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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 39, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/39\hfil Existence of traveling waves]
{Existence of traveling waves for diffusive-dispersive conservation laws}

\author[C. I. Kondo, A. F. Rossini \hfil EJDE-2013/39\hfilneg]
{Cezar I. Kondo, Alex F. Rossini}  % in alphabetical order

\address{Cezar I. Kondo \newline
Federal University of Sao Carlos, Department of Mathematics,
 P. O. Box 676 13565-905, Sao Carlos - SP, Brazil}
\email{dcik@dm.ufscar.br}

\address{Alex F. Rossini  \newline
Federal University of Sao Carlos, Department of Mathematics,
 P. O. Box 676 13565-905, Sao Carlos - SP, Brazil}
\email{alexrossini@dm.ufscar.br}

\thanks{Submitted October 1, 2012. Published February 1, 2013.}
\subjclass[2000]{35L65, 76N10}
\keywords{Scalar conservation law; diffusive-dispersive; weak solution;
\hfill\break\indent  traveling wave;  phase portrait}

\begin{abstract}
 In this work we show the existence existence and uniqueness of
 traveling waves for diffusive-dispersive conservation laws with
 flux function in  $C^{1}(\mathbb{R})$, by using phase plane analysis.
 Also we estimate the domain of attraction of the equilibrium point
 attractor corresponding to the right-hand state.
 The equilibrium point corresponding to the left-hand state is a
 saddle point.  According to the phase portrait close to  the saddle point,
 there are exactly two semi-orbits of the system. We establish that
 only one semi-orbit come in the domain of attraction
 and converges to  $(u_{-},0)$ as $y\to -\infty$.
 This provides the desired saddle-attractor connection.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assertion}[theorem]{Assertion}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}
 In this article, we investigate the existence and uniqueness of
traveling wave solutions, which are smooth functions of the form
$$
u(x,t)=s(x-ct)
$$
where $c$ is a constant,
for the partial differential equation
\begin{equation}\label{EDP}
u_{t}+f(u)_x=(a(u)u_x)_{x}+(b(u)u_x)_{xx}\quad  x\in\mathbb{R},t>0,
\end{equation}
where the diffusion function $a:\mathbb{R} \to \mathbb{R}$
and dispersion function $b:\mathbb{R} \to \mathbb{R}$ are given smooth functions.
Furthermore, we assume $a(u)\,,b(u)>0$ for $u\in \mathbb{R}$ and the flux
function $f:\mathbb{R} \to \mathbb{R}$ is continuously differentiable.

 For the case $f(u)=u^2/2$, where $a$ and $b$ real constants with $ab\neq 0$,
equation \eqref{EDP} reduces to the Korteweg-de Vries-Burgers equation
\begin{equation}\label{KdVB}
u_{t}+uu_x=au_{xx}+bu_{xxx}.
\end{equation}
It is usually considered as a combination of the Burgers equation and KdV
equation since in the limit $b\to 0 $ the equation reduces to the
Burgers equation
\begin{equation}\label{Burgers}
u_{t}+uu_x=au_{xx}
\end{equation}
which is named after its use by Burgers \cite{B} for studying the turbulence,
and if the limit $a\to 0 $ is taken, then the equation becomes the KdV equation
\begin{equation}\label{KdV}
u_{t}+uu_x=bu_{xxx}
\end{equation}
which was first suggested by Korteweg and de Vries \cite{KdV}, who used it
as a nonlinear model to study the change of forms of long waves advancing
in a rectangular channel.

The Korteweg-de Vries-Burgers equation \eqref{KdVB} is the simplest form
of the wave equation in which the non-linearity
$(uu_x)$, the dispersion $u_{xxx}$ and the dissipation $u_{xx}$ occur.

 The existence of traveling waves with linear diffusion and dispersion
 were studied by Bona and Schonbek \cite{Schonbek}; and Jacobs, McKinney
and Shearer \cite{DWM}.
In \cite{Schonbek} was  interested in the limiting behaviour of these waves
when the coefficients $a,b$ tends to zero while the ratio
$\frac{b}{a^2}$ remains bounded. In 1993, Jacobs, McKinney, and Shearer
\cite{DWM} rigorously characterized all week solutions profiles of the
single conservation law $u_t + (u^3)_x=0$.

 Bedjaoui and  LeFloch \cite{NLI}-\cite{NL} and
 Thanh \cite{Th}  studied equations of the type
\begin{equation}\label{MG}
u_{t}+f(u)_x=(R(u,\beta u_x))_{x}+\gamma(c_1(u)(c_2(u)u_x)_{x})_{x}.
\end{equation}
In \cite{NL} the authors considered
$$
R(u,v)=b(u,v)|v|^{p}\quad \text{for } p>0.
$$
 Thanh \cite{Th} studied the case
when $R=R(u,v)$ satisfies
$$
R_{v}(u,0)=R_{u}(u,0)=0,\quad R(u,v)v>0,\quad\forall v\neq 0,\;\forall\,\,u.
$$
This assumption is not satisfied in our case.  Bedjaoui and
LeFloch  \cite{NLI} studied the case
$R(u,v)=b(u,v)v$ with $b(u,v)=a(u)$, where $f : \mathbb{R} \to \mathbb{R} $
is a smooth mapping satisfying
\begin{equation}
uf''(u)>0\quad\text{for all }u\neq 0,\quad \lim_{\pm \infty}f'=+\infty. \label{ast}
\end{equation}
The associated non-linear system \eqref{MG} admits exactly three equilibrium
points for a certain speed interval that
depends the kinetic function (see \cite{LeFLoch}),
thanks the hypothesis \eqref{ast}
on the flux-function $f:\mathbb{R}\to\mathbb{R}$.
For such equilibrium points, the existence and uniqueness of
traveling waves were investigated when a certain speed wave is fixed
in this interval;
 however the connected points do not satisfy the Oleinik entropy criterion,
see \cite[Theorem 3.3]{NLI}. Observe that the work \cite{NLI} establishes
the existence of traveling waves associated with
nonclassical shocks. When the equilibrium point satisfies such criterion
this implies the existence of the trajectories between them
for each speed wave $c$ in a certain interval, see \cite[Theorem 5.1]{NLI}.

In this article, we are interested in traveling waves associated
 with a classical shock (see Definition \ref{shock}).
Given two states $u_-$ and $u_+$, both of them arbitrary constants,
we investigate the existence and uniqueness of traveling waves
when the speed wave is given by $c=\frac{f(u_+)-f(u_-)}{u_{+}-u_{-}}$, under
only the hypothesis that $f$ is smooth. In this context,
the associated system has at least two equilibrium points $(u_{-},0)$
and $(u_{+},0)$ satisfying the Oleinik entropy criterion.
This differs from \cite[Theorem 3.3]{NLI}.

