\documentclass[reqno]{amsart}
\usepackage{amssymb,hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 148, pp. 1--25.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/148\hfil Asymptotic behavior]
{Asymptotic behavior for the centered-rarefaction appearance
problem}
\author[R. Flores-E.,  G. A. Omel'yanov\hfil EJDE-2005/148\hfilneg]
{Ruben Flores-Espinoza,  Georgii A. Omel'yanov}  % in alphabetical order

\address{Ruben Flores Espinoza \hfill\break
Departamento de Matematicas \\
Universidad de Sonora \\
Hermosillo,  83000, Mexico}
\email{rflorese@gauss.mat.uson.mx}

\address{Georgii A. Omel'yanov \hfill\break
Departamento de Matematicas \\
Universidad de Sonora \\
Hermosillo, 83000,  Mexico}
\email{omel@hades.mat.uson.mx}

\date{}
\thanks{Submitted June 9, 2005. Published December 13, 2005.}
\thanks{Supported by grant 41421F from SEP-CONACYT and by
 grant43208 from CONACYT}
\subjclass[2000]{35l65, 35L67}
\keywords{Gas dynamics; shock wave; rarefaction wave; interaction;
\hfill\break\indent weak asymptotic behaviour}

\begin{abstract}
 We construct a uniform in time asymptotic behavior, describing
 the interaction of two isothermal shock waves with the same
 direction of motion.  The main result is that any smooth
 regularization of the problem  implies the realization of the
 stable scenario of interaction.
 In particular, the rarefaction wave appearance is described.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

Let us consider the gas dynamics system in the isothermal case
\begin{equation}
\begin{gathered}
\frac{\partial \rho }{\partial t}+\frac{\partial (\rho u)}{\partial x}=0,
\quad x\in \mathbb{R}^{1},\quad t>0, \\
\frac{\partial (\rho u)}{\partial t}+\frac{\partial }{\partial x}
(\rho u^{2}+c_{0}^{2}\rho )=0.
\end{gathered}   \label{1}
\end{equation}
Let the initial data be two shock waves with the same direction of motion
(to the right):
\begin{equation}
\begin{gathered}
\rho \big\vert_{t=0}=\rho_{0}+e_{1}H(-x+x_{1}^{0})+e_{2}H(-x+x_{2}^{0}), \\
u\big\vert_{t=0}=v_{1}H(-x+x_{1}^{0})+v_{2}H(-x+x_{2}^{0}).
\end{gathered}   \label{2}
\end{equation}
Here $H(x)$ is the Heaviside function, $e_{i}=\rho _{i}-\rho _{i-1}>0$ and
$v_{i}=u_{i}-u_{i-1}>0$, $i=1,2$, are the amplitudes of jumps, $u_{0}=0$,
 and $\rho _{i}$, $u_{i}$, $c_{0}>0$ are constants. For definiteness,
we assume that $x_{1}^{0}>x_{2}^{0}$. The initial shock waves are
assumed to be stable, so that
\begin{equation}
u_{1}=c_{0}\frac{e_{1}}{\sqrt{\rho _{0}\rho _{1}}},\quad
u_{2}=u_{1}+c_{0} \frac{e_{2}}{\sqrt{\rho _{1}\rho _{2}}}.  \label{3}
\end{equation}
The solution of problem (\ref{1}), (\ref{2}) seems to be well-known
nowadays. Indeed, the standard procedure of ``step-by-step''
consideration before and after the interaction time instant $t=t^{\ast }$
shows that the solution is described by the two noninteracting shock waves
for $t<t^{\ast }$, namely
\begin{equation}
\begin{gathered}
\rho =\rho _{0}+e_{1}H\big(-x+\varphi _{10}(t)\big)+e_{2}H\big(-x+\varphi
_{20}(t)\big), \\
u=v_{1}H\big(-x+\varphi _{10}(t)\big)+v_{2}H\big(-x+\varphi _{20}(t)\big),
\end{gathered}\label{4}
\end{equation}
where $\varphi _{i0}=\varphi _{i0_t}t+x_{i}^{0}$ are the phases of the
shocks,
\begin{equation}
\varphi _{10_t}=c_{0}\sqrt{\frac{\rho _{1}}{\rho _{0}}},\quad
\varphi_{20_t}=u_{1}+c_{0}\sqrt{\frac{\rho _{2}}{\rho _{1}}}.  \label{5}
\end{equation}

Next, at the time instant $t^{\ast }$ of the confluence, the initial
conditions (\ref{2}) are replaced by the shock wave with the amplitudes $
\rho _{2}$ and $u_{2}$ of the jumps of $\rho $ and $u$ which are
concentrated at the point
$x^{\ast }:=\varphi _{10}(t^{\ast })=\varphi _{20}(t^{\ast })$.
Solving this Riemann
problem we obtain that the solution for $t>t^{\ast }$ is a uniquely defined
combination of a shock wave and a centered rarefaction (see, for example
\cite{b1,r1}). Let us call this behavior of the solution the
``stable scenario".

However, the uniqueness of weak solutions for hyperbolic systems of
conservation laws has been proved (with additional conditions) only for
sufficiently small amplitudes of shocks (see \cite{b1,d1,l1}). It is
well--known that apart from the above mentioned solution, the Riemann
problem admits a family of artificial solutions
(see, for example, \cite{l1}). Therefore, the described construction
can not be treated as a well-posed
one for the case of arbitrary amplitudes of shocks.

It is clear that the weak point of this scheme is the consideration of shock
waves as noninteracting ones for time close to $t^*$. Moreover, this
conflicts with the physical sense of the problem since the actual gas
dynamics include viscosity phenomena. Therefore, it is necessary to smooth
the solution for time close to $t^*$ and to consider the process of
interaction in detail.

 Whitham \cite{w1} was the first to solve a similar problem for the
inviscid Burgers-Hopf equation
\begin{equation}
\frac{\partial u}{\partial t}+\frac{\partial }{\partial x}f(u)=0
\label{6}
\end{equation}
with the quadratic nonlinearity $f(u)=u^{2}$. Passing to the Burgers
regularization and using the Hopf-Cole transformation,  Whitham found the
exact solution for the initial data similar to (\ref{2}) and, as a result,
established that the regularization implies the choice of a stable scenario
of interaction. However, this procedure can be applied uniquely for the
quadratic nonlinearity. A progress in this problem has been achieved only
recently by  Danilov and  Shelkovich for equation (\ref{6}) with convex
nonlinearities (\cite{d2}; see also \cite{d3,d4}). Since it is impossible to
find exact solutions in the general case, they constructed an asymptotic
solution in the framework of the ``weak asymptotic method"
\cite{d2,d3,d4,d5,g1,k1}. The main point here is the treatment of the solution
$u_{\varepsilon }(x,t)$ of the regularized problem as a
$\mathcal{C}^{\infty}([0,T];\mathcal{C}^{\infty }(\mathbb{R}^{1}))$
 mapping for $\varepsilon$ a positive constant and a
$\mathcal{C}([0,T];\mathcal{D}'(\mathbb{R}^{1}))$
mapping uniformly in $\varepsilon \in [ 0,1]$, where $\varepsilon $ is
a parameter of regularization. Respectively, a family $u_{\varepsilon }(t,x)$
is called an \textit{asymptotic}
$\mod \mathcal{O}_{\mathcal{D}'}(\varepsilon )$ \textit{solution} of
equation (\ref{6}) if the relation
\begin{equation*}
\frac{d}{dt}\int_{-\infty }^{\infty }u_{\varepsilon }\psi dx-\int_{-\infty
}^{\infty }f(u_{\varepsilon })\frac{\partial \psi }{\partial x}dx=
\mathcal{O}(\varepsilon )
\end{equation*}
holds for any test function $\psi =\psi (x)$. The main advantage of this
approach is the possibility to describe the interaction of nonlinear waves
by an ordinary differential equation. Let us note that this method allows
also to describe interactions of solitons for non integrable problems
\cite{d4,k1}.

Our aim is a generalization of the weak asymptotic method for hyperbolic
systems of conservation laws. Using system (\ref{1}) as a simple but
meaningful example we show that this tool easily allows to construct a
uniform in time asymptotic solution. The case of shock waves with opposite
directions of motion has been examined in the paper \cite{g1}. In the
present paper we consider the case of the same direction of shock waves
motion. It is necessary to note that the construction of a weak asymptotic
solution for a system of conservation laws entails the appearance of a
dynamical system (instead of an equation in the scalar case). Analysis of
this system requires the use of the specifics of the original problem. This
can be done for any particular problem and we indicate the way how to do
this in the general case (see Conclusion).

Specifically for problem (\ref{1}), (\ref{2}), we consider the phenomena of
a rarefaction wave appearance as the most interesting effect. That is why we
point out this part of the work in the title of the paper. Note also that
the origin of centered rarefaction is possible exclusively for systems. To
consider rarefaction waves in scalar cases we should set such wave as the
initial data or set discontinuous initial data with appropriate sign of the
jump. Asymptotic behavior for the corresponding problems of weak
discontinuities interaction for the inviscid Burgers-Hopf equation
have been constructed recently in \cite{d3}.

\section{Construction of the asymptotic solution}

\subsection{Definitions and statement of the main result}

Following the ideas sketched above we arrive at:

\begin{definition} \label{def2.1} \rm
Sequences $\rho_\varepsilon(t,x)$ and $u_\varepsilon(t,x)$ are called a {
\normalfont weak asymptotic} $\mod \mathcal{O}_{\mathcal{D}'}(
\varepsilon)$ {\normalfont solution of system (\ref{1})} if $
\rho_\varepsilon(t,x)$ and $u_\varepsilon(t,x)$ belong to $\mathcal{C}
^\infty([0,T]\times\mathbb{R}^1)$ for $\varepsilon$=constant$>0$ and to $
\mathcal{C}(0,T;\mathcal{D}'(\mathbb{R}^1))$ uniformly in $
\varepsilon\in[0,const]$ and if the relations
\begin{equation}  \label{7}
\begin{gathered}
  \frac{d}{dt}\int_{-\infty}^\infty
\rho_\varepsilon\psi_1 dx- \int_{-\infty}^\infty \rho_\varepsilon
u_\varepsilon\frac{\partial\,\psi_1}{\partial\, x} dx=\mathcal{O}
(\varepsilon), \\
 \frac{d}{dt}\int_{-\infty}^\infty \rho_\varepsilon
u_\varepsilon \psi_2 dx- \int_{-\infty}^\infty (\rho_\varepsilon
u_\varepsilon^2+c_0^2\rho_\varepsilon)\frac{\partial\,\psi_2}{\partial\, x}
dx=\mathcal{O}(\varepsilon)
\end{gathered}
\end{equation}
hold for any test function $\psi_i=\psi_i(x)\in\mathcal{D}(\mathbb{R})$, $
i=1,2$.
\end{definition}

It is necessary to note that any $\mathcal{O}(\varepsilon)$ diffusion (or
diffusion--dispersion) regularization (see, for example \cite{l1}) of
equations (\ref{1}) implies the appearance of $\mathcal{O}(\varepsilon)$
corrections in relations (\ref{7}).

To present the asymptotic solution, let us denote $\omega=\omega(\eta)\in
\mathcal{C}^\infty(\mathbb{R}^1)$ an auxiliary function such that
% \label{8}
\begin{equation*}
\lim_{\eta\to -\infty}\omega= 0 \quad{\text{and}}\quad \lim_{\eta\to
\infty}\omega= 1.
\end{equation*}
For simplicity, we assume that $\omega$ tends to its limiting values at an
exponential rate. Moreover, let
\begin{equation}  \label{9}
\omega '_\eta >0,\quad {\text{and}}\quad
\omega(\eta)+\omega(-\eta)=1.
\end{equation}
Obviously this implies that $\omega(\eta)-1/2$ is an odd function and
$\omega((-x+\phi)/\varepsilon)$ tends to the Heaviside function
$H(-x+\phi)$ as $\varepsilon\to 0$.

Now, let us write the weak asymptotic solution for the problem (\ref{1}), (
\ref{2}) in the following form:
\begin{equation}
\begin{gathered}
\rho _{\varepsilon }=\rho
_{0}+\sum _{i=1}^{2}E_{i}\omega _{i}+(R_{3}-R)\omega _{3}-(R_{4}-R)\omega
_{4}, \\
 u_{\varepsilon }=\sum
_{i=1}^{2}(V_{i}\omega _{i}+W_{i}\omega _{i}')+U\{(x-\phi
_{4})\omega _{4}-(x-\phi _{3})\omega _{3}\},
\end{gathered}
\label{10}
\end{equation}
where
\begin{gather}
\omega _{i}=\omega \big(\frac{-x+\phi _{i}}{\varepsilon }\big),\quad \omega
_{i}'=\frac{d\omega (\eta )}{d\eta }\Big|_{\eta =(-x+\phi
_{i})/\varepsilon },\quad R=\rho _{2}e^{-\beta (x-\phi _{3})/c_{0}},
\label{11}
\\
\begin{gathered}
\phi _{i}=\phi _{i0}(\tau
,t)+\varepsilon \varphi _{i2}(\tau ),\quad \phi _{i0}=\varphi _{i0}(t)+\psi
_{0}(t)\varphi _{i1}(\tau )\quad \text{for}\,\,i=1,2, \\
\phi _{j}=\phi _{j0}(\tau
,t)+\varepsilon \varphi _{j2}(\tau ),\quad \phi _{j0}=\varphi _{j0}(t)+\psi
_{1}(t)\varphi _{j1}(\tau )\quad \text{for}\,\,j=3,4,
\end{gathered}
\label{12}
\\
\psi _{0}=\varphi _{20}-\varphi _{10},\quad \psi _{1}=\varphi _{40}-\varphi
_{30},\quad R_{j}=R|_{x=\phi _{j}},\quad \tau =\psi _{0}/\varepsilon \,.
\label{13}
\end{gather}
The functions $\varphi _{10}$ and $\varphi _{20}$ are the phases (\ref{5})
of noninteracting shock waves (\ref{4}) (for $t<t^{\ast }$),
$E_{i}=E_{i}(\tau )$, $V_{i}=V_{i}(\tau )$, $W_{i}=W_{i}(\tau )$,
$\beta=\beta (\tau ,t,\varepsilon )>0$, $U=U(\tau ,t,\varepsilon )>0$,
and $\varphi _{k1}$, $\varphi _{k2}$, $k=1,\dots ,4$, are smooth functions.
 We will assume that
\begin{gather}
E_{i}\to e_{i},\,\,V_{i}\to v_{i},\,\,W_{i}\to
0,\,\,\varphi _{i1}\to 0,\,\,\varphi _{j1}\to \overline{
\varphi }_{j1}\quad \text{as }\tau \to -\infty ,  \label{14}
\\
E_{i}\to \overline{E}_{i},\,\,V_{i}\to \overline{V}
_{i},\,\,W_{i}\to {\overline{W}}_{i},\,\,\varphi _{i1}\to
\overline{\varphi }_{i1},\,\,\varphi _{j1}\to 0\quad \text{as }
\tau \to +\infty ,  \label{15}
\\
\varphi _{k2}\to \overline{\varphi }_{k2,\pm }\quad \text{as }
\tau \to \pm \infty .  \label{15a}
\end{gather}
Here $\overline{E}_{i}$, $\overline{V}_{i}$, ${\overline{W}}_{i}$,
$\overline{\varphi }_{k1}$, $\overline{\varphi }_{k2,\pm }$ are constants,
 $i=1,2$, $j=3,4$, $k=1,2,3,4$.

