\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 145, 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 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/145\hfil Bifurcation of traveling wave solutions]
{Bifurcation of traveling wave solutions of a  generalized $K(n,n)$ equation}

\author[X. Zhao, G. Zhao, L. Peng \hfil EJDE-2014/145\hfilneg]
{Xiaoshan Zhao, Guanhua Zhao, Linping Peng}  % in alphabetical order

\address{Xiaoshan Zhao \newline
School of Science,
Tianjin University of Technology and Education,
Tianjin 300222, China.\newline
School of Electrical, Computer and Energy Engineering,
Arizona State University,  Tempe, Arizona 85287, USA}
\email{xszhao678@126.com}

\address{Guanhua Zhao \newline
Department of Mathematics, Handan College,
Handan, Hebei 056005, China}
\email{zghlds@126.com}

\address{Linping Peng \newline
School of Mathematics and System Sciences,
Beihang University, LIMB of the Ministry
of Education, Beijing 100191, China}
\email{penglp@buaa.edu.cn}

\thanks{Submitted March 14, 2013. Published June 20, 2014.}
\subjclass[2000]{34C25-28, 35B08, 35B10, 35B40}
\keywords{Solitary wave; periodic wave; kink wave; compatons; bifurcation}

\begin{abstract}
  In this article, a generalized $K(n,n)$ equation is studied by the qualitative
  theory of bifurcations and the method of dynamical systems.
  The result shows the existence of the different kinds of traveling solutions
  of the generalized $K(n,n)$ equation,  including solitary waves, kink waves,
  periodic wave and compacton solutions, which depend on different parametric ranges.
  Moreover, various sufficient conditions to guarantee the existence of the
  above traveling solutions are provided under different parameters conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction}

 The well-known $K(m,n)$ equation \cite{Rosenau} takes the form
\begin{equation}\label{1.10}
u_t+a(u^n)_x+(u^m)_{xxx}=0,\quad n>1.
\end{equation}
which generates the so termed compactons: solitary waves with exact compact support.
 Compactons are defined as solitons with finite wavelengths or solitons free 
of exponential tails 
\cite{Ablowitz, Benjamin, Feng, Hereman, LL2, LS3, LO1,
Rosenau, Wadati,  Wazwaz1, Yan, Zabusky}.
 Wazwaz \cite{Wazwaz2} used the $\tanh$ and sine-cosine method to study the 
generalized $K(n,n)$ equation given by
\begin{equation}\label{2.10}
u_t+\alpha(u^n)_x+\beta(u^{2n}(u^{-n})_{xx})_x=0,\quad n>1.
\end{equation}
In this paper, we aim  to consider the bifurcation behavior of the traveling 
wave solutions of equation \eqref{2.10} in the parameter space and obtain 
bifurcations of traveling solutions under different parameter conditions. 
Let $u(x,t)=\psi(\xi),\ \xi=x-ct$, where $c$ is wave speed. Substituting the
above traveling transformation into equation \eqref{2.10} and integrating once,
 we have
\begin{equation}\label{2.11}
n\beta\psi^{n-1}\psi_{\xi\xi}-n(n+1)\beta\psi^{n-2}\psi^2_\xi-\alpha\psi^n+c\psi-g=0,
\end{equation}
where $g$ is an integral constant. equation \eqref{2.11} is equivalent to the 
two dimensional system as follows
\begin{equation} \label{2.12}
\begin{gathered}
 d\psi/d\xi=y,\\
 dy/d\xi =(n(n+1)\psi^{n-2}y^2+\rho(\psi^n-\mu\psi+\nu))/(n\psi^{n-1}),\\
\end{gathered}
\end{equation}
where $\rho=\alpha/\beta$, $\mu=c/\alpha,\nu=g/\alpha$. 
And the system \eqref{2.12}  has the first integral
\begin{equation}\label{3.10}
H(\psi,y)=-1/2\psi^{-2(n+1)}y^2-\rho\psi^{-3n}(1/(2n)\psi^n+\mu/(1-3n)\psi+\nu/(3n))
=h.
\end{equation}
In fact, system \eqref{2.12} is a three-parameter planar dynamical system 
depending on the parameter group $(\rho,\mu,\nu)$. Because the phase orbits 
defined by the vector fields of system \eqref{2.12} determine types of 
traveling wave solutions of equation \eqref{2.10}, we should consider 
the bifurcations of phase portraits of system \eqref{2.12} in the phase plane 
$(\psi,y)$ as the parameters $\rho,\mu,\nu$ change. Since only physical models 
with bounded traveling waves are meaningful, we just focus on the bounded 
solutions of system \eqref{2.12}.

The paper is organized as follows. In Section 2, we discuss bifurcation curves 
and phase portraits of system \eqref{2.12}. In Section 3, we show the existence 
of solitary and periodic kink wave and compacton solutions of \eqref{2.10}. 
In Section 4, we present some exact explicit solutions for \eqref{2.10}.