 In our main result (Theorem \ref{main}) ensures that the points $(u_\pm ,0)$
can be connected by doing an analysis of the phase portrait of the
saddle point $(u_{-},0)$.
We also estimate of the domain attraction of the node $(u_+,0)$.
This technique is different the one in \cite[Theorem 5.1]{NLI},
it does not use of the existence of a kinetic function.

 In the case of traveling waves our manuscript can be used to establish
existence and uniqueness of traveling waves when the flux-function
associated is  $f(u)=u^2/2$ and  when the diffusive-dispersive
coefficients are positive constants. Moreover, we can adapt our
work to establish existence and uniqueness of traveling waves for the equation
\begin{equation}\label{edp1}
u_{t}+f(u)_x=au_{xx}+bu_{xxt}
\end{equation}
 where $a>0$ and $b\neq 0$, see in Appendix II.

 An outline of this article is as follows.
In Section 2,  we recall the concept of traveling wave solution
connecting the states $u\pm$ together with the concept of week
solution. We close the section with the method of linearization
for differential equations.
In section 3, we begin by recalling  well-known concepts and results.
Also, the stability of equilibrium point of the associated differential
equation is established. After declaring an invariance theorem we establish
a result about traveling waves, which is essential to the existence of
trajectories connecting the states $u\pm$.
In section 4, an estimate of domain of attraction is provided.
In Section 5, the analysis of the phase portrait close to the saddle
point shows that there are exactly two semi-orbits of system.
We establish that only one semi-orbit enters  the  attraction domain
of the attracting equilibrium point $(u_{+},0)$, and it converges to
 $(u_{-},0)$ as $y\to -\infty$. This gives the desired
saddle-attractor connection.


\section{traveling waves: a weak solution}

We consider partial differential equation
\begin{equation}\label{edp}
u_{t}+f(u)_x=(a(u)u_x)_{x}+(b(u)u_x)_{xx}
\end{equation}
where  $f:\mathbb{R} \to \mathbb{R}$ is the flux function, the functions
$a=a(u)>0$, $b=b(u)>0$ are  $C^{1}(\mathbb{R})$ and  $C^{2}(\mathbb{R})$
respectively. We seek the existence of traveling wave solution
$u(x,t)=s(x-ct)$, for some constant speed $c\in \mathbb{R}$,
satisfying the following conditions at infinity:
\begin{equation}\label{infinito}
\lim_{y\to \pm\infty}s^{(j)}(y)=0,\quad j=1,2 \quad  \text{and} \quad
\lim_{y\to \pm \infty}s(y)=u_{\pm},\quad u_{-}\neq u_{+}.
\end{equation}

Note that from \eqref{edp} the function  $s=s(y)$  satisfies  the ordinary
differential equation
\begin{equation}\label{edoA}
(f(s(y))-c s(y))'=(a(s(y))s'(y))'+(b(s(y))s'(y))''
\end{equation}
where
$$(\cdot)'=\frac{d}{dy}(\cdot),\quad y\in\mathbb{R}.
$$
Integrating \eqref{edoA} on $]-\infty,y[$ and using the conditions at
infinity \eqref{infinito} we have
\begin{equation}\label{integral}
\begin{aligned}
&-c(s(y)-u_-)+f(s(y))-f(u_-)\\
&= a(s(y))s'(y)-\lim_{y\to-\infty} a(s(y))s'(y)
+(b(s(y))s'(y))'  -\lim_{y\to -\infty}(b(s(y))s'(y))'
\end{aligned}
\end{equation}
Define the function
$h(s)=-c(s-u_-)+f(s)-f(u_-)$ for $s\in\mathbb{R}$. Since
$$
\lim_{y\to-\infty} a(s(y))s'(y)=\lim_{y\to-\infty}b'(s(y))(s'(y))^2=0,
$$
we can  re-write \eqref{integral} in the  form
\begin{equation}\label{edo}
h(s(y))=a(s(y))s'(y) +(b(s(y))s'(y))',\quad y\in\mathbb{R}.
\end{equation}
We obtain the following lemma.

\begin{lemma}\label{weak}
Let $u(x,t)=s(x-ct)$, $c\in\mathbb{R}$, be a \emph{traveling wave} solution
of \eqref{edp} satisfying \eqref{infinito}. Then, in equation \eqref{edo},
letting $y\to +\infty$ we obtain
\begin{equation}\label{RH}
-c(u_{+}-u_{-})+f(u_+)-f(u_-)=0;
\end{equation}
i.e., the triple $(u_{-},u_{+},c)$  satisfies the Rankine-Hugoniot
relation.
\end{lemma}

\begin{remark} \label{rmk2}\rm
In agreement with  Lemma \ref{weak}, it will be assumed that
$$
c=\frac{f(u_{+})-f(u_{-})}{u_{+}-u_{-}}.
$$
\end{remark}

Setting $w(y)=b(s(y))s'(y)$ in \eqref{edo} we have the  second-order
system
\begin{equation}\label{sistemaA}
\begin{gathered}
s'(y)=\frac{w(y)}{b(s(y))} ,\\
w'(y)=h(s(y))-\frac{a(s(y))}{b(s(y))}w(y).
\end{gathered}
\end{equation}
Let $F$ be vector field
$$
F(s,w)=\left(\frac{w}{b(s)},h(s)-\frac{a(s)}{b(s)}w\right),
$$
a point in the $(s,w)$-plane is an equilibrium point of \eqref{sistemaA}
if and only if it has of the form $(s,0)$ with $h(s)=0$.

\begin{proposition}\label{chave}
A point $(s,w)$ is an equilibrium point of the \eqref{sistemaA} if and
only if $w=0$ and  the triple $(s,u_-,c)$ satisfies the Rankine-Hugoniot
 relation for the associate conservation law  $u_t+f(u)_x=0$.
\end{proposition}


\begin{remark} \label{rmk4}\rm
Denote by $\Gamma$ the set of equilibrium points of \eqref{sistemaA} and
let $(u_i,0)\in \Gamma$. Then
$$
h(s)=-c(s-u_{i})+f(s)-f(u_{i}).
$$
Geometrically, $\Gamma$ is the intersection of the straight line connecting
$(u_{\pm}, 0)$ and the graph of $f$.
\end{remark}

\begin{definition}[Weak Solution] \label{shock} \rm
A discontinuous function of the form
\[
u(x,t)=\begin{cases}
u_{-}, & x - ct \leq 0 \\
u_{+}, & x - ct > 0
\end{cases}
\]
is said to be a  weak solution  of the conservation law
$u_t+f(u)_x=0$ if it satisfies the Rankine-Hugoniot relation
$$
-c(u_{+}-u_{-})+f(u_+)-f(u_-)=0.
$$
\end{definition}