Moreover, we assume that
\begin{equation}
\phi _{30}-\phi _{40}\to 0\quad \text{as } \tau \to
-\infty ,\quad \phi _{10}-\phi _{20}\to 0\quad \text{as } \tau
\to +\infty ,  \label{16}
\end{equation}
and set the geometric conditions for the phases of the regularized waves
\begin{gather}
\text{for}\quad t<t^{\ast }:\quad \phi _{2}<\phi _{1}<\phi _{j}\quad {
\mod\mathcal{O}(\varepsilon )},\quad j=3,4,  \label{16a}
\\
\text{for}\quad t>t^{\ast }:\quad \phi _{3}<\phi _{4}<\phi _{i}\quad {
\mod\mathcal{O}(\varepsilon )},\quad i=1,2.  \label{16b}
\end{gather}
The notation $\mod\mathcal{O}(\varepsilon )$  means here that we do not compare
these phases when the distances between the paths are $\mathcal{O}
(\varepsilon )$.

For the functions $\beta $ and $U$ we assume the convergence
\begin{equation}
\beta (\tau ,t,\varepsilon )\to \overline{\beta }^{\,\pm }(t)+\mathcal{
O}(\varepsilon ),\quad U(\tau ,t,\varepsilon )\to \overline{U}^{\,\pm
}(t)+\mathcal{O}(\varepsilon )\quad \text{as } \tau \to \pm
\infty  \label{17}
\end{equation}
to some bounded by constants (in $C$-sense) smooth functions. However, we
suppose the following relations hold:
\begin{equation}
\beta (0,t^{\ast },\varepsilon )=\mathcal{O}(\varepsilon ^{-1}),\quad
U(0,t^{\ast },\varepsilon )=\mathcal{O}(\varepsilon ^{-1}).  \label{17a}
\end{equation}
We will assume also that all convergences mentioned above are of exponential
rate.

Assumptions (\ref{14}) and the first assumption (\ref{16}) imply that the
anzatz (\ref{10}) describes the two noninteracting waves (\ref{4}) before
the interaction $(t<t^{\ast }$, so that for $\tau \to -\infty $ as
$\varepsilon \to 0)$. After the time instant of interaction
$(t>t^{\ast }$, so that for $\tau \to +\infty $ as
$\varepsilon \to 0)$ anzatz (\ref{10}) describes shock waves with
 amplitudes $\overline{E}_{i}$, $\overline{V}_{i}$ completed by weak
discontinuities on the curves $x=\phi _{3}$ and $x=\phi _{4}$.
In fact, we will prove that
$\overline{\beta }^{\,+}(t)=\overline{U}^{\,+}(t)=1/(t-t^{\ast })$.
Therefore, the terms $U(x-\phi _{4})$ and $R-R_{4}$ describe in the
limit as $\tau \to \infty $ a centered rarefaction concentrated between
the curves $x=\phi _{3}$ and $x=\phi _{4}$. The phase corrections
$\varepsilon \varphi_{k2}$ describe small shifts of the trajectories
which appear as a result of the interaction. The terms $W_{i}\omega _{i}'$
describe soliton--type corrections concentrated at the time of the
waves interaction.

Now let us formulate our main result.

\begin{theorem} \label{theorem1}
Any smooth regularization of the problem \eqref{1}, \eqref{2}
 implies the existence of a weak asymptotic solution which describes
uniformly in time the stable scenario of the shock
waves interaction.
\end{theorem}

\begin{remark} \label{rmk2.1} \rm
We choose a regularization $\omega $ of the initial data with additional
properties (\ref{9}) since these assumptions allow to simplify
the analysis. However, it is clear that these assumptions have only a
technical nature, and a similar to Theorem \ref{theorem1} result holds for
any general regularization (see, for example, \cite{d4}).
\end{remark}

In what follows we will use the notation:

\begin{definition}
A sequence $f(t,x,\varepsilon)$ is said to be of the value
$\mathcal{O}_{\mathcal{D}'}(\varepsilon^k)$ if the relation
\begin{equation}  \label{18}
\int_{-\infty}^\infty f(x,t,\varepsilon)\psi(x) dx=\mathcal{O}
(\varepsilon^k)
\end{equation}
holds for any test function $\psi=\psi(x)\in\mathcal{D}(\mathbb{R})$.
\end{definition}

\subsection{Preliminary calculations}

To determine the asymptotic statement (\ref{10}) we should calculate
 weak expansions of $\rho _{\varepsilon }$ and of the products
$\rho _{\varepsilon}u_{\varepsilon }$,
$\rho _{\varepsilon }u_{\varepsilon }^{2}$. Almost
trivial calculations show that
\begin{equation}
\rho _{\varepsilon }=\rho _{0}+\sum
_{i=1}^{2}E_{i}H_{i}+(R_{3}-R)H_{3}-(R_{4}-R)H_{4}+\mathcal{O}_{\mathcal{D}
'}(\varepsilon ^{2}).  \label{19}
\end{equation}
Here and in what follows we have
\begin{equation*}
H_{i}=H(-x+\phi _{i}).
\end{equation*}
Next, products $\omega _{i}\omega _{j}$ appear in the formulas for
 $\rho_{\varepsilon }u_{\varepsilon }$ and
$\rho _{\varepsilon }u_{\varepsilon}^{2}$. For example,
\begin{equation}
\begin{aligned}
\rho _{\varepsilon }u_{\varepsilon }
&=\sum _{i=1}^{2}V_{i}(\rho _{0}\omega _{i}+E_{i}\omega
_{i}^{2})+(E_{1}V_{2}+E_{2}V_{1})\omega _{1}\omega _{2} \\
&\quad +U\sum _{j=3}^{4}(-1)^{j}(x-\phi _{j})(\rho
_{0}\omega _{j}-(-1)^{j}(R_{j}-R)\omega _{j}^{2}) \\
&\quad +U\big((x-\phi _{4})(R_{3}-R)-(x-\phi
_{3})(R_{4}-R)\big)\omega _{3}\omega _{4} \\
&\quad +\sum _{i=1}^{2}\sum _{j=3}^{4}(-1)^{j}\big(
UE_{i}(x-\phi _{j})-V_{i}(R_{j}-R)\big)\omega _{i}\omega _{j}+\sum
_{i=1}^{2}\big\{\rho _{0}+E_{i}\omega _{i} \\
&\quad +(R_{3}-R)\omega _{3}-(R_{4}-R)\omega _{4}\big\}
W_{i}\omega _{i}'+W_{1}E_{2}\omega _{1}'\omega
_{2}+W_{2}E_{1}\omega _{1}\omega _{2}'.
\end{aligned}\label{20}
\end{equation}

\begin{lemma}\label{lemma1}
Under the assumptions mentioned above the following relations
hold
\begin{gather}  \label{21}
\omega_i^k=H_i+\varepsilon\, d_k\, \delta_i+\mathcal{O}_{\mathcal{D}
'}(\varepsilon^2),
\\ \label{22}
\omega_i^k\omega_j^\ell= B_{k\ell}(\sigma_{ji})H_i+B_{\ell
k}(\sigma_{ij})H_j-\varepsilon \{C_{k\ell}(\sigma_{ji})\delta_i+C_{\ell
k}(\sigma_{ij})\delta_j\} +\mathcal{O}_{\mathcal{D}'}(\varepsilon^2),
\\  \label{23}
\omega_j\omega'_i=\varepsilon B_{11}(\sigma_{ji}) \delta_i+\mathcal{O
}_{\mathcal{D}'}(\varepsilon^2),
\end{gather}
where $k, \ell\geqslant 1$, $d_k$ are some constants, $d_1=0$ and
\begin{equation}  \label{24}
\begin{gathered}
 B_{k\ell}(\sigma_{ji})  = k\int_{-\infty}^\infty
\omega^{k-1}(\eta)\omega'(\eta)\omega^\ell(\sigma_{ji}+\eta)d\eta,
\\
C_{k\ell}(\sigma_{ji})  =   k\int_{-\infty}^\infty
\eta\,\omega^{k-1}(\eta)\omega'(\eta)\omega^\ell(\sigma_{ji}+\eta)d
\eta.
\end{gathered}
\end{equation}
\end{lemma}

Here and in what follows
\begin{equation}
\sigma _{ji}:=\frac{\phi _{j}-\phi _{i}}{\varepsilon },\quad
d_{2}:=\int_{-\infty
}^{\infty }\omega (\eta )\omega (-\eta )d\eta ,\quad
\delta _{i}=\delta
(-x+\phi _{i}),  \label{25}
\end{equation}
where $\delta $ is the Dirac delta-function.

\begin{proof}[Proof of Lemma \ref{lemma1}]
Relation (\ref{21}) is almost obvious. Let us note only that the
equality $d_{1}=0$ is a direct consequence of the equality in (\ref{9}).
Furthermore, considering the left-hand side of relation (\ref{22}) in the
weak sense we obtain the following:
\begin{equation}
\begin{aligned}
I  &:=
\int_{-\infty }^{\infty }\omega ^{k}\big( \frac{-x+\phi _{i}}{\varepsilon }
\big) \omega ^{\ell }\big( \frac{-x+\phi _{j}}{\varepsilon }\big) \psi
(x)dx \\
 & =  \int_{-\infty }^{\infty }\omega ^{k}
\big( \frac{-x+\phi _{i}}{\varepsilon }\big) \omega ^{\ell }
\big( \frac{-x+\phi_{j}}{\varepsilon }\big) \frac{d}{dx}
\int_{-\infty }^{x}\psi (x')dx'\,dx \\
& =  -\int_{-\infty }^{\infty }\psi
_{0}(x)\big\{ \omega ^{\ell }\big( \frac{-x+\phi _{j}}{\varepsilon }
\big) \frac{\partial }{\partial x}\omega ^{k}\big( \frac{-x+\phi _{i}
}{\varepsilon }\big) + \\
&  \quad + \omega ^{k}\big( \frac{
-x+\phi _{i}}{\varepsilon }\big) \frac{\partial }{\partial x}\omega
^{\ell }\big( \frac{-x+\phi _{j}}{\varepsilon }\big) \big\} dx \\
& =  \vspace{0.2cm} k\int_{-\infty }^{\infty }\omega
^{k-1}(\eta )\omega '(\eta )\omega ^{\ell }(\sigma _{ji}+\eta
)\psi _{0}(\phi _{i}-\varepsilon \eta )d\eta + \\
&  \quad +\ell \int_{-\infty }^{\infty
}\omega ^{\ell -1}(\eta )\omega '(\eta )\omega ^{k}(\sigma
_{ij}+\eta )\psi _{0}(\phi _{j}-\varepsilon \eta )d\eta ,
\end{aligned}
\label{26}
\end{equation}
where $\psi \in \mathcal{D}(\mathbb{R}^{1})$,
$\psi _{0}(x)=\int_{-\infty}^{x}\psi (x')dx'$ and we took into account the
exponential rate of vanishing of the product
$\omega _{i}\omega _{j}\psi_{0} $ as $x\to \pm \infty $.
Now applying the Taylor expansion and
using the notation (\ref{24}), (\ref{25}) we can rewrite the right-hand side
in (\ref{26}) in the form % \label{27}
\begin{equation*}
I=B_{k\ell }(\sigma _{ji})\psi _{0}(\phi _{i})+B_{\ell k}(\sigma _{ij})\psi
_{0}(\phi _{j})-\varepsilon C_{k\ell }(\sigma _{ji})\psi (\phi
_{i})-\varepsilon C_{\ell k}(\sigma _{ij})\psi (\phi _{j})+\mathcal{O}
(\varepsilon ^{2}).
\end{equation*}
Since $\psi _{0}(\phi _{i})=\int_{-\infty }^{\infty }H(-x+\phi _{i})\psi
(x)dx$, we obtain relation (\ref{22}). Furthermore,
\begin{equation*}
\int_{-\infty }^{\infty }\omega _{j}\omega _{i}'\psi
(x)dx=\varepsilon \int_{-\infty }^{\infty }\omega (\eta +\sigma _{ji})\omega
'(\eta )\psi (\phi _{i}-\varepsilon \eta )d\eta =\varepsilon
B_{11}(\sigma _{ji})\psi (\phi _{i})+\mathcal{O}(\varepsilon ^{2}),
\end{equation*}
as was to be proved.
\end{proof}