\section{Bifurcation set and phase portraits of system \eqref{2.12}}

In this section, we will study all phase portraits and bifurcation set of 
system \eqref{2.12} in the parameter space. Let $d\xi=n\psi^{n-1}d\omega$. 
Thus, system  \eqref{2.12} becomes
\begin{equation} \label{2.20}
 \begin{gathered}
 d\psi/d\omega=n\psi^{n-1}y,\\
 dy/d\omega=n(n+1)\psi^{n-2}y^2+\rho(\psi^n-\mu\psi+\nu).\\
 \end{gathered}
\end{equation}
Except for the straight line $\psi=0$, systems \eqref{2.12} and \eqref{2.20} 
have the same first integral as \eqref{3.10}. Note that for a fixed $h$, 
equation \eqref{3.10} determines a set of invariant curves of system \eqref{2.20},
 which contains more different branches of curves. As $h$ changes, \eqref{3.10} 
defines different families of orbits of system \eqref{2.20} with different 
dynamical behavior.
We assume that $(\psi_i,y_i)$  is an equilibrium point of system \eqref{2.12}. 
At this point, the determinant of the linearized system of system \eqref{2.20} 
has the form 
\begin{equation}
J(\psi_i,y_i)=n^3(n+1)\psi^{2n-4}_iy^2_i-n\rho\psi^{n-1}_i(n\psi^{n-1}_i-\mu).
\end{equation}
By the bifurcation theory of dynamical system \cite{Andronov, Guckenheimer}, 
we know that if $J(\psi_i,y_i)>0\ (or<0)$ , then equilibrium point 
$(\psi_i,y_i)$  is a center (or saddle point); if $J(\psi_i,y_i)=0)$ and 
the Poincare index of $(\psi_i,y_i)$ is zero, then the equilibrium point 
$(\psi_i,y_i)$ is a cusp point. Denote $h_i=H(\psi_i,y_i)$, $h=H(\psi,y)$ 
defined by \eqref{3.10}, $M_i(\psi_i,0)$ and $M(\psi,0)$ are  equilibrium 
points of system \eqref{2.12} and $\psi_i<\psi_{i+1}$. 
$N_\pm(0,\pm1/6\sqrt{-6\rho\nu})$  are the equilibrium points on the straight 
line $\psi=0$. From the above qualitative analysis, we can obtain the bifurcation 
curves and phase portraits with the aid of mathematical software Maple.

\subsection{Bifurcation set and phase portraits of system \eqref{2.20} when $n=2$}

In this case, there are two bifurcation curves on the $(\mu,\nu)$-plane
\begin{equation} \label{conslaws}
\Pi_1 :\nu=0 , \quad
\Pi_2: \nu=\frac{1}{4}\mu^2,
\end{equation}
which divide the $(\mu,\nu)$-parameter plane into four different subregions
(see Figure \ref{fig1}).

\begin{figure}[ht]
  \begin{center}
\includegraphics[width=0.6\textwidth]{fig1}
  \end{center}
\caption{Bifurcation set and curves of system \eqref{2.20} for $n=2$. 
Where $\Pi^{\pm}_1=\{(\mu,\nu)|\mu>0(\mu<0),\nu=0\}$, 
$\Pi^{\pm}_2=\{(\mu,\nu)|\mu>0(\mu<0),\nu=\frac{1}{4}\mu^2\}$, 
$D_1:0<\Pi^+_1<\nu<\Pi^+_2$,  $D_2:\nu>\Pi_2>0$, $D_3: \Pi^-_1<\nu<\Pi^-_2$,  
$D_4:\nu<\Pi_1$.}
\label{fig1}
\end{figure}


By using the above facts to do qualitative analysis, we obtain the 
following results.

\begin{proposition}\label{prop1}  (see Figure \ref{fig2})
Suppose that $\rho>0$.

(1) For $(\mu,\nu)\in\Pi^\pm_1$, system \eqref{2.20} has two heteroclinic 
orbits connecting to the saddle points $ M_1(\psi_1,0)$ and $ O(0,0)$. 
In addition, there is a family of homoclinic orbits to $O$  
for $ h\in(-\infty, h_1)$.

(2)For $(\mu,\nu)\in (D_1 \cup D_3)$, system \eqref{2.20} has a 
homoclinic orbit connecting to the saddle point $ M_2(\psi_2,0)$ 
(or $M_1(\psi_1,0)$). In addition, there is a family of periodic orbits 
surrounding the center $M_1(orM_2)$ for $h\in(h_1, h_2)\ (or h\in(h_2, h_1))$.

(3) For $(\mu,\nu)\in\Pi^\pm_2$, system \eqref{2.20} has a cusp point.

(4) For $(\mu,\nu)\in D_4$, system \eqref{2.20} has four heteroclinic 
 orbits connecting to the saddle points $ M_1(\psi_1,0)$, $ M_2(\psi_2,0)$  
and $N_\pm(0,\pm1/6\sqrt{-6\rho\nu})$ respectively. In addition, there are 
two families of heteroclinic orbits to $ N_\pm(0,\pm1/6\sqrt{-6\rho\nu})$  
for $h \in(-\infty, h_1)$.
\end{proposition}

\begin{proposition}\label{prop2} (see Figure \ref{fig3}).
Suppose that $\rho<0$.

(1) For $(\mu,\nu)\in\Pi^\pm_1$, system \eqref{2.20} has a homoclinic 
orbit connecting to the saddle point $O(0,0)$. In addition, there are a family 
of homoclinic orbits to $O$ and a family of periodic orbits surrounding the 
center $M_1\ (or M_2)$ respectively for $h\in(h_1, 0)$.