 We know that weak solutions are not unique. To choose the only
physically relevant solution we use the
Oleinik entropy criterion, that requires
\begin{equation}\label{Oleinik}
\frac{f(u_{+})-f(u_{-})}{u_{+}-u_{-}}<\frac{f(u)-f(u_{-})}{u-u_{-}},
\quad \forall u\in (u_{+},\,u_{-})\,,
\end{equation}
or equivalently,
\begin{equation}
\frac{f(u)-f(u_{+})}{u-u_{+}}<\frac{f(u_{+})-f(u_{-})}{u_{+}-u_{-}},\quad
 \forall u\in (u_{+},\,u_{-}).
\end{equation}

In view of proposition \ref{chave}, assuming that $(s,0)$ is a equilibrium
point, the function
\begin{equation}\label{16}
u(x,t)=\begin{cases}
u_{-}, & x-ct \leq 0 \\
s, & x-ct > 0
\end{cases}
\end{equation}
is a weak solution of the conservation law $u_t+f(u)_x=0$. Conversely,
if $u(x,t)$ is as in \eqref{16} and also a weak solution then the points
$(s,0)$ e $(u_-,0)$ are equilibrium points of the $F$.

 Another fact that follows from the existence of weak solution physically
relevant is that $h(s)<0$ in $(u_+,u_-)$, we assume without
loss of generality that $u_+<u_{-}$. We can also conclude that
 $f'(u_+)\leq c \leq f'(u_-)$.

\subsection{Linearization}

The Jacobian matrix  is
$$
DF(s,w) =  \begin{pmatrix}
-w\left(\frac{b'(s)}{b^2(s)}\right) & \frac{1}{b(s)}  \\
(f'(s)-c)-w\left(\frac{a'(s)b(s)-a(s)b'(s)}{b^2(s)}\right)
 & -\frac{a(s)}{b(s)}
\end{pmatrix}
$$
We choose $(u_i,0)$ an arbitrary equilibrium point and obtain
$$
DF(u_{i},0) = \begin{pmatrix}
0 & \frac{1}{b(u_{i})} \\
f'(s)-c & -\frac{a(u_{i})}{b(u_{i})}  \\
\end{pmatrix}.
$$
Therefore, the characteristic equation of $DF(u_{i},0)$ is
$$
|DF(u_{i},0)-\lambda Id|=\lambda^{2}+\frac{a(u_{i})}{b(u_{i})}\lambda
-\frac{1}{b(u_{i})}(f'(u_{i})-c)=0,
$$
which admits two roots:
\begin{equation}\label{raiz}
\begin{gathered}
\lambda_{1}=-\frac{a(u_{i})}{2b(u_{i})}
 - \sqrt{\frac{(a(u_{i}))^2}{4(b(u_{i}))^2}+\frac{f'(u_{i})-c}{b(u_{i})}}\,,\\
\lambda_{2}=-\frac{a(u_{i})}{2b(u_{i})}
 + \sqrt{\frac{(a(u_{i}))^2}{4(b(u_{i}))^2}+\frac{f'(u_{i})-c}{b(u_{i})}}
\end{gathered}
\end{equation}
where $i=\pm$.
Next, we recall some concepts and theorems  that will be useful for
the classification of the equilibrium points of field $F$.

\section{Definitions and statement of results}

Consider the nonlinear system
\begin{equation}\label{nonlinear system}
X'=G(X),
\end{equation}
where $G:D\subseteq \mathbb{R}^{2}\to \mathbb{R}$ is continuously
 differentiable and $D$  is a neighborhood  of the  $X=X_{0}$.

\begin{definition} \label{def6}\rm
The equilibrium point  $X=X_0$ of \eqref{nonlinear system} is
\begin{itemize}
\item[(1)] stable, if for each $\epsilon>0$, there is $\delta(\epsilon)>0$
 such that $\|X(t)-X_0\|<\epsilon$ for all $t\geq 0$ whenever
$\|X(0)-X_0\|<\delta$;

\item[(2)] unstable, if not stable;

\item[(3)] asymptotically stable, if it is stable and $\delta$ can be
chosen such that
$$
\lim_{t\to +\infty}\|X(t)-X_0\|=0
$$
 whenever $\|X(0)-X_0\|<\delta$.
\end{itemize}
\end{definition}

\begin{theorem}[\cite{K}]\label{Kalliu}
Let $X=X_{0}$ be an equilibrium point for the  nonlinear system
\eqref{nonlinear system} and let $A=DG(X_0)$. Then
\begin{itemize}
\item[(1)]  If $\operatorname{Re}(\lambda_{i})<0$ for all eigenvalues
$\lambda_{i}$ of $A$ then $X_{0}$ is asymptotically stable relative
to the nonlinear system;
\item[(2)]  If $\operatorname{Re}(\lambda_{i})>0$ for one or more of the
 eigenvalues $\lambda_{i}$ then $X_{0}$ is unstable  relative to the nonlinear
system, where $i=1,2$.
\end{itemize}
\end{theorem}

\subsection{Nonlinear classification system}

From Theorem \ref{Kalliu} and \eqref{raiz}, we have
 the following classification:
\begin{itemize}
\item[(1)] If $f'(u_{i})-c>0$ then $(u_{i},0)$ is a saddle point
(of the linearized system). Thus, the point
$(u_{i},0)$ is unstable.

\item[(2)] If  $f'(u_{i})-c<0$ and  since $a(u_{i})> 0$ then $(u_{i},0)$
is  asymptotically stable.
\end{itemize}

In the remainder of this article is devoted to the case when
 $f'(u_+)<c<f'(u_-)$
and thus shall be ensured the existence and uniqueness of  traveling
wave connecting the states $u_{\pm}$.
Now follow with some more definitions and results of the theory
of differential equations.

\begin{definition}[Invariant set] \label{def8}\rm
(1) A set $M\subset D$ is said to be  \emph{invariant} set with respect to
\eqref{nonlinear system} if $X(0)\in M$ then $X(t)\in M$ for all
$t\in \mathbb{R}$.