A simple analysis of the integrals in (\ref{24}) implies the statement

\begin{lemma} \label{lemma2}
 The convolutions $B_{k\ell}$ and $C_{k\ell}$ exist and have
the following properties:
\begin{gather}  \label{28}
\begin{gathered}
 B_{k\ell}(\sigma_{ij})+B_{\ell k}(\sigma_{ji})=1,
\\
 \lim_{ \sigma\to -\infty}B_{k\ell}(\sigma)=
\lim_{ \sigma\to -\infty}C_{k\ell}(\sigma)=0,\quad \lim_{ \sigma\to
+\infty}B_{k\ell}=1\quad {\text{for }} k,\ell\geqslant 1,
\end{gathered}
\\  \label{29}
\lim_{\sigma\to +\infty}C_{1\ell}=0,\quad \lim_{\sigma\to
+\infty}C_{2\ell}=d_2, \quad {\text{for }}\ell\geqslant 1.
\end{gather}
\end{lemma}

Applying the statement of Lemma \ref{lemma1} we obtain that the weak
asymptotic behavior of the right-hand side in formula (\ref{20}) has the form
\begin{equation}  \label{30}
\rho_\varepsilon
u_\varepsilon=\sum_{i=1}^4M_iH_i+\varepsilon\sum_{i=1}^2N_i\delta_i+
\mathcal{O}_{\mathcal{D}'}(\varepsilon^2),
\end{equation}
where for $i=1,2$:
\begin{equation}  \label{31a}
\begin{gathered}
 M_i=M_{i0}+\sum_{j=3}^4(-1)^j\big(UE_i(x-\phi_j)-V_i(R_j-R)
\big)B_{11}(\sigma_{ji}), \\
 M_{i0}=V_i(\rho_0+E_i)+(E_1V_2+E_2V_1)B_{11}(\sigma_{\bar ii}),
\end{gathered}
\end{equation}

\begin{equation}  \label{31b}
\begin{aligned}
N_i=&d_2V_iE_i-(E_1V_2+E_2V_1)C_{11}(\sigma_{\bar ii})+W_i\big\{\rho_0+E_i/2
\\
&+W_iE_{\overline i}B_{11}(\sigma_{{\overline i}
i}) +(R_3-R)B_{11}(\sigma_{3i})-(R_4-R)B_{11}(\sigma_{4i})\big\}, \\
& \bar i=2\,\,\text{for } i=1,\,\, \bar i=1\,\,\text{for }i=2;
\end{aligned}
\end{equation}
for $i=3,4$,
\begin{equation}  \label{31c}
\begin{aligned}
M_i=&(-1)^i\Big\{\rho_0U(x-\phi_i)+\sum_{j=1}^2
\Big(UE_j(x-\phi_i)
 -V_j(R_i-R)\Big)B_{11}(\sigma_{ji})\Big\}\\
&-U(x-\phi_i)(R_i-R) \\
&+U\Big((x-\phi_4)(R_3-R)+(x-\phi_3)(R_4-R)\Big)
B_{11}(\sigma_{\bar ii}), \\
&\bar i=4\quad\text{for } i=3,\quad \bar i=3\quad\text{for } i=4.
\end{aligned}
\end{equation}
Now let us calculate the time derivatives of $\rho _{\varepsilon }$
and $\rho _{\varepsilon }u_{\varepsilon }$. Since
\begin{equation}
\frac{d\tau (t,\varepsilon )}{dt}=\frac{\psi _{0_{t}}}{\varepsilon },\quad
\psi _{0_{t}}=\varphi _{20_{t}}-\varphi _{10_{t}},  \label{31d}
\end{equation}
to obtain the precision $\mathcal{O}(\varepsilon )$ in the
right-hand side of relations (\ref{7}) we should take into account the terms
of order $\mathcal{O}_{\mathcal{D}'}(\varepsilon )$ in (\ref{19})
and (\ref{30}). At the same time, the phase derivatives do not include
$\mathcal{O}(1/\varepsilon )$ terms since
\begin{equation}
\frac{d\phi _{i0}}{dt}=\varphi _{i0_{t}}+\psi _{0_{t}}\varphi _{i1}+\frac{
\psi _{0}}{\varepsilon }\psi _{0_{t}}\varphi _{i1}'=\varphi
_{i0_{t}}+\psi _{0_{t}}(\tau \varphi _{i1})',\quad i=1,2  \label{32}
\end{equation}
and similarly
\begin{equation}
\frac{d\phi _{j0}}{dt}=\varphi _{j0_{t}}+\alpha (\tau \varphi _{j1})',\quad j=3,4.  \label{33}
\end{equation}
Here and in what follows the apostrophe denotes terms of value $\mathcal{O}
(1)$ of the derivative with respect to $\tau $ (or, what is the same, terms
$\mathcal{O}(1/\varepsilon )$ of the derivative with respect to $t$) and
\begin{equation}
\alpha =\psi _{1_t}/\psi _{0_t}.  \label{34}
\end{equation}
Thus, using formula (\ref{19}), the obvious equalities $(R_{j}-R)\delta
_{j}=0$, $j=3,4$ and notation (\ref{32}) we find that
\begin{equation}
\begin{aligned}
\frac{\partial \rho _{\varepsilon }}{\partial t}
& =  \frac{\psi _{0_{t}}}{\varepsilon }\Big \{\sum
_{i=1}^{2}E_{i}'H_{i}-R'H_{3}+(R'-R_{4}')H_{4}\Big\} \\
 &\quad +  \sum _{i=1}^{2}\frac{d\phi _{i}}{dt}E_{i}\delta _{i}-
\frac{dR}{dt}H_{3}+\frac{d}{dt}(R-R_{4})H_{4}+\mathcal{O}_{\mathcal{D}
'}(\varepsilon ).
\end{aligned}
\label{35}
\end{equation}
Here and in what follows $df/dt$ means all terms of value $\mathcal{O}(1)$
of the partial derivative $\partial f(x,t,\tau (t,\varepsilon ))/\partial t$.

Next, we need to use the following statement:

\begin{lemma}\label{lemma4}
Let $S=S(\tau)$ be a function from the Schwartz space and let
a function $\phi_k=\phi_k(\tau,\varepsilon)\in\mathcal{C}^\infty$ have the
representation
\begin{equation*}
\phi_k=x^*+\varepsilon\chi_k,
\end{equation*}
where $x^*$ is a constant  and $\chi_k=\chi_k(\tau,\varepsilon)$ is a slowly
increasing function. Then
\begin{equation}  \label{35a}
SH(-x+\phi_k)=SH(-x+x^*)+\varepsilon S\chi_k\delta(x-x^*)+\mathcal{O}_{
\mathcal{D}'}(\varepsilon^2).
\end{equation}
Moreover,
\begin{equation}  \label{35b}
S\delta(x-x^*)=S\delta(x-\phi_k)+\mathcal{O}_{\mathcal{D}'}(
\varepsilon) =S\delta(x-\phi_{k0})+\mathcal{O}_{\mathcal{D}
'}(\varepsilon).
\end{equation}
At the same time
\begin{equation}  \label{35c}
H(-x+\phi_k)= H(-x+\phi_{k0})+\varepsilon\varphi_{k2}\delta(x-\phi_{k0})+
\mathcal{O}_{\mathcal{D}'}(\varepsilon^2),
\end{equation}
if $\phi_k=\phi_{k0}+\varepsilon\varphi_{k2}$ and $\varphi_{k2}=
\varphi_{k2}(\tau)$ is a uniformly bounded function.
\end{lemma}

\begin{proof}  For any test function $\psi =\psi (x)$ we have
\begin{align*}
 &S\langle H(-x+\phi _{k}),\psi (x)\rangle \\
&=S\Big\{\int_{-\infty
}^{x^{\ast }}+\int_{x^{\ast }}^{x^{\ast }+\varepsilon \chi _{k}}\Big\}\psi
(x)dx \\
&=S\langle H(-x+x^{\ast }),\psi (x)\rangle +\varepsilon S\Big(
\chi _{k}\psi (x^{\ast })+\int_{0}^{\chi _{k}}\big(\psi (x^{\ast
}+\varepsilon \eta )-\psi (x^{\ast })\big)d\eta \Big).
\end{align*}
The last integral is bounded by $\varepsilon c\chi _{k}$, where $c$
 is a constant.
However, the product $\chi _{k}(\tau )S(\tau )$ remains bounded by a
constant uniformly in $\tau $. This implies relation (\ref{35a}). Next,
\begin{align*}
& S\langle \delta (x-x^{\ast }),\psi (x)\rangle\\
&=S\psi (x^{\ast}\pm \varepsilon \chi _{k})\\
&=S\langle \delta (x-\phi _{k}),\psi (x)\rangle
+\mathcal{O} (\varepsilon )   \\
&=S\langle \delta (x-\phi _{k0}),\psi (x)\rangle
+\varepsilon S\varphi _{k2}\psi '(\phi _{k0}
+\varepsilon \theta \varphi _{k2})+
\mathcal{O}(\varepsilon ),\quad \theta \in [ 0,1].
\end{align*}
Thus, we obtain both equalities (\ref{35b}). To prove relation (\ref{35c})
is enough to take into account the boundedness of the function $\varphi
_{k2} $. Let us take notes that the second equality in (\ref{35b}) holds for
any smooth function $S$ if the difference $\phi _{k}-\phi _{k0}$ is a
function of the value $\mathcal{O}(\varepsilon )$. This completes the proof.
\end{proof}