(2) For $(\mu,\nu)\in (D_1 \cup D_3)$, system \eqref{2.20} has two 
heteroclinic orbits connecting to the saddle points $h\in(-\infty, h_1)$ and 
$M_1$ (or $M_2$) respectively. In addition, there are a family of periodic orbits 
surrounding the center $M_1$ (or $M_2$), a family of heteroclinic orbits connecting 
to the saddle points $ N_\pm(0,\pm1/6\sqrt{-6\rho\nu})$ for $h\in(h_2, 0)$ 
(or $h\in(h_1, 0)$) and a family of heteroclinic orbits connecting to the 
saddle points $ N_\pm(0,\pm1/6\sqrt{-6\rho\nu})$ for 
$h\in(-\infty, h_1)$ (or $h\in(-\infty, h_2)$) respectively. If $H(\psi,0)=h$ 
(here $M(\psi,0)$ is the saddle point) defined by \eqref{3.10} has a zero 
$\psi^*$ satisfying $0<\psi<\psi^*$ (or $0>\psi>\psi^*$), there exist a 
homoclinic orbit connecting to the saddle point $M$  and three heteroclinic 
orbits connecting to the saddle points $M$ and $ N_\pm(0,\pm1/6\sqrt{-6\rho\nu})$ 
respectively. Furthermore, there are a family of periodic orbits surrounding 
the center $M_2\ (or M_1)$ and a family of heteroclinic orbits to 
$ N_\pm(0,\pm1/6\sqrt{-6\rho\nu})$ for $h\in(h_2, h_1)$ (or $h\in(h_1, h_2)$)
 respectively.

(3) For $(\mu,\nu)\in\Pi^\pm_2$, system \eqref{2.20} has four heteroclinic 
orbits connecting to the saddle points $ N_\pm(0,\pm1/6\sqrt{-6\rho\nu})$  
and a cusp point $M(\psi,0)$ respectively. In addition, there are two families 
of heteroclinic orbits to $ N_\pm(0,\pm1/6\sqrt{-6\rho\nu})$  for 
$h\in(-\infty, h_1)$ and $h\in(h_1, 0)$ respectively.

(4) For $(\mu,\nu)\in D_4$, system \eqref{2.20} has two centers $M_1$ 
and $M_2$. When $h\in(h_2, 0)$ (or $h\in(h_1, 0)$, there are two families 
of periodic orbits surrounding the centers $M_2$ and $M_1$ respectively.

\end{proposition}

By the above analysis, we have the following phase portraits of system 
\eqref{2.12} under different parametric conditions shown in figures 
\ref{fig2} and \ref{fig3}. they are mae with the help of mathematical 
software Maple.

\begin{figure}[htpb]
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig2-1}
\includegraphics[width=0.32\textwidth, clip=true]{fig2-2}
\includegraphics[width=0.32\textwidth, clip=true]{fig2-3}\\
(1)\hfil
(2)\hfil
(3)\hfil
\end{center}
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig2-4}
\includegraphics[width=0.32\textwidth, clip=true]{fig2-5}
\includegraphics[width=0.32\textwidth, clip=true]{fig2-6}\\
(4) \hfil
(5) \hfil
(6) \hfil
\end{center}
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig2-7}\\
(7) \hfil
\end{center}
\caption{Phase portraits of \eqref{2.20} when $n=2$ for $\rho>0$.
 (1) $(\mu,\nu)\in \Pi^+_1 $; (2) $(\mu,\nu)\in D_1$; (3)
$(\mu,\nu)\in \Pi^+_2$; (4) $(\mu,\nu)\in \Pi^-_2$; (5) $(\mu,\nu)\in D_3$; 
(6) $(\mu,\nu)\in \Pi^-_1 $;  (7) $(\mu,\nu)\in D_4 $.}
\label{fig2}
\end{figure}

\begin{figure}[htpb]
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig3-1}
\includegraphics[width=0.32\textwidth, clip=true]{fig3-2}
\includegraphics[width=0.32\textwidth, clip=true]{fig3-3}\\
(1)\hfil
(2)\hfil
(3)\hfil
\end{center}
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig3-4}
\includegraphics[width=0.32\textwidth, clip=true]{fig3-5}
\includegraphics[width=0.32\textwidth, clip=true]{fig3-6}\\
(4) \hfil
(5) \hfil
(6) \hfil
\end{center}
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig3-7}
\includegraphics[width=0.32\textwidth, clip=true]{fig3-8}
\includegraphics[width=0.32\textwidth, clip=true]{fig3-9}\\
(7) \hfil
(8) \hfil
(9) \hfil
\end{center}
\caption{Phase portraits of \eqref{2.20} when $n=2$ for $\rho<0$.
 (1) $(\mu,\nu)\in \Pi^+_1 $; (2) $(\mu,\nu)\in D_1$; (3) $(\mu,\nu)\in D_1$;
 (4) $(\mu,\nu)\in \Pi^+_2$; (5) $(\mu,\nu)\in \Pi^-_2$; 
 (6) $(\mu,\nu)\in D_3$; (7) $(\mu,\nu)\in D_3$; 
 (8) $(\mu,\nu)\in \Pi^-_1 $; (9) $(\mu,\nu)\in D_4 $.}
\label{fig3}
\end{figure}

\subsection{Bifurcation set and phase portraits of system \eqref{2.20} when 
$n=2k$ $(k>1)$}

In this case, there are two bifurcation curves on the $(\mu,\nu)$-parameter plane:
\begin{equation}
\label{conslaws2}
\Gamma_1: \nu=0 , \quad
\Gamma_2: \nu=[(\frac{1}{2k})^{\frac{1}{2k-1}}-(\frac{1}{2k})^{\frac{2k}{2k-1}}]
\mu^{\frac{2k}{2k-1}},
\end{equation}
which divide the $(\mu,\nu)$-parameter plane into four different subregions 
(see Figure \ref{fig4}).