(2) A set $M\subset D$ is said to be \emph{positively invariant} set
(negatively invariant set) with respect to
$X'=G(X)$, if $X(0)\in M$ then $X(t)\in M$ for all $t\geq 0$ ($t\leq 0$).
\end{definition}

\begin{definition} \label{def9} \rm
A trajectory $X(t)$ of \eqref{nonlinear system} approaches a set $M\subset D$
as $t\to +\infty$, if for every $\epsilon >0$ there is $T>0$ such that
$$
\operatorname{dist}(X(t),M)\doteq \inf_{p\in M}\|X(t)-p\|<\epsilon,\quad
\forall t>T.
$$
\end{definition}

\begin{theorem}[LaSalle's invariance principle \cite{K}] \label{LaSalle}
Let $V:D\subseteq \mathbb{R}^{2}\to \mathbb{R}$ be continuously differentiable
such that
$$
\dot{V}(s,w)\doteq \nabla V(s,w)\cdot G(s,w)\leq 0
$$
for all $ X=(s,w)\in \Omega$, with $\Omega$  compact positively invariant in $D$.
Let $E$ be the set of all points in $\Omega$ where $\dot{V}(s,w)=0$.
Let  $M$ be the largest invariant set in $E$. Then every trajectory of
$X'=G(X)$, starting  $\Omega$; i.e., $X(0)\in \Omega$,
approaches of $M$ as $t\to +\infty$.
\end{theorem}

Now turning our attention to  system \eqref{sistemaA}, we consider the  function
\begin{equation}\label{Lyapunov}
V(s,w)=-\int_{u_+}^{s}h(x)b(x)\,dx+\frac{w^2}{2},\quad
 (s,w) \in \mathbb{R}^{2}.
\end{equation}
Note that
$$
\dot{V}(s,w)=\frac{w^2}{b(s)}(-a(s))\leq0\quad\text{and}\quad
\dot{V}(s,w)=0\Leftrightarrow w=0.
$$

\begin{proposition}\label{aproxima}
Suppose that there is a compact $\Omega \subset  \mathbb{R}^{2}$
positively invariant  with respect to system \eqref{sistemaA} with vector field
$$
F(s,w)=\Big(\frac{w}{b(s)},h(s)-\frac{a(s)}{b(s)}w\Big).
$$
Then every solution of this system starting in $\Omega$ approaches the
set $\Gamma\cap \Omega$.
\end{proposition}

\begin{proof}
With the quantities \eqref{Lyapunov} at hand, we can rewrite this context
 $E=\{(s,w)\in \Omega; w=0\}$.
Using Theorem \ref{LaSalle}
every solution starting in $\Omega$ approaches the set $M$ the largest
invariant set in  $E$. It is easy to see that
$\Gamma\cap \Omega$ is invariant, here $\Gamma$ is defined in
Remark \ref{rmk4}, so $\Gamma\cap \Omega \subset M$. Let us show that
$M \subset\Gamma\cap \Omega  $. Indeed, given $(u,0)\in M$,
let $(s(y),w(y))$ be the  solution of the system with
initial data $(s(0),w(0))=(u,0)$. We have that
 $(s(y),w(y))\in M$ for all $y\in\mathbb{R}$, since it is $M$
invariant, so $w(y)=0$, $y\in\mathbb{R}$. On the other hand,
follow of the system that
$$
s'(y)=\frac{w(y)}{b(s(y))}=0,
$$
 which implies $s(y)=u$. Thus, $(u,0)$
is an equilibrium  point of the system \eqref{sistemaA}; i.e.,
 $(u,0)\in \Gamma\cap \Omega$.
\end{proof}

\section{Traveling wave solution: domain of attraction}

The goal of this section is to determine a compact $\Omega$ positively invariant
in $\mathbb{R}^2$ and estimate the domain of attraction of the
equilibrium point $(u_+,0)$.
Initially we give the definition of the domain of attraction.


\begin{definition} \label{def12}\rm
Let $X=X_0$ be an equilibrium point asymptotically stable of the system
$X'=G(X)$. Denote by $\phi(t,\overline{X})$
the solution starting at $\overline{X}$ in $t=0$. Then,
the \emph{domain attraction}
corresponding to $X_0$ is the set from $\overline{X}$ such that
$$
\lim_{t\to +\infty}\|\phi(t,\overline{X})-X_{0}\|=0.
$$
\end{definition}

\subsection{Estimation of the domain attraction}

Let $m=\min_{(s,w)\in \partial (D\cup R)} V(s,w)$,
 where the set $D$ and $R$ are given by
\begin{gather*}
D=\Big\{(s,w)\in\mathbb{R}^2:(s-u_{+})^2+\frac{w^2}{\gamma^2}
 \leq (u_{+}-q)^2 ,\; u_{+}\leq s \leq q\Big\},\\
R=\Big\{(s,w)\in\mathbb{R}^2:(s-u_{+})^2+\frac{(u_{+}-p)^2}{(\gamma|u_{+}-q|)^2}
w^2\leq (u_{+}-p)^2 ,\;  p\leq s \leq u_{+}\Big\}
\end{gather*}
(see Figure \ref{fig1}), with
\begin{equation}\label{A}
 \gamma^{2}> (\text{Lip}_{[p,u_{-}]}f+|c|)\max_{[p,u_-]}b(s)
\end{equation}
and
\begin{equation}\label{B}
\int_{p}^{u_{-}}h(x)b(x)\,dx>0\quad \text{and}\quad h>0\text{ in } (p,u_{+}).
\end{equation}
where $p$ was chosen satisfying the inequality $2u_{+}-u_{-}\leq p<u_+$.

 The condition \eqref{B} makes sense since we have $f'(u_+)<c$.
Indeed, from $f'(u_+)<c$, there is a $t>0$ such that
$$
\frac{f(u)-f(u_+)}{u-u_+}< c, \quad \forall u\in(u_{+}-t,u_+).
$$
Hence,  $h(u)>0$ for all $u\in(u_{+}-t,u_+)$ thus
$$
\int_{u}^{u_+}h(x)b(x)\,dx>0,\quad \forall u\in(u_{+}-t,u_+).
$$
Choose $p\in(u_{+}-t,u_+)$. Being $I(v)=\int_{p}^{v}h(x)b(x)\,dx>0$ continuous
positive in $(p,u_{+}]$ there is some $u_{+}< q\leq u_-$ such that $I(q)>0$,
i.e., $\int_{p}^{q}h(x)b(x)\,dx>0$.
In this work we assume that holds $q=u_{-}$.