The relations mentioned in Lemma \ref{lemma4} allow to simplify the
right-hand side in formula (\ref{35}). Indeed, denoting by $x^{\ast
},t^{\ast }$ the point and the time instant of intersection of the paths $
x=\varphi _{i0}(t),i=1,\dots ,4$, we obtain the equality
\begin{equation}
\tau =\frac{\psi _{0}(t)}{\varepsilon }=\psi _{{0}_{t}}\frac{t-t^{\ast }}{
\varepsilon }.  \label{36}
\end{equation}
Thus, for the functions of the form (\ref{12}) we have
\begin{gather*}
\phi _{i}  =   x^{\ast
}+\varepsilon \tau \Big( \frac{\varphi _{i0_{t}}}{\psi _{0_{t}}}+\varphi
_{i1}(\tau )\Big) +\varepsilon \varphi _{i2}
=: x^{\ast }+\varepsilon \chi _{i}(\tau ),\quad i=1,2, \\
\phi _{j}  =   x^{\ast
}+\varepsilon \tau \alpha \Big( \frac{\varphi _{j0_{t}}}{\psi _{1_{t}}}
+\varphi _{j1}(\tau )\Big) +\varepsilon \varphi _{j2}
=: x^{\ast }+\varepsilon \chi _{j}(\tau ),\quad j=3,4.
\end{gather*}
Here and in the sequel $\chi _{k}=\chi _{k0}+\varphi _{k2}$, $k=1,2,3,4$,
\begin{equation}
\chi _{i0}=\tau \Big(\frac{\varphi _{{i0}_{t}}}{\psi _{{0}_{t}}}+\varphi
_{i1}\Big),\quad i=1,2,\quad \chi _{j0}=\alpha \tau \Big(\frac{\varphi
_{{j0}_{t}}}{\psi _{{1}_{t}}}+\varphi _{j1}\Big),\quad j=3,4.  \label{37}
\end{equation}
The assumptions for $E_{i}$, $\phi _{i}$, and $\beta $ allow to use the
statement of Lemma \ref{lemma4}. Therefore,
\begin{equation}
\begin{aligned}
&\frac{1}{\varepsilon }\Big\{ \sum
_{i=1}^{2}E_{i}'H_{i}-R'H_{3}+(R'-R_{4}')H_{4}\Big\}\\
&=\frac{1}{\varepsilon }(\sum
_{i=1}^{2}E_{i}-R_{4})'H(x-x^{\ast })\\
&\quad +\big(\sum _{i=1}^{2}\chi _{i}E_{i}'-\chi _{4}R_{4}'
+(\chi _{4}-\chi _{3})R'\big)\delta
(x-x^{\ast })+\mathcal{O}_{\mathcal{D}'}(\varepsilon ).
\end{aligned}
\label{38}
\end{equation}
Furthermore, let us define the function $\beta $ in the following form:
\begin{equation}
\beta =\{t-t^{\ast }+\varepsilon g_{1}(\tau )/\psi _{0_{t}}\}^{-1}=\psi
_{{0}_t}\{\varepsilon (\tau +g_{1}(\tau ))\}^{-1}.  \label{39}
\end{equation}
Here $g_{1}$ is assumed to be a smooth function such that with an
exponential rate
\begin{equation}
g_{1}\to 0\quad \text{as } \tau \to +\infty \quad \text{
and}\quad g_{1}\to \bar{g}_{1}|\tau |\quad \text{as } \tau
\to -\infty ,  \label{40}
\end{equation}
where $\bar{g}_{1}$ is a constant. Then
\begin{equation}
\beta (\phi _{4}-\phi _{3})=\psi _{0_{t}}\frac{\chi _{4}-\chi _{3}}{\tau
+g_{1}}.  \label{41}
\end{equation}
Respectively, $R_{4}$ does not depend on $t$, whereas the derivative $
R'$ does not include any term of the value $\mathcal{O}(1)$.
Therefore, formulas (\ref{35}), (\ref{38}), and (\ref{41}) imply the
relation
\begin{equation}
\begin{aligned}
\frac{\partial \rho _{\varepsilon }}{\partial t}
&=\frac{\psi _{0_{t}}}{\varepsilon }(E_{1}+E_{2}-R_{4})'H(-x+x^{\ast })
+\frac{dR}{dt}(H_{4}-H_{3}) \\
&\quad +\sum _{i=1}^{2}\frac{d\phi _{i}}{dt}E_{i}\delta
_{i}+\psi _{0_{t}}\{\sum _{i=1}^{2}\chi _{i}E_{i}'-\chi
_{4}R_{4}'\}\delta (x-x^{\ast })+\mathcal{O}_{\mathcal{D}'}(\varepsilon ).
\end{aligned}
\label{42}
\end{equation}
Observe now that
\begin{equation*}
\frac{d\phi _{i}}{dt}=\frac{d\phi _{i0}}{dt}+\psi _{{0}_t}\varphi
_{i2}'
\end{equation*}
and the derivative $\varphi _{i2}'$ is assumed to be a function
from the Schwartz space. Applying the statement of Lemma \ref{lemma4} again
we can transform formula (\ref{42}) to the final form
\begin{equation}
\begin{aligned}
\frac{\partial \rho _{\varepsilon }}{\partial t}
&=\frac{\psi _{0_{t}}}{\varepsilon }(E_{1}+E_{2}-R_{4})'H(-x+x^{\ast })
+\frac{dR}{dt}(H_{4}-H_{3}) \\
&\quad +\sum _{i=1}^{2}\frac{d\phi _{i0}}{dt}E_{i}\delta
_{i}+\psi _{0_{t}}\{\sum _{i=1}^{2}(E_{i}\varphi _{i2})'+L_{1}\}\delta (x-x^{\ast })+\mathcal{O}_{\mathcal{D}'}(\varepsilon ),
\end{aligned}
\label{42a}
\end{equation}
where
\begin{equation}
L_{1}=\sum _{i=1}^{2}\chi _{i}E_{i}'-\chi _{4}R_{4}'.
\label{42b}
\end{equation}
Carrying out similar calculations for the time derivative of
$\rho_\varepsilon u_\varepsilon $ we obtain the formula
\begin{equation}  \label{43}
\begin{aligned}
 \frac{\partial\, \rho_\varepsilon
u_\varepsilon}{\partial\, t}
&= \frac{\psi_{0_t}}{\varepsilon}
\sum_{i=1}^4M'_iH(-x+x^*)+\sum_{i=1}^2\frac{d\phi_{i}}{dt}
M_i\delta_i \\
&\quad +\sum_{i=1}^4\frac{d M_i}{dt}H_i
+L_2\delta(x-x^*) +\mathcal{O}_{\mathcal{D}'}(\varepsilon),
\end{aligned}
\end{equation}
where notation (\ref{31a})-(\ref{31c}) has been used and
\begin{equation}  \label{45}
L_2=\psi_{0_t}\big\{\sum_{i=1}^2N'_i+\sum_{i=1}^4
M'_i\chi_i\big\}.
\end{equation}
The explicit formulas (\ref{31a}), (\ref{31c}) and simple algebra imply the
identity
\begin{equation}  \label{46}
\sum_{i=1}^4M_i=(\rho_0+\rho_2+E_1+E_2-R_4)\big(V_1+V_2+U(\phi_{3}-\phi_4)
\big).
\end{equation}
Since the spatial derivatives of $\rho _{\varepsilon }$,
$\rho _{\varepsilon}u_{\varepsilon }$ and
$\rho _{\varepsilon }u_{\varepsilon }^{2}$ do not
include any term of value
$\mathcal{O}_{\mathcal{D}'}(\varepsilon^{-1})$, the definition
of the asymptotic solution and formulas (\ref{42}), (\ref{43})
and (\ref{46}) require the following equalities:
\begin{equation}
\frac{\partial }{\partial \tau }(E_{1}+E_{2}-R_{4})=0,\quad \frac{
\partial }{\partial \tau }\big(V_{1}+V_{2}+U(\phi _{3}-\phi _{4})\big)=0.
\label{146}
\end{equation}
Firstly, let us define the function $U$ in the form similar to (\ref{39}),
namely
\begin{equation}
U=\{t-t^{\ast }+\varepsilon g_{2}(\tau )/\psi _{{0}_t}\}^{-1}
=\psi_{{0}_t}\{\varepsilon (\tau +g_{2}(\tau ))\}^{-1},
\label{46b}
\end{equation}
where $g_{2}$ is a smooth function with the same properties as $g_{1}$. Then
the product $U(\phi _{3}-\phi _{4})$ does not depend on $t$. Next, to obtain
the constants of integration of equations (\ref{146}) it is enough to
consider the limits of the expressions as $\tau \to -\infty $ and to
use the first assumptions in (\ref{14}) and (\ref{16}).
Therefore, we obtain
the identities
\begin{equation}
\rho _{0}+E_{1}+E_{2}=R_{4},\quad V_{1}+V_{2}+U(\phi _{3}-\phi _{4})=u_{2}.
\label{47}
\end{equation}
Actually, these identities require the continuity of the solution on the
curve $x=\phi _{4}$ that separates the shock waves and the centered
rarefaction after the interaction.

Now let us note that $M_{i}-M_{i0}$, $i=1,2$, are soliton-type functions
with respect to $\tau $. Indeed, $M_{i}-M_{i0}\to 0$ as $\tau
\to \infty $ since $B_{11}(\sigma _{ji})\to 0$ as $\tau
\to \infty $ and $i=1,2$, $j=3,4$. For $\tau \to -\infty $
the convolutions $B_{11}(\sigma _{ji})\to 1$. So under the first
assumption (\ref{16})
\begin{equation*}
M_{i}-M_{i0}\to UE_{i}(\phi _{3}-\phi
_{4})+V_{i}(R_{3}-R_{4})\to 0,\quad i=1,2.
\end{equation*}
Therefore, we can apply the statement of Lemma \ref{lemma4} and rewrite
formula (\ref{43}) in the form
\begin{equation}
\frac{\partial \rho _{\varepsilon }u_{\varepsilon }}{\partial t}=\sum
_{i=1}^{2}\frac{d\phi _{i}}{dt}M_{i0}\delta _{i}+\sum _{i=1}^{4}\frac{
dM_{i}}{dt}H_{i}+L_{3}\delta (x-x^{\ast })+\mathcal{O}_{\mathcal{D}'}(\varepsilon ),  \label{43a}
\end{equation}
where
\begin{equation}
L_{3}=\sum _{i=1}^{2}\frac{d\phi _{i}}{dt}(M_{i}-M_{i0})+L_{2},
\label{43b}
\end{equation}
and $L_{2}$ is defined in (\ref{45}), and the equalities (\ref{47}) have
been took into account.

Let us transform the term $\sum _{i=1}^{4}H_{i}dM_{i}/dt$. Since $
M_{i0}=M_{i0}(\tau )$, $i=1,2$, we can apply the statement of Lemma 2.3
again and obtain the relation
\begin{equation*}
\begin{aligned}
\sum _{i=1}^{4}\frac{dM_{i}}{dt}H_{i}
& =  \sum _{j=3}^{4}
\frac{dM_{j}}{dt}H_{j}+\sum _{i=1}^{2}\frac{dM_{i}}{dt}H(-x+x^{\ast })+
\mathcal{O}_{\mathcal{D}'}(\varepsilon ) \\
 & =  \sum _{i=1}^{3}\frac{dM_{i}}{dt}H_{3}+\frac{dM_{4}}{dt
}H_{4}+\mathcal{O}_{\mathcal{D}'}(\varepsilon ).
\end{aligned}
\end{equation*}
Furthermore, since the equalities (\ref{46}) and (\ref{47}) imply the
identity
\begin{equation*}
\sum_{i=1}^{3}\frac{dM_{i}}{dt}=-\frac{dM_{4}}{dt},
\end{equation*}
we derive the relation
\begin{equation}
\sum _{i=1}^{4}\frac{dM_{i}}{dt}H_{i}=\frac{dM_{4}}{dt}(H_{4}-H_{3})+
\mathcal{O}_{\mathcal{D}'}(\varepsilon ).  \label{491}
\end{equation}
Next, the explicit formula (\ref{31c}) for $i=4$ and the first equality (\ref
{47}) allow to rewrite $M_{4}$ as follows
\begin{align*}
M_{4}&=R\big(V_{1}+V_{2}+U(x-\phi _{4})\big)-\big(
R(V_{1}+V_{2})+U(x-\phi _{4})(E_{1}+E_{2})\big)B_{11}(\sigma _{41}) \\
&\quad +U\big((x-\phi _{4})(R_{3}-R)+(x-\phi
_{3})(R_{4}-R)\Big)B_{11}(\sigma _{34}) \\
&\quad -\big(UE_{2}(x-\phi _{4})+V_{2}R\big)
\big(B_{11}(\sigma _{42})-B_{11}(\sigma _{41})\big) \\
&\quad -R_{4}\big(V_{1}B_{11}(\sigma_{14})+V_{2}B_{11}(\sigma _{24})\big).
\end{align*}
The last term here does not depend on $t$. Furthermore, under the geometric
assumptions (\ref{16a}), (\ref{16b}) the function $B_{11}(\sigma
_{42})-B_{11}(\sigma _{41})$ belongs to the Schwarz space. Next, $d\varphi
_{k2}/dt$ belong to the Schwarz space also whereas $\phi _{k}-\phi _{k0}$
have the value of $\mathcal{O}(\varepsilon )$, $k=1,\dots ,4$. This and the
statement of Lemma \ref{lemma4} imply the equality
\begin{equation}
\frac{dM_{4}}{dt}(H_{4}-H_{3})=\frac{dM_{40}}{dt}(H_{4}-H_{3})+L_{4}\delta
(x-x^{\ast })+\mathcal{O}_{\mathcal{D}'}(\varepsilon ),  \label{492}
\end{equation}
where
\begin{equation}
\begin{aligned}
M_{40} & =  R\big(V_{1}+V_{2}+U(x-\phi _{40})\big)
+M_{401}B_{11}(\sigma _{34}) \\
 &\quad -  \{R(V_{1}+V_{2})+U(E_{1}+E_{2})(x-\phi
_{40})\}B_{11}(\sigma _{41}), \\
M_{401} & =  U\big((x-\phi _{40})(R_{3}-R)+(x-\phi
_{30})(R_{4}-R)\big)
\end{aligned}
\label{493}
\end{equation}
and
\begin{equation}
\begin{aligned}
L_{4}&=  -  \frac{d}{dt}\big\{\big(UE_{2}(x-\phi _{4})+V_{2}R
\big)\big(B_{11}(\sigma _{42})-B_{11}(\sigma _{41})\big) \\
 &\quad +  \varepsilon \varphi _{42}U\big(R-(E_{1}+E_{2})B_{11}(
\sigma _{41})+(R_{3}-R)B_{11}(\sigma _{34})\big) \\
 &\quad +  \varepsilon \varphi _{32}U(R_{4}-R)B_{11}(\sigma _{34})
\big\}.
\end{aligned}
\label{494}
\end{equation}
Next, by similar reasons we derive the equality
\begin{equation}
\sum _{i=1}^{2}\frac{d\phi _{i}}{dt}M_{i0}\delta _{i}=\sum _{i=1}^{2}
\frac{d\phi _{i0}}{dt}M_{i0}\delta _{i}+\psi _{0_t}\sum _{i=1}^{2}\frac{
d\varphi _{i2}}{d\tau }M_{i0}\delta (x-x^{\ast })+\mathcal{O}_{\mathcal{D}
'}(\varepsilon ).  \label{495}
\end{equation}
Combining (\ref{491}) - (\ref{495}) we rewrite formula (\ref{43a}) in the
resulting form
\begin{equation}
\frac{\partial \rho _{\varepsilon }u_{\varepsilon }}{\partial t}=\sum
_{i=1}^{2}\frac{d\phi _{i0}}{dt}M_{i0}\delta _{i}+\frac{dM_{40}}{dt}
(H_{4}-H_{3})+L_{5}\delta (x-x^{\ast })+\mathcal{O}_{\mathcal{D}'}(\varepsilon ),  \label{49}
\end{equation}
where
\begin{equation}
L_{5}=L_{3}+L_{4}+\psi _{0_t}\sum _{i=1}^{2}\frac{d\varphi _{i2}}{
d\tau }M_{i0},  \label{496}
\end{equation}
and $L_{3}$, $L_{4}$ are presented in (\ref{43b}), (\ref{494}).

Now let us pass to the calculation of spatial derivatives. Carrying out
similar as above analysis we obtain the statement

\begin{lemma} \label{lemma5}
Under the assumptions mentioned above the following relations
hold
\begin{equation}  \label{50}
\begin{gathered}
  \frac{\partial\, \rho_\varepsilon}{\partial\, x}
=-\sum_{i=1}^2E_i\delta_i+\frac{\partial\, R}{\partial\, x}(H_4-H_3) +
\mathcal{O}_{\mathcal{D}'}(\varepsilon), \\
  \frac{\partial\, \rho_\varepsilon u_\varepsilon}{
\partial\, x}= -\sum_{i=1}^2M_{i0}\delta_i+\frac{\partial\, M_{40}}{
\partial\, x}(H_4-H_3) +M_*\delta(x-x^*)+\mathcal{O}_{\mathcal{D}
'}(\varepsilon), \\
  \frac{\partial\, \rho_\varepsilon u_\varepsilon^2
}{\partial\, x}= -\sum_{i=1}^2K_i\delta_i+\frac{\partial\, K_{4}}{
\partial\, x}(H_4-H_3) +K_*\delta(x-x^*)+\mathcal{O}_{\mathcal{D}
'}(\varepsilon),
\end{gathered}
\end{equation}
where
\begin{align*}
  K_i=&V_i^2(\rho_0+E_i)+2\rho_0V_1V_2B_{11}(
\sigma_{\bar ii}) +V_1(V_1E_2+2V_2E_1) B_{\bar ii}(\sigma_{\bar ii}) \\
&  +V_2(V_2E_1+2V_1E_2)B_{i\bar i}(\sigma_{\bar ii}), \quad i=1,2,
\\
  K_4=&R\big(U(x-\varphi_{40})+V_1+V_2\big)
^2B_{11}(\sigma_{14}) \\
 &-U^2(x-\varphi_{40})\big(
(E_1+E_2-R)(x-\varphi_{40})B_{11}(\sigma_{41})\\
&+2\rho_0(x-\varphi_{30})B_{11}(\sigma_{34})\big)
\end{align*}
and $K_*$, $M_*$ are smooth functions from the Schwarz space.
\end{lemma}

We do not present the explicit formulas of $K_*$ and $M_*$ since they are
huge. It is important only that $K_*$ and $M_*$ depend on $\varphi_{i2}$,
$i=1,2$ via convolutions $B_{kl}(\sigma_{ij})$ only since
$\sigma_{ij}=(\phi_{i0}-\phi_{j0})/\varepsilon+\varphi_{i2}-\varphi_{j2}$.