\begin{figure}[ht]
  \begin{center}
\includegraphics[width=0.45\textwidth]{fig4}
  \end{center}
\caption{Bifurcation set and curves of system \eqref{2.20} 
for $n=2k,k>1$. Where $\Gamma^{\pm}_1=\{(\mu,\nu)|\mu>0(\mu<0),\nu=0\}$, 
$\Gamma^{\pm}_2=\{(\mu,\nu)|\mu>0(\mu<0),\nu=[\frac{1}{(2k)^{\frac{1}{2k-1}}}
-\frac{1}{(2k)^{\frac{2k}{2k-1}}}]\mu^{\frac{2k}{2k-1}}\}$, 
$G_1:0<\Gamma^+_1<\nu<\Gamma+_2$,  $G_2:\nu>\Gamma_2>0$, 
$G_3: \Gamma^-_1<\nu<\Gamma^-_2$,  $G_4:\nu<\Gamma_1$.}
\label{fig4}
\end{figure}
By applying the above facts to do qualitative analysis, we obtain the following 
results.

\begin{proposition}\label{prop3}
Suppose that $\rho>0$ (see Figure \ref{fig5}).

(1) For $(\mu,\nu)\in\Gamma^\pm_1$, system \eqref{2.20} has one saddle point. 
 system \eqref{2.20} has two heteroclinic orbits connecting to the saddle points 
 $ M_1(\psi_1,0)$ and $ O(0,0)$. In addition, there is a family of homoclinic 
 orbits to $O$  for $ h\in(-\infty, h_1)$ .

(2) For $(\mu,\nu)\in (G_1 \cup G_3)$, system \eqref{2.20} has a 
 homoclinic orbit connecting to the saddle point $ M_2(\psi_2,0)$ 
(or $M_1(\psi_1,0)$). In addition, there is a family of periodic orbits surrounding 
the center $M_1$ (or $M_2$) for $h\in(h_1, h_2)$ (or $h\in(h_2, h_1)$).

(3) For $(\mu,\nu)\in\Gamma^\pm_2$, system \eqref{2.20} has a cusp point.

(4) For $(\mu,\nu)\in G_4$, system \eqref{2.20} has two saddle points.

\end{proposition}

\begin{proposition}\label{prop4}
Suppose that $\rho<0$ (see Figure \ref{fig6}).

(1) For $(\mu,\nu)\in\Gamma^\pm_1$, system \eqref{2.20} has a center $M_1$. 
 In addition, there are a family of periodic orbits surrounding the center  $M_1$  
 for $h\in(h_1, 0)$.

(2) For $(\mu,\nu)\in (G_1 \cup G_3)$, system \eqref{2.20} has one saddle 
 point and one center. In addition, there are a family of periodic orbits 
 surrounding the center for $h\in(h_2, 0)$ (or $h\in(h_1, 0)$). If $H(\psi,0)=h$ 
(here $M(\psi,0)$ is the saddle point) defined by \eqref{3.10} has a zero 
 $\psi^*$ satisfying $0<\psi<\psi^*$ (or $0>\psi>\psi^*$),  there exists a 
 homoclinic orbit connecting to the saddle point $M$ , and there are a family 
 of periodic orbits surrounding the center for $h\in(h_2, h_1)$ 
 (or $h\in(h_1, h_2)$).

(3) For $(\mu,\nu)\in\Gamma^\pm_2$,  system \eqref{2.20} has a cusp point 
 $M(\psi,0)$.

(4) For $(\mu,\nu)\in G_4$, system \eqref{2.20} has two centers $M_1$ and 
 $M_2$. When $h\in(h_1, 0)$ (or $h\in(h_2, 0)$), there are two families of 
periodic orbits surrounding the centers $M_1$ and $M_2$ respectively.

\end{proposition}

By means of the above analysis, we have the following phase portraits of 
system \eqref{2.20} under different parametric conditions shown in figures
 \ref{fig5} and \ref{fig6}. they were made  with the aid of Mathematical 
software Maple.

\begin{figure}[htpb]
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig5-1}
\includegraphics[width=0.32\textwidth, clip=true]{fig5-2}
\includegraphics[width=0.32\textwidth, clip=true]{fig5-3}\\
(1)\hfil
(2)\hfil
(3)\hfil
\end{center}
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig5-4}
\includegraphics[width=0.32\textwidth, clip=true]{fig5-5}
\includegraphics[width=0.32\textwidth, clip=true]{fig5-6}\\
(4) \hfil
(5) \hfil
(6) \hfil
\end{center}
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig5-7}\\
(7) \hfil
\end{center}
\caption{Phase portraits of \eqref{2.20} when $n=2k$, $k>1$ for $\rho>0$.
 (1) $(\mu,\nu)\in \Gamma^+_1 $; (2) $(\mu,\nu)\in G_1$; 
(3) $(\mu,\nu)\in \Gamma^+_2$; (4) $(\mu,\nu)\in \Gamma^-_2$; 
(5) $(\mu,\nu)\in G_3$; (6) $(\mu,\nu)\in \Gamma^-_1$; 
(7) $(\mu,\nu)\in G_4$.}
\label{fig5}
\end{figure}