\begin{figure}[ht] 
\begin{center}
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\put(7,17){$p$}
\put(25,17){$u_+$}
\put(46,17){$q$}
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\qbezier(10,20)(12,2)(30,0)
\qbezier(30,0)(48,2)(50,20)
\end{picture}
\end{center} 
\caption{Region containing the domain of attraction} \label{fig1}
\end{figure}

\begin{lemma}\label{m}
Let $\Sigma =D\cup R$ be defined as above and  $\partial \Sigma$ denote
its boundary.  Let $\gamma$ be given by \eqref{A}. Then  we have
$$
0<m=\min_{(s,w)\in \partial \Sigma} V(s,w)=V(u_{-},0)
=\int_{u_{+}}^{u_{-}}-h(x)b(x)\,dx.
$$
\end{lemma}

\begin{proof}
Let $E=\partial D\cap \partial \Sigma$. Then
$$
E=\{(s,w)\in\mathbb{R}^2:(s-u_{+})^2+\frac{w^2}{\gamma^2} =(u_{+}-u_{-})^2,\;
u_+\leq s \leq u_{-}\}
$$
and we have
$$
w^2=\gamma^2((u_{+}-u_{-})^2-(s-u_{+})^2),
$$
we replace $w$ in $V(\cdot,\cdot)$ in \eqref{Lyapunov} we have
$$
\min_{(s,w)\in E}V(s,w)=\min_{s\in[u_+,u_{-}]}
\Big[-\int_{u_+}^{s}h(x)b(x)\,dx+\frac{\gamma^2}{2}((u_{+}-u_{-})^{2}
-(s-u_{+})^2)\Big].
$$
Define $g(s)=-\int_{u_+}^{s}h(x)b(x)\,dx+\frac{\gamma^2}{2}((u_{+}-u_{-})^{2}
-(s-u_{+})^2)$, for $s\in[u_{+},u_{-}]$.
It follows that
\begin{align*}
g'(s) & = -h(s)b(s)-\gamma^{2}(s-u_{+})\\
    & =  -(s-u_{+})\Big( \gamma^2 + b(s)\Big(\frac{f(s)-f(u_+)}{s-u_+}-c\Big)\Big)
< 0,\quad s\in (u_{+},u_{-})
\end{align*}
where the inequality above comes from \eqref{A}, because,
since $-(s-u_{+})<0$ we can deduce that
$\Big( \gamma^2 + b(s)\big(\frac{f(s)-f(u_+)}{s-u_+}-c\big)  \Big) >0$, in fact
\begin{equation}\label{lipchitz}
\begin{aligned}
cb(s) -b(s)\Big(\frac{f(s)-f(u_+)}{s-u_+}\Big)
&\leq |c|b(s)+b(s)\big|\frac{f(s)-f(u_+)}{s-u_+}\big|  \\
& \leq (\text{Lip}_{[p,u_{-}]}f+|c|)\max_{[p,u_-]}b(s)
< \gamma^2.
\end{aligned}
\end{equation}
Therefore, the function $g$ is strictly decreasing in $[u_+,u_{-}]$ and
realizes its minimum value
in $s=u_{-}$. Thus,
 $$
\min_{(s,w)\in  E}V(s,w)=V(u_{-},0).
$$

On the other hand, on   $F=\partial R\cap \partial (D\cup R)$
$$
F=\Big\{(s,w)\in\mathbb{R}^2:(s-u_{+})^2
 +\frac{(u_{+}-p)^2}{(\gamma|u_{+}-u_{-}|)^2} w^2
= (u_{+}-p)^2 ,\; p\leq s \leq u_{+}\Big\}
$$
one has
$$
w^2=\gamma^2\Big((u_{+}-u_{-})^{2}-(s-u_{+})^{2}
\frac{(u_{+}-u_{-})^2}{(u_{+}-p)^2}\Big),
$$
we replace $w$ in $V(\cdot,\cdot)$ in \eqref{Lyapunov} we have
\begin{align*}
\min_{(s,w)\in  F}V(s,w)
&= \min_{s\in[p,u_{+}]}\Big[-\int_{u_+}^{s}h(x)b(x)\,dx\\
&\quad +\frac{\gamma^2}{2}\Big((u_{+}-u_{-})^2-(s-u_{+})^{2}
 \frac{(u_{+}-u_{-})^2}{(u_{+}-p)^2}\Big)\Big].
\end{align*}
Setting
$$
G(s)=-\int_{u_+}^{s}h(x)b(x)\,dx+\frac{\gamma^2}{2}
\left((u_{+}-u_{-})^2-(s-u_{+})^{2}\frac{(u_{+}-u_{-})^2}{(u_{+}-p)^2}\right),
$$
 for $s\in[p,u_{+}]$, we have
\begin{align*}
G'(s)& =-h(s)b(s)-\gamma^{2}(s-u_{+})\frac{(u_{+}-u_{-})^2}{(u_{+}-p)^2}\\
 & =  -(s-u_{+})\Big( \gamma^{2}\frac{(u_{+}-u_{-})^2}{(u_{+}-p)^2}
+ b(s)\Big(\frac{f(s)-f(u_+)}{s-u_+}-c\Big)    \Big),\quad s\in (p,u_{+}).
\end{align*}
Note that
$$
\frac{(u_{+}-u_{-})^2}{(u_{+}-p)^2}>1\quad\text{ and}\quad -(s-u_{+})>0
$$
so
$$
 \gamma^{2}\frac{(u_{+}-u_{-})^2}{(u_{+}-p)^2} + b(s)
\Big(\frac{f(s)-f(u_+)}{s-u_+}-c\Big)    >
\gamma^2 + b(s)\Big(\frac{f(s)-f(u_+)}{s-u_+}-c\Big).
$$
But, we show that $\gamma^2 + b(s)\big(\frac{f(s)-f(u_+)}{s-u_+}-c\big) > 0 $
similarly  as in \eqref{lipchitz}. Therefore
$G$ is  strictly decreasing in $[p,u_+]$, and
$$
\min_{(s,w)\in  F}V(s,w)=V(p,0).
$$

Now let us compare the values $V(u_{-},0)$  and $V(p,0)$.
Clearly from \eqref{B} it follows that
$$
V(p,0)=\int_{p}^{u_{+}}h(x)b(x)\,dx
=\int_{p}^{u_{-}}h(x)b(x)\,dx-\int_{u_+}^{u_{-}}h(x)b(x)\,dx > V(u_{-},0).
$$
So the lemma is proved.
\end{proof}


 We are now able to build a compact positively invariant.
Consider $l\in(0,m)$, where $m$ is given in Lemma \ref{m}, and define
a set
$$
\Omega_{l}=\{(s,w)\in D\cup R : V(s,w)\leq l\}.
$$

\begin{assertion}\label{incluso}
$\Omega_{l}\subset \operatorname{int} (D\cup R)$.
\end{assertion}

\begin{proof}
To prove this, we suppose to the contrary, then there exists
$(s_0,w_0)\in\Omega_{l} \cap \partial (D\cup R)$, therefore $V(s_0,w_0)\geq m>l$
producing a contradiction.
\end{proof}

\begin{assertion}\label{compacto}
Let $\Omega_{l}$ be o set above. Then, it is compact positively invariant.
\end{assertion}

For proof of the assertion \ref{compacto} we need of Lemma \ref{global},
whose proof can be found at \cite{K}.