Substituting the expressions (\ref{42a}), (\ref{49}) and (\ref{50}) into the
definition (\ref{7}), we derive our main relations for obtaining of the
asymptotic solution parameters
\begin{equation}  \label{51}
\begin{aligned}
 \sum_{i=1}^2(\frac{d\phi_{i0}}{dt}
E_i-M_{i0})\delta_i+\Big(\frac {d R}{d t} +\frac{\partial\, M_{40}}{
\partial\, x}\Big)(H_4-H_3)& \\
  +\big(\psi_{0_t}\sum_{i=1}^2(E_i
\varphi_{i2})'+L_1+M_*\big)\delta(x-x^*)&=\mathcal{O}_{\mathcal{D}
'}(\varepsilon),
\end{aligned}
\end{equation}
\begin{equation}  \label{52}
\begin{aligned}
  \sum_{i=1}^2(\frac{d\phi_{i0}}{dt}
M_{i0}-K_{i}-c_0^2E_i)\delta_i+\Big(\frac {d M_{40}}{d t} +\frac{\partial\,
K_{4}}{\partial\, x}+c_0^2\frac{\partial\, R}{\partial\, x}\Big)(H_4-H_3) &\\
 +(L_5+K_*)\delta(x-x^*) &=\mathcal{O}
_{\mathcal{D}'}(\varepsilon).
\end{aligned}
\end{equation}

\section{Analysis of the shock wave dynamics}

Let us consider the system obtained by setting equal to zero the
coefficients of the functions $\delta _{i}$
\begin{equation}
\frac{d\phi _{i0}}{dt}E_{i}=V_{i}(\rho
_{0}+E_{i})+(E_{1}V_{2}+E_{2}V_{1})B_{11}(\sigma _{\bar{\imath}i}),
\label{53}
\end{equation}
\begin{equation}
\begin{aligned}
&\frac{d\phi _{i0}}{dt}(V_{i}(\rho
_{0}+E_{i})+(E_{1}V_{2}+E_{2}V_{1})B_{11}(\sigma _{\bar{\imath}i}))\\
&=V_{i}^{2}(\rho _{0}+E_{i}) +2\rho _{0}V_{1}V_{2}B_{11}(\sigma _{
\bar{\imath}i})+V_{1}(V_{1}E_{2}+2V_{2}E_{1})B_{\bar{\imath}i}(\sigma _{\bar{
\imath}i}) \\
&\quad +V_{2}(V_{2}E_{1}+2V_{1}E_{2})B_{i
\bar{\imath}}(\sigma _{\bar{\imath}i})+c_{0}^{2}E_{i},\quad i=1,2.
\end{aligned}\label{54}
\end{equation}

\begin{lemma} \label{lemma6}
Let the assumptions mentioned above hold. Then the second
assumption \eqref{16} holds and system
\eqref{53}, \eqref{54} describes the confluence of the shock waves for
$\tau\to\infty$.
\end{lemma}

\begin{proof}  Let us denote
\begin{equation}  \label{56}
z_i=V_i/E_i, \quad i=1,2
\end{equation}
and divide equations (\ref{53}) by $E_i$. Obviously, we obtain
\begin{equation}  \label{57}
\frac{d\phi_{i0}}{dt}=z_i(\rho_0+E_i)+E_{\bar i}(z_1+z_2)B_{11}(\sigma_{\bar
ii}), \quad i=1,2.
\end{equation}
Subtracting equation (\ref{57}) for $i=1$ from (\ref{57}) for $i=2$ and
changing the coefficient of the $\delta(x-x^*)$ function in (\ref{51}) we
derive the following equation for the difference $\sigma_{21}=(\phi_2-
\phi_1)/\varepsilon$
\begin{equation}  \label{58}
\psi_{0_t}\frac{d\sigma_{21}}{d\tau}=z_2\Big\{\rho_0\Big(1-\frac{z_1}{z_2}
\Big)+ (E_{1}+E_2)\Big(1-\Big(1+\frac{z_1}{z_2}\Big)B_{11}(\sigma_{21})\Big)
\Big\},
\end{equation}
since
\begin{equation*}
\frac{d}{d\,t}(\phi_2-\phi_1)=\psi_{0_t}\frac{d\sigma_{21}}{d\tau}.
\end{equation*}
Note that the assumptions (\ref{14}) require the scattering--type "initial"
datum
\begin{equation}  \label{581}
\sigma_{21}/\tau|_{\tau\to-\infty}\to 1.
\end{equation}
Next, using notation (\ref{56}) and equations (\ref{53}) again, we can
rewrite equations (\ref{54}) in the form
\begin{equation}  \label{59}
\begin{aligned}
&\big(z_i(\rho_0+E_i)+E_{\bar i}(z_1+z_2)B_{11}(\sigma_{\bar ii})\big)^2\\
&=c_0^2 +E_iz_i^2(\rho_0+E_i)
 +2\rho_0z_1z_2E_{\bar i}B_{11}(\sigma_{\bar ii})
+E^2_{\bar i}(z^2_{\bar i}+2z_1z_2)
B_{12}(\sigma_{\bar ii}) \\
&\quad +E_1E_2(z^2_{i}+2z_1z_2)B_{21}(\sigma_{\bar ii}), \quad  i=1,2.
\end{aligned}
\end{equation}
Therefore, treating $E_i$ as known coefficients we obtain two algebraic
equations for the functions $z_1$ and $z_2$. To solve them let us subtract
one equation (\ref{59}) from another one. Obviously, we obtain a homogeneous
quadratic equation. Solving this equation and choosing the sign using the
first two assumptions (\ref{14}), we pass to the equality
\begin{equation}  \label{60}
z_1=\mathcal{L}z_2,
\end{equation}
where %\label{61}
\begin{gather*}
\mathcal{L}=\big(\sqrt{b^2+a_1a_2}-b\big)/a_1,
\\
 a_j  =  \big(\rho_0+E_j+E_{\bar j}B_{11}(\sigma_{\bar jj})
\big)^2 -E_j\big(\rho_0+E_jB_{11}(\sigma_{j\bar
j})^2+\sum_{i=1}^2E_iB_{21}(\sigma_{\bar jj})\big), \\
\begin{aligned}
b  =&  \big(E_1+E_2B_{11}(\sigma_{21})\big)^2-\big(
E_2+E_1B_{11}(\sigma_{12})\big)^2 +E_1E_2(1-2B_{11}(\sigma_{21})) \\
&   -(E_1+E_2)\big(E_1B_{21}(\sigma_{21})-E_2B_{21}(
\sigma_{12})\big).
\end{aligned}
\end{gather*}
A simple analysis of the convolutions $B_{ij}(\sigma)$ and the H\"{o}lder
inequality result in the statement
%\end{proof}

\begin{lemma} \label{lemma7}
The function $B_{11}^2(\sigma)-B_{12}(\sigma)$ is even and
negative. The inequality
\begin{equation*}
2B_{11}(\sigma)-B_{21}(\sigma)>0
\end{equation*}
holds uniformly in $\sigma$.
\end{lemma}

Therefore,
\begin{equation*}
B_{11}^2(\sigma_{j\bar j})+B_{21}(\sigma_{\bar jj})=1+B_{11}^2(\sigma_{j\bar
j}) -B_{12}(\sigma_{j\bar j})\leqslant 1.
\end{equation*}
This implies the inequality
\begin{align*}
a_j  = & \rho_0\big(\rho_0+E_j+2E_{\bar j}B_{11}(\sigma_{\bar
jj})\big) +E_1E_2\big(2B_{11}(\sigma_{\bar jj})^2-B_{21}(\sigma_{\bar jj})
\big) \\
 & +  E_j^2\big(1-B^2_{11}(\sigma_{j\bar j})-B_{21}(\sigma_{\bar jj})
\big)>0, \quad j=1,2,
\end{align*}
and the existence of the function $\mathcal{L}$.

Now we can write out the solutions $z_j$ of equations (\ref{59}) as
functions of $E_1$, $E_2$, and $\sigma_{21}$,
\begin{equation}  \label{62}
z_1=c_0\mathcal{L}\mathcal{M}, \quad z_2=c_0\mathcal{M},
\end{equation}
where
\begin{gather*}
\mathcal{M}=\{\alpha_1\mathcal{L}^2+2\alpha_2\mathcal{L}+\alpha_3\}^{-1/2},
\\
\begin{aligned}
\alpha_1  = & \rho_0\big(\rho_0+E_1+2E_{2}B_{11}(\sigma_{21})
\big) \\
  &  +E_1E_2\big(2B_{11}(\sigma_{21})-B_{21}(
\sigma_{21})\big)+E_2^2B_{11}^2(\sigma_{21}),
\end{aligned}\\
 \alpha_2  =  E_1E_2\big(B_{11}(\sigma_{21})-B_{21}(
\sigma_{21})\big)+ E_2^2\big(B_{11}^2(\sigma_{21})-B_{12}(\sigma_{21})\big),
\\
 \alpha_3  =  E_2^2\big(B_{11}^2(\sigma_{21})-B_{12}(
\sigma_{21})\big).
\end{gather*}
Furthermore, considering $\mathcal{L}$ and $\mathcal{M}$ as functions of
$\sigma_{21}$ and using the properties of the convolutions
$B_{ij}(\sigma_{21})$, one can prove the following statement.

\begin{lemma} \label{lemma7a}
 Let $E_i(\tau)>0$ uniformly in $\tau$. Then the function
$\mathcal{L}=\mathcal{L}(\sigma_{21})$ decreases from the value
$\mathcal{L}_{-\infty}>1$ to the value $\mathcal{L}_\infty<1$ as
$\sigma_{21}$ goes from
$-\infty$ to $\infty$ and $\mathcal{L}|_{\sigma_{21}=0}=1$, where
\begin{equation*}
\mathcal{L}_{-\infty}=\frac{\sqrt{\rho_0+E_1+E_2}}{\sqrt{\rho_0}}, \quad
\mathcal{L}_{\infty}=\frac{\sqrt{\rho_0}}{\sqrt{\rho_0+E_1+E_2}}.
\end{equation*}
Conversely, the function $\mathcal{M}=\mathcal{M}(\sigma_{21})$ increases
from the value $\mathcal{M}_{-\infty}$ to the value $\mathcal{M}_\infty$
as $\sigma_{21}$ goes from $-\infty$ to $\infty$, where
\begin{equation*}
\mathcal{M}_{-\infty}=\big((\rho_0+E_1)(\rho_0+\sum_{i=1}^2E_i)\big)
^{-\frac12}, \,\, \mathcal{M}_{\infty}=\frac{\sqrt{\rho_0+E_1+E_2}}{\sqrt{
\rho_0\big(\rho_0E_1 +(\rho_0+E_2)^2\big)}}.
\end{equation*}
At the point $\sigma_{21}=0$, $\mathcal{M}=\mathcal{M}_0$, where
\begin{equation}  \label{62c}
\mathcal{M}_0=\big(\rho_0(\rho_0+\sum_{i=1}^2E_i)\big)^{-\frac12}.
\end{equation}
\end{lemma}

Using the first two assumptions (\ref{14}) we find now the limiting values
$\bar{\mathcal{M}}$ and $\bar{z}_{i}$ of $\mathcal{M}$ and $z_{i}$
\begin{equation}
\bar{\mathcal{M}}=1/\sqrt{\rho _{1}\rho _{2}},\quad \bar{z}_{1}=c_{0}/\sqrt{
\rho _{0}\rho _{1}},\quad \bar{z}_{2}=c_{0}/\sqrt{\rho _{1}\rho _{2}}\quad
\text{as } \sigma _{21}\to -\infty .  \label{63}
\end{equation}
Therefore, we obtain the limiting value of the right-hand side
$F=F(\tau,\sigma _{21})$ of equation (\ref{58}), namely
\begin{equation}
F\to \bar{F}=c_{0}(\sqrt{\rho _{2}}-\sqrt{\rho _{0}})/\sqrt{\rho _{1}
}\quad \text{as } \sigma _{21}\to -\infty .  \label{64}
\end{equation}
According to formulae (\ref{3}), (\ref{5}) and (\ref{31d}), the limiting
right-hand side (\ref{64}) is equal to $\psi _{0t}$. So that
\begin{equation}
\sigma _{21}\to \tau \quad \text{as } \tau \to -\infty ,
\label{65}
\end{equation}
which justifies the existence of a solution with the property (\ref{581}).