\begin{figure}[htpb]
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig6-1}
\includegraphics[width=0.32\textwidth, clip=true]{fig6-2}
\includegraphics[width=0.32\textwidth, clip=true]{fig6-3}\\
(1)\hfil
(2)\hfil
(3)\hfil
\end{center}
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig6-4}
\includegraphics[width=0.32\textwidth, clip=true]{fig6-5}
\includegraphics[width=0.32\textwidth, clip=true]{fig6-6}\\
(4) \hfil
(5) \hfil
(6) \hfil
\end{center}
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig6-7}
\includegraphics[width=0.32\textwidth, clip=true]{fig6-8}
\includegraphics[width=0.32\textwidth, clip=true]{fig6-9}\\
(7) \hfil
(8) \hfil
(9) \hfil
\end{center}
\caption{Phase portraits of \eqref{2.20} when $n=2k$, $k>1$ for $\rho<0$.
 (1) $(\mu,\nu)\in \Gamma^+_1 $; (2) $(\mu,\nu)\in G_1$;
 (3) $(\mu,\nu)\in G_1$; (4) $(\mu,\nu)\in \Gamma^+_2$; 
 (5) $(\mu,\nu)\in \Gamma^-_2$; (6) $(\mu,\nu)\in G_3$; 
 (7) $(\mu,\nu)\in G_3$; (8) $(\mu,\nu)\in \Gamma-_1 $; (9) $(\mu,\nu)\in G_4 $.}
\label{fig6}
\end{figure}

\subsection{Bifurcation set and phase portraits of system \eqref{2.20} when 
$n=2k+1$ ($k\geq1$)}


In this case, there are two bifurcation curves on the $(\mu,\nu)$-parameter plane:
\begin{equation}  \label{conslaws3}
\Upsilon_1: \nu=0 , \quad
\Upsilon_2: \nu=[(\frac{1}{2k+1})^{\frac{1}{2k}}-(\frac{1}{2k+1})^{\frac{2k+1}{2k}}]
\mu^{\frac{2k+1}{2k}},
\end{equation}
which divide the $(\mu,\nu)$-parameter plane into four different subregions
(see Figure \ref{fig7}).

\begin{figure}[ht]
  \begin{center}
\includegraphics[width=0.51\textwidth]{fig7}
  \end{center}
\caption{Bifurcation set and curves of \eqref{2.20} for 
$n=2k+1(k\geq1)$. $\Upsilon^{\pm}_1=\{(\mu,\nu)|\mu>0(\mu<0),\nu=0\}$,
 $\Upsilon^{\pm}_2=\{(\mu,\nu)|\mu>0(\mu<0),\nu=[\frac{1}{(2k+1)^{\frac{1}{2k}}}
-\frac{1}{(2k+1)^{\frac{2k+1}{2k}}}]\mu^{\frac{2k+1}{2k}}\}$, 
$Z_1:0<\Upsilon^+_1<\nu<\Upsilon^+_2$,  $Z_2:\nu>\Upsilon^+_2>0$, 
$Z_3:\nu<\Upsilon^-_2<0$,  $Z_4:\Upsilon^-_2<\nu<\Upsilon^+_1$.}
\label{fig7}
\end{figure}
By using the above facts to do qualitative analysis, we obtain the following results.

\begin{proposition}\label{prop5}
Suppose that $\rho>0$ (see Figure \ref{fig8}).

(1) For $(\mu,\nu)\in\Upsilon^-_1$, system \eqref{2.20} has two 
 saddle points on the axis of abscissa and two saddle points on the axis
  of ordinates. And there exist four heteroclinic orbits connecting to the 
 saddle points $ M_1(\psi_1,0)$, $M_2(\psi_2,0)$ and 
 $ N_\pm(0,\pm1/6\sqrt{-6\rho\nu})$ respectively, for $h=h_1$ (or $h_2$).

(2) For $(\mu,\nu)\in(Z_1\cup Z_4)$, system \eqref{2.20} has two saddle points
 $ M_1(\psi_1,0)$, $M_3(\psi_3,0)$, and one center $ M_2(\psi_2,0)$. And there 
exist a homoclinic orbit connecting to the saddle point $M_1$ or $M_3$ for 
$h=h_1$  (or $h_3)$ and a family of periodic orbits surrounding the center 
$M_2$ for $h\in(h_2, h_1)\ (or h\in(h_2, h_3))$.

(3) For $(\mu,\nu)\in\Upsilon^\pm_2$, system \eqref{2.20} has a saddle 
point and a cusp point.

(4) For $(\mu,\nu)\in(Z_2\cup Z_3)$, system \eqref{2.20} has a saddle point.

\end{proposition}


\begin{proposition} \label{prop6} (see Figure \ref{fig9}).
Suppose that $\rho<0$ 

(1) For $(\mu,\nu)\in\Upsilon^+_1$,  system \eqref{2.20} has two centers. 
 In addition, there are two families of periodic orbits surrounding the centers 
 $M_1\ (or M_2)$ for $h \in(h_1, 0)$, respectively.

(2) For $(\mu,\nu)\in(Z_1 \cup Z_4)$, system \eqref{2.20} has one saddle 
 point and two centers. In addition, there are two families of periodic orbits 
 surrounding the two centers for $h\in(h_1, 0)$ and $h\in(h_3, 0)$  respectively. 
 If $H(\psi,0)=h$ (here $M(\psi,0)$ is the saddle point) defined by \eqref{3.10} 
 has a zero $\psi^*$ satisfying $0<\psi^*<\psi$ (or $0>\psi^*>\psi$), there exists 
 a homoclinic orbit connecting to the saddle point $M$.

