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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 42, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2017/42\hfil Semiconductors with strong spatial dispersion]
{Symmetry analysis, bifurcation and exact solutions of nonlinear
wave equation in semiconductors with strong spatial dispersion}
\author[B. Gao, C. He \hfil EJDE-2017/42\hfilneg]
{Ben Gao, Chunfang He}
\address{Ben Gao \newline
College of Mathematics, Taiyuan University of Technology,
Taiyuan 030024, China}
\email{benzi0116@163.com}
\address{Chunfang He \newline
College of Mathematics,
Taiyuan University of Technology,
Taiyuan 030024, China}
\email{lxmath123@163.com}
\thanks{Submitted December 28, 2016. Published Febrary 8, 2017.}
\subjclass[2010]{35J05, 35E05, 43A80, 26A18}
\keywords{Symmetry analysis; bifurcation; exact solutions;
\hfill\break\indent nonlinear wave equation in semiconductors}
\begin{abstract}
Based on Lie symmetry analysis and steady bifurcation method, we study
the nonlinear wave equation in semiconductors with strong spatial dispersion.
The similarity reductions and exact solutions are obtained based on the
optimal system and power series method. Then, steady bifurcation and
solitary waves are presented. Especially, the existence and solvability of
solitary or period wave are discussed, and all kinds of the solitary and
period wave solutions are given by direct integration.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\allowdisplaybreaks
\section{Introduction}
The celebrated nonlinear wave model called Korteweg
de-Vries (KdV) equation \cite{c1} is
\begin{equation}\label{}
u_t+uu_x+u_{xxx}=0.
\end{equation}
This equation can be used in shallow water waves models, and a lot of other fields
such as fluid mechanics, optical fibers, electromagnetic waves, acoustic waves
in plasmas and so on \cite{c2}.
Several noticeable attempts to improve the KdV model were taken over the years.
In 2008, Al'shin et al \cite{c3} give the following nonlinear wave equation,
known as the improved KdV equation, and is based on the theory of electromagnetism.
\begin{equation}\label{a1}
(u-u_{xx})_t+uu_x+u_{xxx}=0,
\end{equation}
which describes waves in semiconductors with strong spatial dispersion.
This equation can be derived from the nonstationary processes in semiconductors
that are described by systems consisting of stationary field equation,
continuity equation and constitutive equation.
1-soliton solution and conservation laws of \eqref{a1} are given by Anjan Biswas
and Kara\cite{c4}.
The main purpose of this work is to study the symmetry analysis,
bifurcation and exact solutions of \eqref{a1}.
The similarity reductions and exact solutions are obtained based on the optimal
system and power series method.
Steady bifurcation and solitary waves are presented based on the ideas in
\cite{c9,c5,c6,c7,c8,c10,c101}. Our result may be of great interest for
both mathematician and physicist.
The rest of this article is organized as follows.
In Section \ref{sec2}, the vector fields and the optimal systems are
obtained by employing Lie symmetry analysis method.
In Section \ref{sec3}, the similarity reductions and exact solutions are obtained.
In Section \ref{sec4}, steady bifurcation are analyzed by selecting the
integration constant as the bifurcation control parameter, then the existence
and solvability of solitary or period wave are discussed.
Conclusions and remarks are presented at the end of the paper.
\section{Lie symmetry analysis of \eqref{a1}} \label{sec2}
In this section, we perform Lie symmetry analysis of \eqref{a1}.
Lie symmetry analysis method is described in many books, e.g.\ \cite{c11,c12}.
First of all, let us consider a one-parameter group of infinitesimal
transformation:
\begin{equation}\label{a2}
\begin{gathered}
\overline{t}= t+\epsilon\tau(x,t,u)+O(\epsilon^2), \\
\overline{x} = x+\epsilon\xi(x,t,u)+O(\epsilon^2), \\
\overline{u}=u+\epsilon\eta(x,t,u)+O(\epsilon^2),
\end{gathered}
\end{equation}
where $\epsilon$ is a group parameter. The vector field associated
with the above group of transformations can be written as
\begin{equation}\label{a3}
V=\tau(x,t,u)\frac{\partial}{\partial t}+\xi(x,t,u)
\frac{\partial}{\partial x}+\eta(x,t,u)\frac{\partial}{\partial u}.
\end{equation}
Applying the third prolongation $Pr^{(3)}V$ to \eqref{a1}, we find that
the coefficient functions $\tau(x,t,u), \xi(x,t,u)$ and $ \eta(x,t,u) $ must
satisfy the invariant condition
\begin{equation}\label{a4}
\eta^{t}+u\eta^x+u_x\eta+\eta^{xxx}-\eta^{xxt}=0,
\end{equation}
where
\begin{equation}\label{a5}
\begin{gathered}
\eta^t=D_t(\eta)-u_tD_t(\tau)-u_xD_t(\xi), \\
\eta^x=D_x(\eta)-u_tD_x(\tau)-u_xD_x(\xi), \\
\eta^{xxx}=D_x(\eta^{xx})-u_{xxt}D_x(\tau)-u_{xxx}D_x(\xi),\\
\eta^{xxt}=D_t(\eta^{xx})-u_{xtt}D_t(\tau)-u_{xxt}D_t(\xi).