Next, $z_{1}=z_{2}$ at the point $\sigma _{21}=0$. Thus, $F=0$ at this point
since $B_{11}(0)=1/2$. Moreover, close to this point
\begin{equation*}
F=\sigma _{21}F'(\tau ,0),\quad F'(\tau ,0)<0.
\end{equation*}
Therefore, the value $\sigma _{21}=0$ can be reached only as $\tau
\to \infty $. Furthermore, stabilization of $E_{i}$ requires the
vanishing of the derivative $F_{\tau }'$ as $\tau \to
\infty $. Respectively, one can prove that all derivatives of $F$
vanish as $\tau \to \infty $. Hence,
\begin{equation}
\sigma _{21}\to 0\quad \text{as } \tau \to \infty .
\label{67}
\end{equation}
It remains to show that system (\ref{53}), (\ref{54}) coincides with the
standard Rankine-Hugoniot conditions as $\tau \to \pm\infty
$. Indeed, let $\tau \to -\infty $. Since ${\bar{z}}_{i}=e_{i}/v_{i}$,
the last two equalities in (\ref{63}) present the standard formulas
(\ref{3}) for the velocities $u_{1}$ and $u_{2}$. It is obvious
now that formulas (\ref{57}) define in this limit the same values
of $\varphi _{i0_t}$ as by standard formulas (\ref{5}).

Let $\tau \to \infty $. Using the limiting value (\ref{62c}) of
$\mathcal{M}$ it is easy to derive the following relations:
\begin{gather}
z_{i}\to {\overline{z}}=:\frac{c_{0}
}{\sqrt{\rho _{0}(\rho _{0}+{\overline{E}}_{1}+{\overline{E}}_{2})}},\quad
\frac{d\phi _{i0}}{dt}\to {\overline{\phi }}_{0_{t}}
=: c_{0}\sqrt{\frac{\rho _{0}+{\overline{E}}_{1}+{
\overline{E}}_{2}}{\rho _{0}}},  \label{68}
\\
V_{1}+V_{2}\to {\overline{V}}_{1}+{\overline{V}}_{2}
=:c_{0}\frac{{\overline{E}}_{1}+{\overline{E}}_{2}}{
\sqrt{\rho _{0}(\rho _{0}+{\overline{E}}_{1}+{\overline{E}}_{2})}}.
\label{69}
\end{gather}
Therefore, we obtain again the standard formulas but for an alone shock
wave. This completes the proof of Lemma \ref{lemma6}.
\end{proof}

The statement of this lemma describes qualitatively the behavior of the
system (\ref{53}), (\ref{54}) solution and justifies our main assumptions (
\ref{16}). However, actually we do not need to solve this system exactly.
Indeed, according to formulas (\ref{56}), (\ref{68}) and (\ref{69}) we can
specify
\begin{equation}
V_{i}(\tau )=v_{i}+(\overline{V}_{i}-v_{i})\zeta _{1}(\tau ),\quad \varphi
_{i1}(\tau )=\overline{\varphi }_{i1}\xi _{i}(\tau ),\quad i=1,2,
\label{681}
\end{equation}
where
\begin{equation}
\overline{V}_{i}=\overline{z}\overline{E}_{i},\quad \overline{\varphi }
_{i1}=(\overline{\phi}_{{0}_t}-\varphi _{{i0}_t})/\psi _{{0}_t},
\label{682}
\end{equation}
and $\zeta _{1}(\tau )$, $\xi _{i}(\tau )$ are smooth functions that
increase monotonically from zero to unit as $\tau $ goes from $-\infty $ to $
\infty $. Obviously, functions (\ref{681}), (\ref{682}) satisfy the
corresponding assumptions (\ref{14}), (\ref{15}) and (\ref{16}). Moreover,
the analysis drawn above imply that the differences of the left and right
hand sides of equalities (\ref{53}), (\ref{54}) become soliton-type
functions after substitution $V_{i}$ and $\varphi _{i1}$ of the form (\ref
{681}), (\ref{682}). Therefore, according to the statement of Lemma \ref
{lemma4} these differences can be supplemented to the coefficients of the $
\delta (x-x^{\ast })$ functions in relations (\ref{51}) and (\ref{52}).

So we define $V_i$ and $\varphi_{i1}$, $i=1,2$, by formulas (\ref{681}), (
\ref{682}). Consequently, we obtain the equalities
\begin{equation}  \label{683}
\Big(\frac{d\phi_{i0}}{dt}E_i-M_{i0}\Big)\Big|_*\overset{\normalfont\text{def
}}{=} G_{i1},\quad \Big(\frac{d\phi_{i0}}{dt}M_{i0}-K_{i}-c_0^2E_i\Big)\Big|
_*:= G_{i2},
\end{equation}
where the stars mean the substitution of $V_j$, $\varphi_{j1}$, $j=1,2$,
mentioned above and $G_{ik}=G_{ik}(\tau)$ are soliton-type functions. As
for the amplitudes $E_i$, they remain indeterminate on this stage.

\section{Analysis of the centered rarefaction}

Considering the centered rarefaction terms in (\ref{51}) and (\ref{52}) we
will use the statement of Lemma 2.3 again. However, we can not neglect
soliton type terms now, since the functions $U$ and $\beta$ are not bounded
uniformly in $t$ as $\varepsilon\to0$. Hence, we should take into account
some soliton terms as the coefficients of $\delta(x-x^*)$ functions.

Let us start with some algebraic transformations.

\begin{lemma} \label{lemma8}
Let functions $U$, $\beta$ and $\phi_j$, $j=3,4$ be such that
\begin{equation}  \label{70}
\frac {d R}{d t} +\frac{\partial\, M_{40}}{\partial\, x}=F_1,
\end{equation}
where $F_1$ is a soliton-type function. Then
\begin{equation}  \label{71}
\begin{aligned}
&  \frac {d M_{40}}{d t}+\frac{
\partial\, K_{4}}{\partial\, x}+c_0^2\frac{\partial\, R}{\partial\, x} \\
&  =R\Big\{\big(\frac{d U}{d t}+U^2
\big)(x-\phi_4)-U\big(\frac{d \phi_4}{d t}-V_1-V_2\big)\Big\} +c_0^2\frac{
\partial\, R}{\partial\, x} \\
&\quad  -\Big\{(E_1+E_2)\Big(\big(\frac{d U}{d t}+U^2
\big)(x-\phi_4) -U\frac{d \phi_4}{d t}\Big) \\
&\quad  +RU(V_1+V_2)\Big\}
B_{11}(\sigma_{41})+F_2=0.
\end{aligned}
\end{equation}
Here $F_2$ is the soliton-type function defined by
\begin{align*}
 F_2= & -  \Big\{U(2x-\phi_3-\phi_4)\Big(U\big(
E_1+E_2-R+2\rho_0-2\frac{\partial\, R}{\partial\, x}(x-\phi_4)\big) \\
 & -  N_2B_{11}(\sigma_{34})\Big) +U^2(x-\phi_4)(R_3+R_4-2R) +
\frac{d M_{401}}{d t}\Big\}B_{11}(\sigma_{34}) \\
 & +  F_1\{V_1+V_2+U\big((x-\phi_4)B_{11}(\sigma_{43})-(x-
\phi_3)B_{11}(\sigma_{34})\big)\},
\end{align*}
and $N_2$, $M_{40}$, $M_{401}$ are defined in formulas
\eqref{31a}, \eqref{493}.
\end{lemma}

To prove this statement it is sufficient to eliminate the term $d R/d t$
from the derivative $d M_{40}/d t$ using the equality (\ref{70}).

Let us analyze equations (\ref{70}) and (\ref{71}) beforehand. To this aim,
consider equation {\normalfont(\ref{70})} on the curve $x=\phi _{3}$.
Neglecting soliton - type functions we obtain the equation
\begin{equation}
\frac{d\phi _{30}}{dt}+c_{0}\frac{U}{\beta }-U(\phi _{3}-\phi
_{4})-(V_{1}+V_{2})B_{11}(\sigma _{14})-c_{0}\frac{U}{\beta \rho _{2}}\sum
_{i=1}^{2}E_{i}=0.  \label{72}
\end{equation}
Considering the left-hand side of equation (\ref{72}) for $\tau \to
\pm\infty $ and taking into account the geometric assumptions (\ref
{16a}), (\ref{16b}), we derive
\begin{gather}
\frac{d\varphi _{30}}{dt}+c_{0}\frac{{\overline{U}}^{\,+}}{{\overline{\beta }}
^{\,+}}-{\overline{U}}^{\,+}(\varphi _{30}-\varphi _{40})-({\overline{V}}_{1}+{
\overline{V}}_{2})=0\quad \text{as } \tau \to \infty ,
\label{73}
\\
\frac{d{\overline{\phi }}_{30}}{dt}+c_{0}\frac{\rho _{0}}{\rho _{2}}\frac{{
\overline{U}}^{\,-}}{{\overline{\beta }}^{\,-}}=0\quad \text{as } \tau
\to -\infty .  \label{74}
\end{gather}
Next, consider equation \eqref{71} on the curve $x=\phi _{4}$.
Neglecting soliton - type functions we obtain
\begin{equation}
\Big(1-\frac{E_{1}+E_{2}}{R_{4}}B_{11}(\sigma _{41})\Big)\frac{d\phi _{40}}{
dt}+c_{0}\frac{\beta }{U}-(V_{1}+V_{2})B_{11}(\sigma _{14})=0.  \label{75}
\end{equation}
Therefore,
\begin{gather}
\frac{d\varphi _{40}}{dt}+c_{0}\frac{{\overline{\beta }}^{\,+}}{{\overline{U}}
^{\,+}}-({\overline{V}}_{1}+{\overline{V}}_{2})=0\quad \text{as } \tau
\to \infty ,  \label{76}
\\
\frac{d{\overline{\phi }}_{40}}{dt}+c_{0}\frac{\rho _{2}}{\rho _{0}}\frac{{
\overline{\beta }}^{\,-}}{{\overline{U}}^{\,-}}=0\quad \text{as } \tau
\to -\infty .  \label{77}
\end{gather}
Now let us consider equations (\ref{70}) and (\ref{71}) on the curves $
x=\phi _{4}$ and $x=\phi _{3}$ respectively. Neglecting soliton - type
functions again and using the equalities (\ref{72}), (\ref{75}) we derive
the following equalitites:
\begin{equation}
\begin{gathered}
(\phi _{30}  -  \phi _{40})\Big\{\frac{d\beta }{
dt}+U\beta \big(1-2B_{11}(\sigma _{34})\big)\Big\}=0, \\
(\phi _{30}  -  \phi _{40})\Big\{\frac{dU}{dt} +U^{2}\Big\}=0.
\end{gathered}
\label{78}
\end{equation}
This implies the desired equations for the limiting functions
\begin{equation}
\frac{d{\overline{\beta }}^{\,+}}{dt}+{\overline{U}}^{\,+}{\overline{\beta }}
^{\,+}=0,\quad \frac{d{\overline{U}}^{\,+}}{dt}+{{\overline{U}}^{\,+}}^{2}=0\quad
\text{as } \tau \to \infty .  \label{79}
\end{equation}
Therefore,
\begin{equation}
{\overline{\beta }}^{\,+}={\overline{U}}^{\,+}=(t-t^{\ast })^{-1}\quad \text{as}
\quad \tau \to \infty ,  \label{80}
\end{equation}
which coincides with formulas (\ref{39}), (\ref{40}) and (\ref{46b}).
Moreover, formulas (\ref{47}) and (\ref{80}) allow to rewrite equations
(\ref{73}) and (\ref{76}) as follows:
\begin{equation}
\frac{d\varphi _{30}}{dt}=u_{2}-c_{0},\quad \frac{d\varphi _{40}}{dt}={
\overline{V}}_{1}+{\overline{V}}_{2}-c_{0}\quad \text{as } \tau
\to \infty .  \label{81}
\end{equation}
It is clear now that $\varphi _{30}$ and $\varphi _{40}$ are the standard
characteristics which go to the left in the media with the velocities
$u=u_{2}$ and $u={\overline{V}}_{1}+{\overline{V}}_{2}$ respectively.

Now let us consider the situation as $\tau\to-\infty$. Equations (\ref{74})
and (\ref{77}) imply the congruence of the curves $x={\overline\phi}_{30}$
and $x={\overline\phi}_{40}$ if and only if
\begin{equation*}
\Big(\frac{{\overline U}^{\,-}}{{\overline\beta}^{\,-}}\Big)^2=\Big(\frac{\rho_2}{
\rho_0}\Big)^2.
\end{equation*}
Extracting the root and taking into account geometric assumption (\ref{16a})
we obtain the condition
\begin{equation}  \label{82}
\frac{U}{\beta}\to\frac{\rho_2}{\rho_0} \quad \text{as } \tau\to-\infty.
\end{equation}
At the same time, equations (\ref{78}) do not prescribe anything as $
\tau\to-\infty$. Therefore, we can try to define the functions $\beta$ and $
U $ in an arbitrary manner preserving the properties (\ref{80}) and (\ref{82}
) to within $\mathcal{O}(\varepsilon)$. To this aim we use the
representation (\ref{39}) and (\ref{46b}) setting
\begin{equation}  \label{84}
g_1=(1+\rho_2)\sqrt{\tau^2+\gamma_1^2}\,\zeta_3(-\tau),\quad g_2=(1+\rho_0)
\sqrt{\tau^2+\gamma_2^2}\,\zeta_4(-\tau),
\end{equation}
where $\gamma_i>0$ are arbitrary constants, $\zeta_i(\tau)>0$ are smooth
functions such that $\zeta_i(\tau)\to0$ as $\tau\to-\infty$ and $
\zeta_i(\tau)\to1$ as $\tau\to\infty$, $i=3,4$. Moreover, let
\begin{equation}  \label{83}
\zeta_3(0)=(1+\rho_2)^{-1},\quad \zeta_4(0)=(1+\rho_0)^{-1}.
\end{equation}
It is easy to check that this choice of $g_i$ guarantees the fulfillment of
both the inequalities $\beta^{-1}(\tau)\geqslant \mbox{const}>0$ and
$U^{-1}(\tau)\geqslant \mbox{const}>0$ for any $\tau\in\mathbb{R}^1$, and the
limiting relations (\ref{80}), (\ref{82}).

