$$
w_t=(1-DV)w, \quad w_x=D w,
$$
$$
w|_{t=x=0}=w_0,
$$
where $w_{0}$ is a constant,
satisfies to Blokhintsev equation \textup{(\ref{eq-25-aan})}, coefficients $P(x,t)$, $\nu(x,t)$ of which  are defined by formulas above.
\end{theorem}


\section{Final remarks}
Methods of construction of explicit solutions to
mathematical physics equations 
based on  entering of supplementary differential relation  are the basis of differential constrains method
 \cite{2}
and has wide application area. So, for example, construction of invariant, partly-invariant solutions
 and so on by methods of group analysis of 
  differential equations is direct application of differential constrains method with the use of algebraic structure of  symmetry
  group of considered equation \cite{6}.
   Differential constrains which have natural origin and  are connected
with  assumed equation directly, are of the main interest.
In this context, we formulate a problem for Blokhintsev equation  
which in our opinion be of interest,  be connected with differen\-tial constrains methods, and directly enlarge 
analogous problem for classic wave equation
\begin{equation}\label{eq-40-aan}
w_{tt}= \Delta w.
\end{equation}

{\bf Problem.}
To describe all functional-invariant solutions to equation
 (\ref{eq-01-aan}).

For equation (\ref{eq-40-aan}), requirement of functional
invariant property of a solution $w$
leads to supplementary
nonlinear differential equation of the first order
\begin{equation}\label{eq-41-aan}
w_{t}^2= |\nabla w|^2.
\end{equation}
The system (\ref{eq-40-aan}), (\ref{eq-41-aan})
was studied intensively by many authors \cite{7} -- \cite{10}.
The final description for real-valued solutions was obtained in paper \cite{17}.
There are not a final description for complex-valued  solutions,
there are  only special results
  \cite{9}.


Note, the equation (\ref{eq-01-aan}) has variable coefficients 
$\nu$, $V$, $P$ contrary to wave equation (\ref{eq-40-aan}).
In this context, we note the problem setting by R. Courant: 
to find all linear hyperbolic equations of the second order 
which have collection of generalized func\-tional-invariant 
solutions  (GFIS).
In two-dimensional case, a	necessary and sufficient condition
for coefficients of linear differential equation of the second order with two independent variables for existing GFIS  was obtained \cite{9}.
In high dimensional cases, it is known only special classes of these equations,
and in general case, this problem leaves unsolved.
In papers
\cite{18,19},
special classes of equations  and second-order systems with 
variable
coefficients admitting GFIS are constructed and obtained  
some applications of these results  to inverse problems.







\bigskip
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% 
% 

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