(3) For $(\mu,\nu)\in\Upsilon^\pm_2$, system \eqref{2.20} has a cusp point 
 and a center. When $h\in(h_1,0)$  (or $h\in(h_2, 0)$), there exist a family of 
 periodic orbits surrounding the center.

(4) For $(\mu,\nu)\in(Z_2 \cup Z_3)$, system \eqref{2.20} has one center. 
When $h\in(h_1, 0)$, there is a family of periodic orbits surrounding the center.

\end{proposition}

From the above analysis, we have the following phase portraits of 
system \eqref{2.12} under different parametric conditions shown in 
figures \ref{fig8} and \ref{fig9}.

\begin{figure}[htpb]
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig8-1}
\includegraphics[width=0.32\textwidth, clip=true]{fig8-2}
\includegraphics[width=0.32\textwidth, clip=true]{fig8-3}\\
(1)\hfil
(2)\hfil
(3)\hfil
\end{center}
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig8-4}
\includegraphics[width=0.32\textwidth, clip=true]{fig8-5}
\includegraphics[width=0.32\textwidth, clip=true]{fig8-6}\\
(4) \hfil
(5) \hfil
(6) \hfil
\end{center}
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig8-7}\\
(7) \hfil
\end{center}
\caption{Phase portraits of \eqref{2.20} when $n=2k+1,k\geq1$ for $\rho>0$.
 (1) $(\mu,\nu)\in\Upsilon^+_1 $; (2) $(\mu,\nu)\in Z_1$; 
(3) $(\mu,\nu)\in\Upsilon^+_2$; (4) $(\mu,\nu)\in Z^-_2$; 
(5) $(\mu,\nu)\in Z_3$; (6) $(\mu,\nu)\in\Upsilon^-_2$; (7) $(\mu,\nu)\in Z_4$.}
\label{fig8}
\end{figure}

\begin{figure}[htpb]
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig9-1}
\includegraphics[width=0.32\textwidth, clip=true]{fig9-2}
\includegraphics[width=0.32\textwidth, clip=true]{fig9-3}\\
(1)\hfil
(2)\hfil
(3)\hfil
\end{center}
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig9-4}
\includegraphics[width=0.32\textwidth, clip=true]{fig9-5}
\includegraphics[width=0.32\textwidth, clip=true]{fig9-6}\\
(4) \hfil
(5) \hfil
(6) \hfil
\end{center}
\begin{center}
\includegraphics[width=0.32\textwidth, clip=true]{fig9-7}
\includegraphics[width=0.32\textwidth, clip=true]{fig9-8}
\includegraphics[width=0.32\textwidth, clip=true]{fig9-9}\\
(7) \hfil
(8) \hfil
(9) \hfil
\end{center}
\caption{Phase portraits of \eqref{2.20} when $n=2k+1,k\geq1$ for $\rho<0$.
(1) $(\mu,\nu)\in\Upsilon^+_1 $; (2) $(\mu,\nu)\in Z_1$; 
(3) $(\mu,\nu)\in Z_1$; (4) $(\mu,\nu)\in\Upsilon^+_2$; 
(5) $(\mu,\nu)\in Z_2$; (6) $(\mu,\nu)\in Z_3$; 
(7) $(\mu,\nu)\in\Upsilon^-_2$; (8) $(\mu,\nu)\in Z_4 $; (9) $(\mu,\nu)\in Z_4 $.}
\label{fig9}
\end{figure}

\section{Existence of traveling wave solutions of equation \eqref{2.10})}

In this section, we consider the existence of smooth and non-smooth solitary 
traveling wave and periodic traveling wave solutions of equation \eqref{2.10}.
 Obviously, the system \eqref{2.12} has the same orbits as the system \eqref{2.20}, 
except for $\psi=0$ . The transformation of variables $d\xi=n\psi^{n-1}d\omega$ 
only derives the difference of the parametric representations of orbits of the 
systems \eqref{2.12} and \eqref{2.20} when $\psi=0$. If an orbit of \eqref{2.20} 
has no intersection point with the straight line  $\psi=0$, then, it is well 
defined in \eqref{2.12}. It follows that the profile defined by this orbit  
on the $(\psi,y)$-plane is smooth. If an orbit of \eqref{2.20} has intersection 
point with the straight line $\psi=0$, then it is not defined in \eqref{2.12}. 
It follows that  the profile defined by this orbit on the $(\psi,y)$-plane may 
be non-smooth.

According to the previous discussions, we deduce the following conclusions  
from figures \ref{fig2}, \ref{fig3}, \ref{fig5}, \ref{fig6}, \ref{fig8} and \ref{fig9}.


\begin{theorem}\label{thm1} (see figures \ref{fig2} and  \ref{fig3})
Suppose that $n=2$.

(1) For $(\mu,\nu)\in\Pi^\pm_1$. When $\rho>0$, equation \eqref{2.10} 
has a couple of kink and anti-kink wave solutions for $h=h_1(h_2)$, and has 
a family of uncountably infinite many compactons solutions for $h\in(-\infty, h_1)$ 
(or $h\in(-\infty, h_2)$). And when $\rho<0$ , equation \eqref{2.10} has a 
compacton solution for  $h=h_1$, and has a family of uncountably infinite many 
compactons solutions and a family of smooth periodic wave solutions whose 
amplitudes tend to $\infty$  for $h\in(h_1, 0)$.