This representation of $\beta$, $U$ and formulas (\ref{74}), (\ref{77}) have
as consequence the equations
\begin{equation}  \label{85}
\frac{d{\overline\phi}_{30}}{d t}=\frac{d{\overline\phi}_{40}}{d t}=-c_0
\quad \text{as } \tau\to-\infty.
\end{equation}
Obviously we obtain that ${\overline\phi}_{i0}$, $i=3,4$, are the standard
characteristics which go to the left in the media with the velocity $u=0$.

Completing the preliminary analysis we define the functions $\phi _{i0}$, $
i=3,4$, by formulas (\ref{12}), where $\varphi _{i0}$ are determined in (\ref
{81}) and
\begin{equation}
\varphi _{31}=-\frac{u_{2}}{\psi _{{1}_t}}\xi _{3}(-\tau ),\quad \varphi
_{41}=-\frac{{\overline{V}}_{1}+{\overline{V}}_{2}}{\psi _{{1}_t}}\xi
_{4}(-\tau ).  \label{86}
\end{equation}
Here $\xi _{i}(\tau )$, $i=3,4$, are smooth functions which increase from
zero to unit.

Now we need to justify the choice (\ref{81}), (\ref{84}) and (\ref{86}). To
this aim let us substitute these formulas into the left-hand side of
equality (\ref{70}) and denote the result of substitution by $F_1$.

\begin{lemma} \label{lemma9}
Under the choice \eqref{81}, \eqref{84} and \eqref{86}
 the following relation holds:
\begin{equation}  \label{861}
F_1(H_4-H_3)=F^*_1\delta(x-x^*)+\mathcal{O}_{\mathcal{D}'}(
\varepsilon),
\end{equation}
where $F^*_1=F^*_1(\tau,\varepsilon)$ is a function from the Schwarz space.
\end{lemma}

\begin{proof}  Obviously,
\begin{equation*}
F_1(H_4-H_3)=f_1(H_4-H_3)+\mathcal{O}_{\mathcal{D}'}(\varepsilon),
\,\, f_1=F_1\big(\omega\big(\frac{x-\phi_4}{\varepsilon}\big)-\omega\big(
\frac{x-\phi_3}{\varepsilon}\big)\big).
\end{equation*}
Next, using the representation (\ref{11}) of $R$ we rewrite $F_1$ in the
form
\begin{equation}  \label{87}
\begin{aligned}
  F_1= & -  \frac{R}{c_0}\Big\{\frac{d }{d t}\big(
\beta(x-\phi_3)\big) +\beta(V_1+V_2)B_{11}(\sigma_{14})+\beta U(x-\phi_4)
\Big\} \\
  & +  RU-U(E_1+E_2)B_{11}(\sigma_{41}) \\
  & +  U\Big\{R(2x-\phi_3-\phi_4)\frac{\beta}{c_0}
+R_3+R_4-2R\Big\}B_{11}(\sigma_{34}).
\end{aligned}
\end{equation}
Let us consider $F_1$ for $\tau >>1$. Since the convolutions
$B_{11}(\sigma_{41})$, $B_{11}(\sigma_{34})$ vanish we derive the relation
for $\tau\to\infty$,
\begin{equation*}
F_1=-R\frac{{\overline\beta}^{\,+}}{c_0} \Big\{\frac{(x-\varphi_{30})}{{
\overline\beta}^{\,+}} \frac{d{\overline\beta}^{\,+}}{dt}-\frac{d\varphi_{30}}{dt} +{
\overline U}^{\,+}(x-\varphi_{40})+{\overline V}_1+{\overline V}_2-c_0\Big\}
+\dots ,
\end{equation*}
where the dots mean  vanishing terms and terms of value $\mathcal{O}
(\varepsilon)$. Since $\beta\to U\to (t-t^*)^{-1}$ as $\tau\to\infty$, it is
clear now that formulas (\ref{81}) and (\ref{84}) imply the vanishing of $
F_1 $ as $\tau\to\infty$.

Next, considering $F_1$ for $\tau\ll -1$ we rewrite the equality (\ref{87}) as
follows:
\begin{equation*}
F_1=-(x-\phi_3)\frac{R}{c_0}\frac{d\beta}{d t} +(R-\rho_2)\frac{\beta}{c_0}
\frac{d\phi_{30}}{d t}+F_{11},
\end{equation*}
where
\begin{align*}
 F_{11}=&\beta\Big(\frac{\rho_2}{c_0}
\frac{d\phi_3}{dt}+\rho_0\frac{U}{\beta}\Big) +\beta U(\phi_{40}-\phi_{30})
\frac{R}{2c_0} \\
& +\frac12U\big(\rho_2+R_4-2(
\rho_0+E_1+E_2)\big) +\dots,
\end{align*}
and the dots denote small terms as $\tau\to-\infty$.
Under the choice (\ref{84}) and (\ref{86}) the function $F_{11}$ vanishes.
Furthermore,
\begin{equation*}
|x-\phi_3|\big(\omega\big(\frac{x-\phi_4}{\varepsilon}\big)-\omega\big(\frac{
x-\phi_3}{\varepsilon}\big)\big) \leqslant |\phi_{40}-\phi_{30}|\to0 \quad
\text{as }\tau\to-\infty.
\end{equation*}
Estimating $R-\rho_2$ similarly we obtain that $f_1\to0$ as $\tau\to -\infty$
. Thus, we can apply the statement of Lemma \ref{lemma4} and obtain relation
(\ref{861}), where $F_1^*=\varepsilon(\chi_4-\chi_3)f_1|_{x=x^*}$.

It remains to estimate $F_1^*$ for $|\tau|\leqslant \mbox{const}$.
Obviously, to this aim it is enough to estimate the maximum values of $\beta$
and $(x^*-\phi_3)d\beta/d t$. Formulas (\ref{39}), (\ref{84}) and (\ref{83})
imply the inequalities
\begin{equation*}
\varepsilon\beta\leqslant\frac{\psi_{0_t}}{\gamma_1},\quad \varepsilon\Big|
(x^*-\phi_3)\frac{d\beta}{d t}\Big| =|\varepsilon^2\beta^2(1+g'_{1
\tau})\chi_3|\leqslant\mbox{const}.
\end{equation*}
This completes the proof of Lemma \ref{lemma9}.
\end{proof}

Considering in the same way the coefficients of the Heaviside functions in
relation (\ref{52}) we obtain the similar to Lemma \ref{lemma9} statement

\begin{lemma} \label{lemma10}
 Under the choice \eqref{81}, \eqref{84} and
\eqref{86} the following relation holds:
\begin{equation}  \label{88}
\Big\{\frac {d M_{40}}{d t}+\frac{\partial\, K_{4}}{\partial\, x}+c_0^2\frac{
\partial\, R}{\partial\, x}\Big\}(H_4-H_3) =F_3\delta(x-x^*)+\mathcal{O}_{
\mathcal{D}'}(\varepsilon),
\end{equation}
where $F_3$ is a function from the Schwarz space.
\end{lemma}

Therefore, we define the functions $\beta$, $U$, and $\phi_{j0}$, $j=3,4$,
in accordance with formulas (\ref{81}), (\ref{84}), (\ref{86}) and
supplement the corresponding corrections to the coefficients of $
\delta(x-x^*)$ functions in relations (\ref{51}), (\ref{52}). We stress that
the limiting values ${\overline V}_i$ remain undefined and conditions (\ref
{83}) remain unique ones for the functions $\zeta_3$ and $\zeta_4$.

\section{Completion of the construction}

Now we can complete the construction of the shock waves and the centered
rarefaction defining the functions $E_i$, $\xi_i$, and $\zeta_j$. To this
aim let us consider the continuity conditions (\ref{47}). For the exact
solutions $V_i=V_i(E_1,E_2,\tau)$ of the Rankine--Hugoniot conditions (\ref
{53}), (\ref{54}) (see formulas (\ref{56}), (\ref{62})) system (\ref{47})
implies the appearance of a transcendental algebraic equation for $E_i$.
However, we can simplify the procedure using the arbitrariness in
representation of the functions $E_i$, $V_i$, $g_i$ and $\phi_j$. Indeed,
taking into account the exact formulas (\ref{11}), (\ref{13}) for $R_4$ we
pass to the algebraic equations
\begin{equation}  \label{89}
\begin{gathered}
 \rho_0+E_1+E_2=\rho_2 e^{-\frac{\beta}{c_0}
(\phi_4-\phi_3)}, \\
  V_1+V_2=u_2+U(\phi_{4}-\phi_3).
\end{gathered}
\end{equation}
Let us eliminate the term $\phi_{4}-\phi_3$ from this system. Then we obtain
the equation
\begin{equation}  \label{90}
\rho_0+E_1+E_2 =\rho_2 e^{-\frac{\beta}{c_0U}(V_1+V_2-u_2)}.
\end{equation}

\begin{lemma} \label{lemma11}
Under the choice \eqref{681}, \eqref{81}, \eqref{84} and
\eqref{86} there exist monotonic functions $E_i$,
$i=1,2$, which satisfy equation \eqref{90} and
\begin{equation}  \label{91}
E_i\to e_i\quad\text{as }\tau\to-\infty,\quad
E_1+E_2\to\rho_0(M^2_0-1)\quad\text{as }\tau\to\infty.
\end{equation}
Here the Mach number $M_0$ is the solution of the equation
\begin{equation}  \label{92}
\rho_0 M^2=\rho_2 e^{-\big\{ M-\frac 1{M}-\frac{u_2}{c_0}\big\}}
\end{equation}
and satisfies the inequalities
\begin{equation}  \label{921}
\sqrt{\frac {\rho_1}{\rho_0}}<M_0<\sqrt{\frac {\rho_2}{\rho_0}}.
\end{equation}
\end{lemma}

\begin{proof}
 First of all consider equation (\ref{90}) in the limit as $\tau\to
\infty$. Representation (\ref{681}) of $V_i$ and formula (\ref{69}) imply
the dependence of the right-hand side of equation (\ref{90}) on ${\overline E
}_1+{\overline E}_2$. Let us use the standard notation $M_0$ for the Mach
number calculated before the shock wave front. In the case under
consideration
\begin{equation*}
M_0=\frac{\sqrt{\rho_0+\overline{E}_1+\overline{E}_2}}{\sqrt{\rho_0}}.
\end{equation*}
Since ${\overline V}_1+{\overline V}_2=c_0(M_0+1/M_0)$ and $\beta/U\to1$ as $
\tau\to\infty$, we see that equation (\ref{90}) takes the form (\ref{92}) in
the limit. The left-hand side of this equation is the monotonically
increasing function whereas the right-hand side monotonically decreases with
respect to $M\geqslant1$. This implies the existence of a unique root $
M=M_0>1$ of the equation since $\rho_2>\rho_0$ and $u_2>0$. Respectively, we
derive the desired limiting amplitudes
\begin{equation}  \label{922}
{\overline E}_1+{\overline E}_2=\rho_0(M_0^2-1).
\end{equation}
Note also that inequality $M_0>1$ is the stability condition for shock waves.

Next, let us set $M=M^*:=\sqrt{\rho_1/\rho_0}
$. Direct calculations and the exact formula (\ref{3}) of $u_2$ show that
\begin{equation*}
M^*-\frac1{M^*}-\frac{u_2}{c_0}=-\frac{e_2}{\sqrt{\rho_1\rho_2}}.
\end{equation*}
Thus, the right-hand side of  (\ref{92}) is greater than $\rho_2$
whereas $\rho_0{M^*}^2=\rho_1<\rho_2$. Therefore, $M^*<M_0$ and we derive
the first inequality in (\ref{921}). To prove the second inequality in (\ref
{921}) consider the number $M_*>0$ such that
\begin{equation*}
M_*-\frac{1}{M_*}=\frac{u_2}{c_0}.
\end{equation*}
Solving this equation and using formula (\ref{3}) again we deduce $M_*$ as a
function with respect to $\rho_0$, $\rho_1$, and $\rho_2$. Next, simple
algebra shows that $\rho_0M_*^2<\rho_2$. Thus, $M_0>M_*$. Therefore,
\begin{equation}  \label{923}
{\overline V}_1+{\overline V}_2=c_0(M_0-\frac{1}{M_0})>u_2.
\end{equation}
This implies the desired inequality
\begin{equation}  \label{924}
\rho_0M_0^2=\rho_0+{\overline E}_1+{\overline E}_2<\rho_2.
\end{equation}
We stress that inequalities (\ref{923}) and (\ref{924}) guarantee the
stability of the limiting rarefaction wave.

Furthermore, in accordance with (\ref{14}) we have
\begin{equation*}
\rho_0+E_1+E_2\to\rho_0+e_1+e_2=\rho_2,\quad V_1+V_2\to v_1+v_2=u_2\quad
\text{as }\tau\to-\infty.
\end{equation*}
This implies the consistency of equation (\ref{90}) and assumptions
(\ref{14}).