\begin{lemma}\label{global}
Suppose that there exists a compact set  $W \subset \mathbb{R}^2$
such that every local solution of
$X'=G(X)$, $y>0$, $X(0)=X_{0}=(s(0),w(0))\in W$,
lies entirely in $W$. Then, there is a unique solution passing through
 $X_{0}$ defined in $(0,\infty)$.
\end{lemma}

\begin{proof}[Proof of Assertion \ref{compacto}]
We clearly have that $\Omega_{l}$ is compact, since
$\Omega_{l}=(D\cup R)\cap V^{-1}((-\infty,l])$.
Let $(s(y),w(y))$ be a solution for system  starting
in $\Omega_{l}$, i.e, $(s(0),w(0))\in\Omega_{l}$. We saw earlier
that $\dot{V}(s,w)\leq 0$, then
$$
\frac{d}{dy}(V(s(y),w(y)))\leq 0
$$
so the function $V(s(y),w(y))$ is decreasing for $y\in J=(0,\omega_{+})$,
we denote by $J$ the maximal interval associated to the maximal solution
$(s(y),w(y))$. Thus,
$$
V(s(y),w(y))\leq V(s(0),w(0))\leq l(< m).
$$
Consequently,
$$
(s(y),w(y))\in \Omega_{l}\quad  \text{for }\quad  y\in J,
$$
provided that $(s(y),w(y))\in\Sigma$ for all $y\in J$, since
$m=\min_{(s,w)\in \partial \Sigma} V(s,w)$.
It then follows from Lemma \ref{global}  that $\omega_{+}=\infty$.
So the assertion \ref{compacto}  follows.
\end{proof}

We have proven $(u_+,0)\in \Omega_{l}$ for all $l\in(0,m)$ and is the
only equilibrium point of the system \eqref{sistemaA}
in $\Omega_{l}$, once $\Omega_{l}$
lies entirely in the interior of  $D\cup R$ and the function $h$ is
 positive  in $(p,u_{+})$ and  negative in $(u_{+},u_{-})$.

 Therefore, according to Proposition \ref{aproxima} every solution starting
in $\Omega_{l}$ tends toward $(u_+,0)$ when $y\to +\infty$.
Furthermore, sets $\Omega_{l}$ are an approximation of the domain
of attraction of the point $(u_{+},0)$ which is the subject of
the following lemma.

\begin{lemma} \label{lem17}
The domain of attraction of the equilibrium point $(u_{+},0)$ contains the set
$$
W=\{(s,w)\in D\cup R: V(s,w)< V(u_{-},0)\}.
$$
Moreover, the line segment  $[u_{+},u_{-})\times \{0\}$ is contained in $W$.
\end{lemma}

\begin{proof}
It is sufficient to prove that $W=\cup_{0<l<m} \Omega_{l}$.
It is easy to see that $ \Omega_{l}\subset W$. It remains to check
that $W\subset\cup_{0<l<m} \Omega_{l}$.
For this, let $(l_{n})$ be a sequence in $(0,m)$ such that  $l_{n}\to m $
as $n\to \infty$.
 As $ V(u_{-},0)-V(s,w)>0$ for $(s,w)\in W $ follows from the definition
of limit that
$$
l_{n}>\int_{u_+}^{s}-h(x)b(x)\,dx+\frac{w^2}{2}
$$
for some $n=n(V(u_{-},0)-V(s,w))$. Thus, $(s,w)\in \Omega_{l_{n}}$
and the identity $W=\cup_{0<l<m} \Omega_{l}$ holds.
For $u_{+}<u<u_{-}$ we have
$$
V(u,0)=\int_{u_+}^{u}-h(x)b(x)\,dx<\int_{u_+}^{u_-}-h(x)b(x)\,dx=V(u_{-},0)=m
$$
and then $(u,0)\in\omega$.
\end{proof}


\section{Existence of semi-orbits}

In this  section we recall the basic results and concepts.
The reader is referred to \cite{LP} and \cite{HW} for more details.

\begin{definition}\rm
Let $X_{0}$ a equilibrium point of a $(s,w)$-planar $C^r$ vector
 field $G=(G_1,G_2)$. We say that
$$
DG(X_{0}) =\begin{pmatrix}
\frac{\partial G_1}{\partial s}(X_{0}) & \frac{\partial G_1}{\partial w}(X_{0})  \\
\frac{\partial G_2}{\partial s}(X_{0}) & \frac{\partial G_2}{\partial w}(X_{0})
\end{pmatrix}
$$
is the \emph{linear part} of the vector field $G$ at the equilibrium point
 $X_{0}$. The equilibrium point $X_{0}$
is called \emph{hyperbolic} if the two eigenvalues de $DG(X_{0})$
 have real part different from $0$.
\end{definition}


\begin{theorem}[The Stable Manifold Theorem \cite{HW}].\label{T2}
Assume $A=DG(X_0)$ has eigenvalues $\lambda_{1}$, $\lambda_{2}$ with
$\lambda_{1} < 0 < \lambda_{2}$. Then there are two orbits of
$X'=G(X)$ that approach $X_0$ as $y \to +\infty$,
along a smooth curve tangent at $X_0$ to the
eigenvectors for $\lambda_{1}$  and two orbits
that approach $X_0$ as $y \to -\infty$, along a smooth
curve tangent at $X_0$ to the eigenvectors of $\lambda_{2}$.
\end{theorem}

\subsection{Semi-orbits}

We now return to the system \eqref{sistemaA}. We recall that the
equilibrium point $(u_{-},0)$ is a saddle point, so it
is \emph{hyperbolic}. It follows from Theorem \ref{T2} that there
are two orbits of \eqref{sistemaA}, that approach $(u_{-},0)$ as
$y \to -\infty$, along a smooth
curve tangent at $(u_{-},0)$ to the eigenvectors of $\lambda_{2}$,
given in \eqref{raiz}.