(2) For $(\mu,\nu)\in (D_1 \cup D_3)$. When $\rho>0$, equation \eqref{2.10}
 has a smooth solitary wave solutions with valley (peak) form for $h=h_2(h_1)$,
 and has a family of uncountably infinite many smooth periodic wave solutions 
 for $h\in(h_1, h_2)\ (or\ h\in(h_2, h_1))$. And when $\rho<0$,
 equation \eqref{2.10} has a smooth solitary wave solutions with valley (peak) 
form for $h=h_1$ (or $h_2)$, and has a family of uncountably infinite many 
smooth periodic wave solutions and a family of uncountably infinite many 
solitary wave solutions with valley (peak) wave solutions for $h\in(h_1, 0)$
 (or $h\in(h_2, 0)$) respectively and has a family of uncountably infinite many 
 periodic wave solutions for $h\in(-\infty, h_1)$ (or $h\in(-\infty, h_2)$); 
if $H(\psi,0)=h$ (here $M(\psi,0)$ is the saddle point) defined by \eqref{3.10} 
has a zero $\psi^*$ satisfying $0<\psi<\psi^*$ (or $0>\psi>\psi^*$, then 
equation \eqref{2.10} has a couple of solitary wave solutions with peak and 
valley form for $h=h_2(h_1)$, and has a family of uncountably infinite many 
smooth periodic wave solutions and a family of uncountably infinite many solitary
 wave solutions with peak (valley) form for $h\in(h_2, 0)$ (or $h\in(h_1, 0))$ 
respectively, and has a family of uncountably infinite many periodic wave solutions 
form for $ h\in(-\infty, h_1)$ or $ h\in(-\infty, h_2)$.

(3) For $(\mu,\nu)\in\Pi^\pm_2$. When $\rho<0$, equation \eqref{2.10} 
has a solitary wave solution with valley (or peak) form for $h=h_1$, and has 
a family of uncountably infinite many periodic wave solutions for $h\in(h_1, 0)$, 
and has a family of uncountably infinite many periodic wave solutions form for 
$ h\in(-\infty, h_1)$.

(4) For $(\mu,\nu)\in D_4$, When $\rho>0$, equation \eqref{2.10} has a 
couple of kink and anti-kink wave solutions for $h=h_2(h_1)$, and has a family
 of uncountably infinite many periodic wave solutions for $h\in(-\infty, h_1)$. 
And when $\rho<0$, equation \eqref{2.10} has two families of uncountably infinite
 many smooth periodic wave solutions for $h\in(h_2, 0)$, and their amplitudes 
tend to $\infty$ for $h\to 0$.

\end{theorem}

\begin{theorem}\label{thm2} (see figures \ref{fig5} and \ref{fig6}).
Suppose that $n=2k$.

(1) For $(\mu,\nu)\in\Gamma^\pm_1$. When $\rho<0$, equation \eqref{2.10} 
has a family of uncountably infinite many periodic solutions for $h\in(h_1, 0)$, 
and their amplitudes tend to $\infty$ for $h\to 0$.

(2) For $(\mu,\nu)\in G_1$. When $\rho>0$, equation \eqref{2.10} has a smooth 
solitary wave solutions with valley form for $h=h_2$  , and has a family of 
uncountably infinite many smooth periodic solutions for $h\in(h_1, h_2)$. 
And when $\rho<0$, equation \eqref{2.10} has a family of uncountably 
infinite many smooth periodic solutions for $h\in(h_2, 0)$, and their amplitudes 
tend to $\infty$ for $h\to 0$; if  $H(\psi_1,0)=h$  (here $M(\psi_1,0)$  is the 
saddle point) defined by \eqref{3.10} has a zero $\psi^*$ satisfying 
$0<\psi_1<\psi^*$, equation \eqref{2.10} has a smooth solitary wave solutions
 with peak form for $h=h_1$ and a family of uncountably infinite many 
smooth periodic solutions for $h\in(h_2, h_1)$.

(3) For $(\mu,\nu)\in G_3$. When $\rho>0$, equation \eqref{2.10} has a smooth 
solitary wave solutions with peak form for $h=h_1$, and has a family of 
uncountably infinite many smooth periodic solutions for $h\in(h_2, h_1)$.
 And when $\rho<0$, equation \eqref{2.10} has a family of uncountably infinite 
many smooth periodic solutions for $h\in(h_1, 0)$, and their amplitudes tend to 
$\infty$ for $h\to 0$; if $H(\psi_2,0)=h$ (here $M(\psi_2,0)$ is the saddle point) 
defined by \eqref{3.10} has a zero $\psi^*$ satisfying $0>\psi_2>\psi^*$, 
equation \eqref{2.10} has a solitary wave solutions with valley form for $h=h_2$ 
and a family of uncountably infinite many smooth periodic solutions for 
$h\in(h_1, h_2)$.

(4) For $(\mu,\nu)\in G_4$. When $\rho<0$, equation \eqref{2.10} has two 
families of uncountably infinite many smooth periodic solutions for $h\in(h_1, 0)$, 
and their amplitudes tend to $\infty$ for $h\to 0$.

\end{theorem}


\begin{theorem}\label{thm3} (see figures \ref{fig8} and \ref{fig9}).
Suppose that $n=2k+1, k\geq1$.