Now we can use the arbitrariness of the functions $E_i$, $V_i$, $g_i$
and $\phi_j$. We set
\begin{equation}  \label{925}
\gamma_1=\theta\gamma_2, \quad \theta>1
\end{equation}
and fix a monotonic function $\zeta_4$. Then there exists a function $
\zeta_3 $ such that the ratio $\beta/U$ monotonically increases from $
\rho_0/\rho_2$ to 1. Let us fix a monotonic function $\zeta_1$ in
representation (\ref{681}) of $V_i$ and define the amplitudes $E_i$ as
follows
\begin{equation}  \label{926}
E_i=e_i+({\overline E}_i-e_i)\zeta_2(\tau), \quad i=1,2,
\end{equation}
where $0<{\overline E}_i<e_i$ are numbers such that equality (\ref{922})
holds. Then formulas (\ref{922}), (\ref{923}) and equation (\ref{90}) allow
to define the function $\zeta_2$, namely
\begin{equation}  \label{93}
\Big(1-\frac{\rho_0}{\rho_2}M_0^2\Big)\zeta_2(\tau)
 =1-\exp\Big(-\frac{\beta}{U}
\big(\frac{M_0^2-1}{M_0}-\frac{u_2}{c_0}\big)\zeta_1(\tau)\Big).
\end{equation}
It is easy to check that $\zeta_2$ is the smooth function which increases
monotonically from zero to unit. This completes the proof of Lemma
\ref{lemma11}.
\end{proof}

Furthermore, the second equation in (\ref{89}) allows to define the
difference $\phi_4-\phi_3$. Indeed, this equation and
(\ref{681}), (\ref{923}) imply the equality
\begin{equation}  \label{94}
\phi_{4}-\phi_3=U^{-1}(V_1+V_2-u_2) =U^{-1}\big(c_0(M_0-\frac{1}{M_0})-u_2
\big)\zeta_1(\tau).
\end{equation}
Thus, the difference $\phi_4-\phi_3$ is positive and tends to zero as $
\tau\to-\infty$. Moreover, in accordance with (\ref{12}), (\ref{81}), (\ref
{86}) we have
\begin{equation}  \label{95}
\phi_{4}-\phi_3=\big(({\overline V}_1+{\overline V}_2)(1-\xi_4(-
\tau))-u_2(1-\xi_3(-\tau))\big)(t-t^*)
+\varepsilon(\varphi_{42}-\varphi_{32}).
\end{equation}
The function $U^{-1}$ varies rapidly in a small neighborhood of the point
$t=t^*$ whereas $U^{-1}\sim t-t^*$ and $U^{-1}\sim \rho_0|t-t^*|$ for
$t-t^*\sim \pm\varepsilon^{1-\gamma}$, $\gamma>0$, respectively. It
is clear now that we can set $\varphi_{32}=-\varphi_{42}$ and choose
$\varphi_{42}$ as a soliton--type function. The equality
\begin{equation}  \label{96}
(\phi_{4}-\phi_3)|_{t=t^*}=\varepsilon\gamma_2\big(c_0(M_0
-\frac{1}{M_0})-u_2
\big)\zeta_1(0)
\end{equation}
shows that the amplitude of $\varphi_{42}$ is positive and proportional to
$\gamma_2$.

The last step of the construction is consideration of the coefficients of
$\delta(x-x^*)$ functions in the relations (\ref{51}) and (\ref{52}).
Taking into account the statements of Lemmas \ref{lemma9}, \ref{lemma10} and
formulas (\ref{683}) we obtain the equations
\begin{equation}  \label{98}
\begin{gathered}
  \psi_{0_t}(\sum_{i=1}^2E_{i}\varphi_{i2})'+L_1+M_*+F^*_1
+\sum_{i=1}^2G_{i1}=0, \\
  L_5+K_*+F_3+\sum_{i=1}^2G_{i2}=0.
\end{gathered}
\end{equation}
Setting $\varphi_{12}=\varphi_{22}$, $W_{1}=W_{2}$ and using formulas
(\ref{42b}), (\ref{496}) we rewrite equations (\ref{98})
in the standard form
\begin{equation}  \label{99}
\sum_{i=1}^2E_i\frac{d\varphi_{12}}{d\tau}=f_1(\tau,\varphi_{12}),\quad 
\frac{d(aW_{1})}{d\tau}=f_2(\tau),
\end{equation}
where
\begin{equation}  \label{100}
\begin{aligned}
  a=&2\rho_0+\frac12\sum_{i=1}^2E_i\big(
1+2B_{11}(\sigma_{i\bar i})\big) \\
&  +(R_3-R_*)\sum_{i=1}^2B_{11}(
\sigma_{3i})+(R_*-R_4)\sum_{i=1}^2B_{11}(\sigma_{4i}),
\end{aligned}
\end{equation}
and $f_i$ are soliton-type functions.

\begin{lemma} \label{lemma12}
There exist such numbers $\gamma_2$, ${\overline E}_2$ and
functions $\zeta_1$, $\xi_i$, $i=1,2,3$, that equations
\eqref{99} have uniformly bounded in $\tau$ solutions.
\end{lemma}

\begin{proof} The solvability of the first equation (\ref{99}) is
obvious since $
E_1+E_2\geqslant\mbox{const}>0$ uniformly in $\tau$. We specify
$\varphi_{12} $ by the condition $\varphi_{12}|_{\tau=0}=0$.
 Note that the
choice $\varphi_{12}=\varphi_{22}$ implies the independence $\sigma_{21}$ of
this correction and allows to write out the difference $\phi_1-\phi_2$ in
the form
\begin{equation}  \label{100a}
\phi_1-\phi_2=\big\{\psi_{0_t}(1-\xi_2) +({\overline \phi}
_{0_t}-\varphi_{10_t})(\xi_2-\xi_1)\big\}(t^*-t).
\end{equation}
Let $\xi_2(\tau)\geqslant\xi_1(\tau)$. Then $(t^*-t)(\phi_1-\phi_2)>0$
accordingly to (\ref{100a}) and the first inequality in (\ref{921}).

To prove the inequality $a\geqslant\mbox{const}>0$ let us denote $\tau_i^*$,
$i=1,2$, the time instants of the intersections of the line $x=x^*$ and the
curves $x=\phi_3$ and $x=\phi_4$ respectively. Using the explicit formula (
\ref{11}) of $R$ it is easy to check that the inequalities $R_3\geqslant
R_*\geqslant R_4$ hold for $\tau\in[\tau_1^*,\tau_2^*]$, where $
R_*=R|_{x=x^*}$. Thus, all terms in (\ref{100}) are positive for such $\tau$.

Furthermore, let us note the fulfillment of the relations
\begin{equation*}
\phi_3|_{\tau=0}=x^*-\varepsilon\varphi_{42}(0)<x^*<x^*+\varepsilon
\varphi_{42}(0)=\phi_4|_{\tau=0}.
\end{equation*}
Hence, $\phi_3|_{\tau=0}\to-0$ and $\phi_4|_{\tau=0}\to+0$ as $\gamma_2\to0$
, whereas $\phi_1|_{\tau=0}=x^*$ and this position does not depend on $
\gamma_2$. Therefore, there exist such number $\gamma_2$ and functions $
\xi_1 $, $\xi_3$ that the inequalities
\begin{equation}  \label{100b}
\phi_1|_{\tau=\tau_1^*}\leqslant x^*, \quad
\phi_1|_{\tau=\tau_2^*}\geqslant x^*
\end{equation}
hold for any fixed $\zeta_1$. Then the formulas for $\phi_i$, $i=1,\dots,4$
and (\ref{100a}) imply the fulfillment of more precise version of geometric conditions (
\ref{16a}) and (\ref{16b})
\begin{equation}  \label{100c}
\begin{gathered}
  \phi_2<\phi_1<\phi_3<\phi_4 \quad\text{for }\tau_1^*>\tau>-T, \\
  \phi_3<\phi_4<\phi_1<\phi_2 \quad\text{for }\tau_2^*<\tau<T,
\end{gathered}
\end{equation}
where $T>\max\{-\tau_1^*,\tau_2^*\}$ is a constant and $\phi_2-\phi_1\to+0$,
$\phi_4-\phi_3\to+0$ as $\tau\to\pm\infty$ respectively.

Let $\tau\in(\tau_2^*,T)$. Then $R_3>R_4> R_*$. Simple transformations and
the first equality (\ref{47}) allow to rewrite formula (\ref{100}) as
follows:
\begin{equation}  \label{101}
\begin{aligned}
  a=&\rho_0\big(\frac12+B_{11}(\sigma_{21})\big)
+(R_3-R_*)\sum_{i=1}^2B_{11}(\sigma_{3i})
+R_*\sum_{i=1}^2B_{11}(\sigma_{4i}) \\
 &+E_2\big(1-2B_{11}(\sigma_{12})\big)
+R_4\big(\frac12+B_{11}(\sigma_{12})-\sum_{i=1}^2B_{11}(\sigma_{4i})\big).
\end{aligned}
\end{equation}
The second inequalities in (\ref{100c}) imply that $B_{11}(\sigma_{12})<1/2$
and $B_{11}(\sigma_{41})<1/2$. Moreover, $B_{11}(\sigma_{12})>B_{11}(
\sigma_{42})$ since $\sigma_{42}=\sigma_{41}+\sigma_{12}$. Hence,
$a\geqslant \mbox{const}>0$ for such $\tau$. For $\tau\geqslant T$ and
sufficiently
large $T$ we obtain the desired estimate $a\sim\rho_0+R_4>0$ since
$B_{11}(\sigma_{12})\sim B_{11}(\sigma_{21})\sim1/2$ and
$B_{11}(\sigma_{ji})\sim 0$, $j=3,4$, $i=1,2$.

Let $\tau\in(-T,\tau_1^*)$. Then $R_*> R_3>R_4$. Using the first equality (
\ref{47}) again, we rewrite formula (\ref{100}) in the form
\begin{align*}
  a=&\rho_0\big(\frac12+B_{11}(\sigma_{21})\big)
+R_*\sum_{i=1}^2\big(B_{11}(\sigma_{i3})-B_{11}(\sigma_{i4})\big) \\
&+R_4\big(B_{11}(\sigma_{12})+
\sum_{i=1}^2B_{11}(\sigma_{i4})-\frac12\big) \\
&+\rho_2\big(1-\sum_{i=1}^2B_{11}(
\sigma_{i3})\big)-E_2\big(1-2B_{11}(\sigma_{21})\big).
\end{align*}
The location (\ref{100c}) of the phases $\phi_i$ implies the inequalities
$B_{11}(\sigma_{i3})>B_{11}(\sigma_{i4})$ and $B_{11}(\sigma_{12})>1/2$.
Thus,
\begin{equation*}
a\geqslant\frac12\rho_0 +\rho_2\big(1-2B_{11}(\sigma_{13})\big) -E_2\big(
1-2B_{11}(\sigma_{21})\big).
\end{equation*}
Since $\rho_2>E_2(\tau)>0$ uniformly in $\tau$, the last estimate can be
transformed to the following form:
\begin{equation*}
a\geqslant\frac12\rho_0 +2E_2\big(B_{11}(\sigma_{21})-B_{11}(\sigma_{13})
\big).
\end{equation*}
Now we use the remained arbitrariness in the choice of ${\overline E}_2$ and
$\zeta_1$. Accordingly to (\ref{93}), $\zeta_2(\tau_1^*)$ will be an
arbitrarily small number if $\zeta_1(\tau_1^*)\ll 1$. Therefore, we can choose
${\overline E}_2$ and $\zeta_1(\tau_1^*)$ such that the inequality
\begin{equation}  \label{104}
E_2\big|B_{11}(\sigma_{21})-B_{11}(\sigma_{13})\big|\Big|_{\tau=\tau_1^*}
\leqslant\frac18\rho_0
\end{equation}
holds. The trajectories $\phi_1$ and $\phi_2$ of the shock waves go to the
right and have different velocities, whereas the characteristic $\phi_3$
goes to the left. So the distances $\phi_1-\phi_2$ and $\phi_3-\phi_1$
increase with time when $\tau$ decreases from $\tau_1^*$ to $-T$. Therefore,
the convolutions $B_{11}(\sigma_{21})$ and $B_{11}(\sigma_{13})$ vanish. The
function $E_2$ increases for such $\tau$. However, we can specify $
\zeta_1(\tau)$ such that the rate of increasing of $E_2$ will be smaller as
the rate of decreasing of $B_{11}(\sigma_{21})$ and $B_{11}(\sigma_{13})$.
Obviously, the estimate similar to (\ref{104}) holds for any $
\tau\in(-T,\tau_1^*)$ under this choice. Consequently, we obtain the desired
estimate $a\geqslant\mbox{const}>0$. Evidently, $a\geqslant\rho_0/2$ for $
\tau\leqslant-T$ and sufficiently large $T$. It remains to specify $W_1$ by
the condition $W_1\to 0$ as $\tau\to-\infty$. This completes the proof of
Lemma \ref{lemma12} and Theorem \ref{theorem1}.
\end{proof}

\subsection*{Conclusion}

Let us indicate a way of generalization of the presented above construction
for strictly hyperbolic systems of conservation laws. Let, for simplicity,
the initial data be a superposition of two stable shock waves with constant
amplitudes. Obviously, to construct a uniform in time asymptotic solution we
need to involve into anzatz regularizations of all possible shock waves,
contact discontinuities and centered rarefaction. More or less simple
algebra allows to represent the nonlinearity again as a linear combination
of these regularizations (see, for example, Subsection 2.2). Therefore, to
determine parameters of the anzatz we derive a dynamical system with
scattering--type initial data. The proof of the existence of a special
solution of this problem is the main obstacle to the construction. For
example, the direct substitution of the anzatz (\ref{10}) into the gas
dynamics equations does not allow to understand for problem (\ref{1}), (\ref
{2}) almost anything. Many simplifications based on the statement of Lemma
\ref{lemma4} allow to transform the corresponding dynamical system to more
reasonable form. However, the resulting system remains very complicated.
Another example is the interaction of shock waves with opposite directions
of motion \cite{g1}. We can prove there the solvability of the corresponding
scattering--type problem for a special case of initial amplitudes, whereas
it remains unclear till now, how to prove the same in the general case. It
is clear that the similar problem for general systems of conservation laws
will be an insuperable obstacle. So the approach proposed in the present
paper can be the main tool to avoid this difficulty. Indeed, changing the
desired solution of the dynamical system to appropriate approximation, we
pass to a simpler existence problem for small corrections. We guess that
this problem can be solved in the general case similarly to the example
considered above.

As specifically for rarefaction wave in the problem (\ref{1}), (\ref{2}), we
observe a soliton--type deformation of the trajectories close to the
interaction point as the most unexpected result. This phenomenon shows that
the mechanism of the rarefaction wave appearance is realized via formation
of a regularization of a step--type function with negative amplitude a
little bit before the shock waves pasting together.

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\end{document}