 More specifically, according to the analysis of phase portrait close
to the equilibrium point $(u-,0)$ [see Appendix I] there are exactly two
semi-orbits of system \eqref{sistemaA}
that converge to  $(u_{-},0)$ as $y\to -\infty$,
one orbit approaches the  $(u-,0)$ from region
$S=\{(s,w)\in\mathbb{R}^2:\,s<u_{-},w<0\}$, while the other approaches
from region $T=\{(s,w)\in\mathbb{R}^2:s>u_{-},w>0\}$ (see Figure \ref{fig2}).

\begin{figure}[ht] 
\begin{center}
\setlength{\unitlength}{.8mm}
\scriptsize
\begin{picture}(71,56)(0,0)
\put(0,20){\line(1,0){70}}
\put(69.7,19.2){$\rightarrow$}
\put(20,0){\line(0,1){55}}
\put(19.1,55.2){$\uparrow$}
\dashline{1}(50,0)(50,55)
\qbezier(0,15)(50,-2)(65,45)
\put(15,55){$w$}
\put(60,45){$T$}
\put(30,15){$S$}
\put(52,16){$u_-$}
\put(70,16){$s$}
\end{picture}
\end{center} 
\caption{The two semi-orbits connection $u_-$} \label{fig2}
\end{figure}

\begin{proposition}\label{unicidade}
Let $(s(y),w(y))$ be a trajectory of system \eqref{sistemaA}
globally defined entering the saddle point $(u_{-},0)$ from the region
$T$ as $y\to -\infty$, then
such  trajectory cannot tend to $(u_{+},0)$ as $y\to +\infty$.
\end{proposition}

\begin{proof}
Multiply \eqref{edo} by $b(s(y))s'(y)$ and integrate on $(-\infty,z)$,
\begin{align*}
\int_{-\infty}^{z}\,h(s(y)b(s(y))s'(y)\,dy
&= \int_{-\infty}^{z}\,a(s(y)b(s(y))(s'(y))^2\,dy\\
&\quad + \frac{1}{2}\int_{-\infty}^{z}\,[(b(s(y))s'(y))^2]'\,dy
\end{align*}
equivalently,
\begin{equation}\label{salva}
\int_{u_{-}}^{s(z)}\,h(x)b(x)\,dx
= \int_{-\infty}^{z}\,a(s(y)b(s(y))(s'(y))^2\,dy
+ \frac{1}{2}[b(s(z))s'(z)]^2.
\end{equation}
Suppose now that $(s(y),w(y)) \to (u_{+},0)$ as $y\to \infty$,
then there is some $z_{0}$ such that
$s(z_{0})=u_{-}$. Choosing $z=z_{0}$ and replacing it in  \eqref{salva},
 we have
$$
0=\int_{-\infty}^{z_0}\,a(s(y)b(s(y))(s'(y))^2\,dy
+ \frac{1}{2}[b(u_{-})s'(z_{0})]^2.
$$
As each term is non-negative we must have
$$
\int_{-\infty}^{z_0}\,a(s(y)b(s(y))(s'(y))^2\,dy=0
$$
and by continuity it follows that $s'(y)=0$ for $y\in(-\infty,z_{0}]$. Then
$s(y)=u_{-}$ for  $y\in(-\infty,z_{0}]$. Thus,
$(s(z_{0}),w(z_{0}))=(u_{-},0)$ and by uniqueness of solutions
$(s(y),w(y))=(u_{-},0)$ for every $y\in\mathbb{R}$.
Therefore, $(s(y),w(y))\to (u_{-},0)$ as $y\to \infty$,
which contradicts our assumption since $u_{-} \neq u_{+}$.
\end{proof}

 We conclude that the unique semi-orbit that can tend to $(u_+,0)$ as
 $y\to \infty$ is
the one that enters the region $S$.
In the next section we will prove that this semi-orbit is the traveling
wave desired.

\section{Existence of traveling wave solutions}

Let $(s(z),w(z))$ be a solution \eqref{sistemaA} starting in
 $D\cup R$ and defined at least for values of $y$ sufficiently
negative such that
\[
(s(y),w(y))\to (u_{-},0),\quad  y\to -\infty
\]
and that $s(y)<u_{-}$, $w(y)<0$  for values of $y$ sufficiently negative,
that corresponds to semi-orbit that approach the
$(u_{-},0)$ from region $S$. We can assure that $(s(y),w(y))$
 belong $D\cup R$ for $y$ sufficiently negative, this follows from
convergence mentioned above .

Let us multiply the second equation of  \eqref{sistemaA} by $w(z)=b(s(z))s'(z)$
and integrate from  $-\infty$ to $y$
\begin{equation}\label{eq1}
\int_{-\infty}^{y}w(z)w'(z)\,dz
=\int_{-\infty}^{y}w(z)h(s(z))\,dz
-\int_{-\infty}^{y}\frac{a(s(z))}{b(s(z))}(w(z))^2\,dz.
\end{equation}
Note that the second term on the right of \eqref{eq1} is  non-negative.
We prove that
$$
I=\int_{-\infty}^{y}\frac{a(s(z))}{b(s(z))}(w(z))^2\,dz>0.
$$
Suppose, by contradiction, that $I=0$  then, by continuity,
 $w(z)=0$ for every $z\in(-\infty, y]$, so $s'(z)=0$, which means precisely that
$s(z)$ is  constant for $z\in(-\infty, y]$. However, we have
 $s(z)\to u_{-}$ as $z\to -\infty$
it then follows that $s(z)=u_{-}$ on $(-\infty, y]$ and thus
 $(s(y),w(y))=(u_{-},0)$ which contradicts the uniqueness of solutions.
From this fact it follows that
\begin{equation}\label{eq2}
\frac{(w(y))^{2}}{2}<\int_{-\infty}^{y}b(s(z))s'(z)h(s(z))\,dz
=\int_{s(y)}^{u_{-}}-b(x)h(x)\,dx\,.
\end{equation}
We can write
$$
\int_{s(y)}^{u_{-}}-b(x)h(x)\,dx=\int_{u_{+}}^{u_{-}}-b(x)h(x)\,dx
- \int_{u_{+}}^{s(y)}-b(x)h(x)\,dx
$$
so rewriting \eqref{eq2} we have
$$
\int_{u_{+}}^{s(y)}-b(x)h(x)\,dx+\frac {w^{2}(y)}{2}
<\int_{u_{+}}^{u_{-}}-b(x)h(x)\,dx.
$$
Thus,
$$
V(s(y),w(y))<V(u_{-},0).
$$
Consequently, we have $(s(y),w(y))\in W$, and
$$
(s(z),w(z))\to (u_{+},0)\quad \text{as } z\to \infty.
$$

Using the fact that $(s(y),w(y))\in W$, then $(s(y),w(y))\in \Omega_{l}$,
for some $l\in(0,m)$. We formulate the Cauchy problem
\begin{equation}\label{PVI}
\begin{gathered}
X'(t)=F(X(t)) \\
X(y)=(s(y),w(y)).
\end{gathered}
\end{equation}
Thus, $(s(t),w(t))$ is a solution and $(s(t),w(t))\in \Omega_{l}$,
for $t\geq y$, recalling once $\Omega_l$
is invariant positively. Moreover,
$$
(s(t),w(t))\to (u_{+},0)\quad \text{as }t\to \infty\,,
$$
recalling once again the fact $(s(y),w(y))$ belong to
domain of attraction of the equilibrium point $(u_{+},0)$.