(1) For $(\mu,\nu)\in\Upsilon^+_1$. When $\rho>0$, equation \eqref{2.10} 
 has a couple of kink and anti-kink wave solutions for $h=h_2$ (or $h_1)$, 
and has a family of uncountably infinite many periodic wave solutions for 
$h\in (h_1, +\infty)$. And when $\rho<0$, equation \eqref{2.10} has two 
families of uncountably infinite many smooth periodic wave solutions for 
$h\in(h_1, 0)$, and their amplitudes tend to $\infty$ for $h\to 0$.

(2) For $(\mu,\nu)\in (Z_1 \cup Z_4)$. When $\rho>0$, equation \eqref{2.10} 
has a smooth solitary wave solutions with valley (peak) form for  $h=h_3$ 
(or $h_1)$, and has a family of uncountably infinite many smooth periodic 
solutions for $h\in(h_2, h_3)(or h\in(h_2, h_1))$. And when $\rho<0$, equation 
\eqref{2.10} has two families of uncountably infinite many smooth periodic 
solutions for $h\in(h_2, 0)$ and $h\in(h_3, 0)$ respectively, and their 
amplitudes tend to $\infty$ for $h\to 0$; if $H(\psi_2,0)=h$ 
(here $M(\psi_2,0)$ is the saddle point) defined by \eqref{3.10} has a zero 
$\psi^*$ satisfying $0<\psi_2<\psi^*$ (or $0>\psi_2>\psi^*$), equation \eqref{2.10} 
has a smooth solitary wave solutions with peak (or valley) form for $h=h_2$ 
and two families of uncountably infinite many smooth periodic solutions 
for $h\in(h_3, h_2)$ and $h\in(h_1, 0)$ respectively.

(3) For $(\mu,\nu)\in\Upsilon^\pm_2$. When $\rho<0$, equation \eqref{2.10} 
has a family of uncountably infinite many smooth periodic solutions for 
$h\in(h_2, 0)$ (or $h\in(h_1, 0)$), and their amplitudes tend to $\infty$ for 
$h\to 0$.

(4) For $(\mu,\nu)\in(Z_2 \cup Z_3)$. When $\rho<0$, equation \eqref{2.10} 
has two families of uncountably infinite many smooth periodic solutions for 
$h\in(h_1, 0)$, and their amplitudes tend to $\infty$ for $h\to 0$.

\end{theorem}

\section{Exact traveling wave solutions of equation \eqref{2.10}}

In this section, to further reveal above results, we provide some exact 
solutions of equation \eqref{2.10} for $n=2$ by the bifurcation theory. 
We denote that $h=H(\psi, 0)$ and $M(\psi, 0)$ are equilibrium points 
of \eqref{2.12}.

1. The case of $\rho>0$ and $h=0$:

(1) When $\nu=0$, corresponding to figures \ref{fig2}(1) and \ref{fig2}(6) 
respectively, equation \eqref{2.10} has the following compacton solutions
\[
u(x, t)=\begin{cases}
\frac{4}{5}\mu\cos[\frac{1}{4}\sqrt{\rho}(x-ct)], 
& |x-ct|<\frac{2\pi}{\sqrt{\rho}},\\
0, & \text{otherwise.} \end{cases}
\]

(2) When $\mu=0$ and $\nu<0$, corresponding to Figure \ref{fig2}(7),
equation \eqref{2.10} has the following  periodic solutions
\[
u(x,t)=\pm\sqrt{\frac{-2\nu}{3}}\sin[\frac{\sqrt{\rho}}{2}(x-ct)].
\]

2. The case of $\rho<0$ and $h=0$:

(1) When $\nu=0$, corresponding to figures \ref{fig3}(7) and \ref{fig3}(8), 
equation \eqref{2.10} has the following explicit formula of solitary patterns 
solutions
\[
u(x,t))=\frac{2}{5}\mu\{1+\cosh[\frac{\sqrt{-\rho}}{2}(x-ct)]\}.
\]

(2) When $\mu=0$ and $\nu<0$, corresponding to Figure \ref{fig3}(9),
equation \eqref{2.10} has the following solitary patterns solutions
\[
u(x,t))=\sqrt{\frac{-2\nu}{3}}\cosh[\frac{\sqrt{-\rho}}{2}(x-ct)].
\]

\subsection*{conclusion}

Employing the bifurcation theory of nonlinear dynamic system, 
we have studied the bifurcations and dynamic behaviors of traveling 
wave solutions of   equation \ref{2.10}. The obtained results show 
that equation \eqref{2.10} has infinite many periodic wave, solitary wave, 
kink wave and compacton solutions under some parameters' conditions. 
Therefore, the results in this work clearly demonstrate the effect of 
the purely nonlinear dispersion and the qualitative change made in the
 genuinely nonlinear phenomenon.

\subsection*{Acknowledgements}
The authors would like to thank the anonymous referees for their
carefully reading of the original manuscript, and for their valuable
comments and suggestions for improving the results as well as the
exposition of this article.

This work is supported by the National Natural Science Foundation
of China (Grants nos. 11302148, 11302158 and  11102132)
and the National  Natural Science Foundation of Tianjin
(Grant no. 12JCYBJC10600) and Natural Science Foundation of 
Tianjin University of Technology and Education (no. KYQD09006),
and NSFC Scholarship project foundation of China.

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\end{document}