\end{gathered}
\end{equation}
Here, $D_x$, $D_t$ denote the total derivative operators with respect to $x$ and $t$,
respectively.
Substituting \eqref{a5} in the invariant condition \eqref{a4}, one can get
\begin{equation}\label{a6}
\xi=-c_1t+c_3,\quad \tau=c_1t+c_2,\quad \eta=-c_1-c_1u,
\end{equation}
where $c_1$, $c_2$ and $c_3$ are arbitrary constants.
Hence the Lie algebra of infinitesimal symmetries of \eqref{a1} is spanned
by the vector fields
\begin{equation}\label{a7}
V_1=-t\frac{\partial}{\partial x}+t\frac{\partial}{\partial t}
-(1+u)\frac{\partial}{\partial u},\quad V_2=\frac{\partial}{\partial t},\quad
V_3=\frac{\partial}{\partial x}.
\end{equation}
Then, all of the infinitesimal generators of \eqref{a1} can be expressed as
\begin{equation}\label{a8}
V=c_1V_1+c_2V_2+c_3V_3.
\end{equation}
The commutation relations of Lie algebra determined by $V_1,V_2,V_3$, are shown
in Table 1. It is obvious that $\{V_1,V_2,V_3\}$ is commute under the Lie bracket.
\begin{table}[ht]
\caption{ Commutation table of Lie algebra}\label{table1}
\begin{center}
\begin{tabular}{|cccc|}
\hline
$[V_i,V_j]$ & $V_1$ & $V_2$ & $V_3$\\\hline
$V_1$ & 0 & $V_3-V_2$ & 0\\
$V_2$ & $V_2-V_3$ & 0 &0 \\
$V_3$ &0 & 0& 0 \\\hline
\end{tabular}
\end{center}
\end{table}
To compute the adjoint representation, we use the commutation Table
\ref{table1} and following Lie series
\begin{equation}\label{a9}
Ad(\exp(\varepsilon V_i))V_j=V_j-\varepsilon[V_i,V_j]
+\frac{1}{2}\varepsilon^2[V_i,[V_i,V_j]]+\cdots,
\end{equation}
we have the following results
\begin{equation}\label{a10}
\begin{gathered} Ad(\exp(\varepsilon V_i))V_i=V_i, \quad i=1,2,3\\
Ad(\exp(\varepsilon V_1))V_2=(1+\varepsilon)V_2-\varepsilon V_3, \quad
Ad(\exp(\varepsilon V_1))V_3=V_3, \\
Ad(\exp(\varepsilon V_2))V_1=V_1-\varepsilon(V_2-V_3),\quad
Ad(\exp(\varepsilon V_2))V_3=V_3,\\
Ad(\exp(\varepsilon V_3))V_1=V_1,\quad Ad(\exp(\varepsilon V_3))V_2=V_2.
\end{gathered}
\end{equation}
Based on the adjoint representations of the vector fields, we obtain the optimal
systems of the \eqref{a1} as follows:
\begin{equation}\label{}
\{V_1+V_2,\; V_1+V_3\}
\end{equation}
\section{Symmetry reductions and power series solutions}\label{sec3}
In the previous section, we obtained the vector fields and the optimal system of
\eqref{a1}. In this section, we will deal with the symmetry reductions and exact
solutions based on the optimal system and power series method.
\subsection{Generator $V_1+V_2$}
The similarity variables are $\xi=(t+1)e^{-t-x}, f(\xi)=(t+1)u+t$, and the
group-invariant solution is $u=\frac{f(\xi)-t}{t+1}$. Substituting the
group-invariant solution in \eqref{a1}, we obtain the following reduction equation
\begin{equation}\label{a11}
1-f-\xi ff'-\xi^3f'''-2\xi^2f''=0,
\end{equation}
where $f'=\frac{df}{d\xi}$.
we can not get exact solutions of reduction equation \eqref{a11} by using
the elementary functions or some already well known mathematical functions,
but we know that the power series can be used to deal with differential equations,
including many complicated nonlinear differential equations with nonconstant
coefficients\cite{c13}. Next, we will consider the exact analytic solution of
the reduction equation \eqref{a11} by using the power series method.
We seek a solution of \eqref{a11} in the power series of the form
\begin{equation}\label{a12}
f(\xi)=\sum_{n=0}^{\infty}c_n\xi^n.