 Therefore, we prove the existence of a single orbit
(up to translation of the independent variable)
connecting the states $(u_{\pm},0)$ with flux function
$f:\mathbb{R}\to\mathbb{R}$ de class $C^{1}$.
We then checked  our main result.

\begin{theorem}\label{main}
Let $f:\mathbb{R} \to \mathbb{R}$ be continuously differentiable.
Suppose there is a weak solution connecting the states $u_{\pm}$
with constant speed $c$, given by \eqref{RH}. Further, assume \eqref{Oleinik}
and $f'(u_+)<c<f'(u_-)$.
Then there is (up to translation of the independent variable $y$)
a unique solution $s(y)$ of \eqref{edoA} satisfying \eqref{infinito}.
\end{theorem}

\section{Appendix I}
Here we will carry out an analysis of the phase portrait close
to the equilibrium point $(u-,0)$. Let us eliminate the possibilities
for the behavior of semi-orbits near the saddle point.

{\bf Case 1:}
If $w(y)>0$ and $s(y)< u_{-}$ for $y\in (-\infty, 0]=I$, then of
\eqref{sistemaA} we have $s'(y)>0$ in $I$, so
$s(y)$ is  increscent. Consequently,  $\lim_{y \to -\infty} s(y)\neq u_{-}$.
 Therefore  $(s(y),w(y)) \nrightarrow (u_{-},0) $ as
$y \to -\infty $ and the semi-orbits do not tend to $(u_{-},0)$ the region
$S=\{(s,w)\in\mathbb{R}^2:\,s<u_{-},w>0\}$.

{\bf Case 2:}
If $w(y)<0$ and $s(y)> u_{-}$ for $y\in (-\infty, 0]=I$, then of
\eqref{sistemaA} we have $s'(y)<0$ in $I$, so
$s(y)$ is decreasing in $I$. Consequently,
$\lim_{y \to -\infty} s(y)\neq u_{-}$.
Therefore, the semi-orbits do not tend to $(u_{-},0)$ the region
$S=\{(s,w)\in\mathbb{R}^2:\,s>u_{-},w<0\}$.

{\bf Case 3:}
If $w(y)=0$ and $s(y)> u_{-}$ in $I$. It follows from \eqref{sistemaA}
that $s'(y)=0$ in $I$, so $s(y)$ is constant.
If $\lim_{y \to -\infty} s(y)= u_{-}$,
we have $s(y)=u_{-}$, contradiction. Therefore the semi-orbits
 do not tend to $(u_{-},0)$ the region
$S=\{(s,w)\in\mathbb{R}^2:\,s>u_{-},w=0\}$.

{\bf Case 4:}
If $w(y)=0$ and $s(y)< u_{-}$ in $I$, is entirely analogous to the
 previous item.
Therefore, the semi-orbits do not tend to $(u_{-},0)$ the region
$S=\{(s,w)\in\mathbb{R}^2:\,s<u_{-},w=0\}$.

{\bf Case 5:}
If  $s(y)= u_{-}$ for $y\in (-\infty, 0]=I$, is  similar.
Therefore the semi-orbits do not tend to $(u_{-},0)$ the region
$S=\{(s,w)\in\mathbb{R}^2:\,s=u_{-},w=0\}$.

{\bf Case-possible:}
Theorem \ref{T2} ensures that there are two semi-orbit along a smooth
curve tangent in $(u_-,0)$ to the eigenvectors of $\lambda_{2}$
(up to translation). Therefore, we have a semi-orbit tending to
the point $(u_-,0)$ of the region
$S=\{(s,w)\in\mathbb{R}^2:\,s<u_{-},w<0\}$, while the other approaches
 from region $T=\{(s,w)\in\mathbb{R}^2:s>u_{-},w>0\}$.

\section{Appendix II}

We consider an partial differential equations
\begin{equation}\label{edp3}
u_{t}+uu_x=au_{xx}+bu_{xxx}
\end{equation}
where   $a>0, b>0$.
We seek existence of traveling waves solution $u(x,t)=s(x-ct)$,
for some constant speed $c\in \mathbb{R}$,
satisfying the following conditions at infinity:
\begin{equation}\label{infinito25}
\lim_{y\to \pm\infty}s^{(j)}(y)=0,\; j=1,2 \quad  \text{and}\quad
 \lim_{y\to \pm \infty}s(y)=u_{\pm},\; u_{-}\neq u_{+}.
\end{equation}
Following \cite{Schonbek}, the following problem is
equivalent to \eqref{edp3}-\eqref{infinito25}. Now consider
\begin{equation}\label{edp4}
u_{t}+uu_x=au_{xx}+bu_{xxx}
\end{equation}
where   $a>0, b>0$,
and the new conditions at infinity are
\begin{equation}\label{infinito27}
\lim_{y\to \pm\infty}s^{(j)}(y)=0,\; j=1,2,\quad
\lim_{y\to - \infty}s(y)=2\eta >0 \quad\text{and}\quad
\lim_{y\to  \infty}s(y)=0.
\end{equation}

 Now we must  determine $p$ as in \eqref{B}; i.e,  obtain
$p$ such that
\begin{equation}
\int_{p}^{2\eta}bh(x)\,dx>0
\end{equation}
with $-2\eta < p < 0$ and $h(x)=x^{2}-2\eta x$. As we have $b>0$ just take
\begin{equation}
\int_{p}^{2\eta}h(x)\,dx=1/3(p-2\eta)( -p^{2}+p{\eta}+2\eta^2) >0.
\end{equation}
So choosing $p\in(-2\eta, -\eta)$ we have
\begin{equation}
\int_{p}^{2\eta}bh(x)\,dx>0.
\end{equation}
 Therefore we are able to apply our work and we have existence and
uniqueness of traveling waves when the flux-function associated is
given by $f(u)=u^2/2$.

 For the BBM-Burgers equation
\begin{equation}\label{edp5}
u_{t}+f(u)_x=au_{xx}+bu_{xxt}
\end{equation}
where $a>0$ and $b\in \mathbb{R}$, we establish the classification of
equilibrium points as in section 3.1 keeping $c>0$ and the sign
of $b$ was studied separately to establish the desired connection.
 When $c<0$ proceed in the same manner.
For $b=0$ we refer to \cite{LeFLoch} for existence.

 For  \eqref{edp5} the associated system becomes
\begin{equation}\label{sistemaII}
\begin{gathered}
s'(y)=\frac{w(y)}{-bc} \\
w'(y)= h(s(y))-\frac{a}{-bc}w(y)
\end{gathered}
\end{equation}
 and $V(s,w)=-\int_{u_+}^{s}-bch(x)\,dx+\frac{w^2}{2}$.

\subsection*{Acknowledgements}
Alex F. Rossini was supported by a grant from Capes, Brazil.

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\bibitem{NL} N. Bedjaoui, P.G. LeFloch;
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