\end{equation}
Substituting \eqref{a12} in \eqref{a11}, we have
\begin{equation}\label{a13}
\begin{split}
&1-c_0-(c_1+c_0c_1)\xi-(c_2+2c_0+c_1^2+4c_1)\xi^2
-\sum_{n=0}^\infty c_{n+3}\xi^{n+3}\\
&-\sum_{n=0}^\infty
\Big(\sum_{k=0}^{n+2}(n+3-k)c_kc_{n-k+3}\Big)\xi^{n+3}
-\sum_{n=0}^\infty(n+1)(n+2)(n+3)c_{n+3}\xi^{n+3} \\
& -2\sum_{n=0}^\infty(n+2)(n+3)c_{n+3}\xi^{n+3}=0.
\end{split}
\end{equation}
Through comparing coefficients of $\xi,\xi^2$ and a constant, we have
\begin{equation}\label{a14}
c_0=1,\quad c_1=0,\quad c_2=-2.
\end{equation}
When $n\geq0$, we have
\begin{equation*} %\label{a15}
c_{n+3}=\frac{-1}{1+2(n+2)(n+3)+(n+1)(n+2)(n+3)}
\Big[\sum_{k=0}^{n+2}(n+3-k)c_kc_{n-k+3}\Big].
\end{equation*}
Therefore, the power series solution for \eqref{a11} can be written as
\begin{equation}\label{a16}
\begin{aligned}
f(\xi)&=1-2\xi^2-\sum_{n=0}^\infty
\frac{1}{1+2(n+2)(n+3)+(n+1)(n+2)(n+3)}\\
&\quad\times \Big(\sum_{k=0}^{n+2}(n+3-k)c_kc_{n-k+3}\Big)\xi^{n+3}.
\end{aligned}
\end{equation}
\subsection{Generator $V_1+V_3$}
The similarity variables are $\xi=\ln(t)-x-t, f(\xi)=(1+u)t$, and the
group-invariant solution is $u=\frac{f(\xi)}{t}-1$. Substituting the
group-invariant solution in \eqref{a1}, we obtain the reduction equation
\begin{equation}\label{a17}
f-f'+ff'-f''+f'''=0.
\end{equation}
where $f'=\frac{df}{d\xi}$. Similarly, we will consider the exact analytic
solutions of the reduction equation \eqref{a17} by using the power series method.
We seek a solution of \eqref{a17} in the power series of the form
\begin{equation}\label{a18}
f(\xi)=\sum_{n=0}^{\infty}c_n\xi^n.
\end{equation}
Substituting \eqref{a18} into \eqref{a17}, we have
\begin{equation}\label{a19}
\begin{aligned}
&\sum_{n=0}^\infty c_n\xi^n-\sum_{n=0}^\infty(n+1)c_{n+1}\xi^n
+\sum_{n=0}^\infty\Big(\sum_{k=0}^n(n-k+1)c_kc_{n-k+1} \Big)\xi^n\\
&-\sum_{n=0}^\infty(n+1)(n+2)c_{n+2}\xi^n+\sum_{n=0}^\infty(n+1)(n+2)(n+3)
c_{n+3}\xi^n=0.
\end{aligned}
\end{equation}
Through comparing the coefficients of $\xi^i(i=0,1,2,\dots)$, we have
\begin{equation}\label{a20}
\begin{aligned}
c_{n+3}&=\frac{1}{(n+1)(n+2)(n+3)}\Big( (n+1)c_{n+1} +(n+1)(n+2)c_{n+2}-c_n\\
&\quad -\sum_{k=0}^n(n-k+1)c_kc_{n-k+1} \Big).
\end{aligned}
\end{equation}
Therefore, the power series solution for Eq.\eqref{a17} can be written as follows
\begin{equation}\label{a21}
\begin{aligned}
f(\xi)&=\sum_{n=0}^\infty\frac{1}{(n+1)(n+2)(n+3)}
\Big( (n+1)c_{n+1} +(n+1)(n+2)c_{n+2}-c_n \\
&\quad -\sum_{k=0}^n(n-k+1)c_kc_{n-k+1}\Big)\xi^n.
\end{aligned}
\end{equation}
\section{Bifurcation and solitary waves of \eqref{a1}}\label{sec4}
In this section, we consider the linear combination of generators $V_2+cV_3$,
gives rise to the traveling wave solutions, and study the bifurcation of the
two-dimensional dynamic system which satisfied the traveling wave transformation,
then existence and solvability of solitary or period wave of Eq.\eqref{a1} by
analyzing the homoclinic orbits or period orbits of two-dimensional dynamic system
based on the bifurcation diagram.
The linear combination of generators $V_2+cV_3$ leads to the group-invariant
solution $u(x,t)=u(\xi)$, where $\xi=x-ct$ is the similarity variables of $V_2+cV_3$.
Substituting the group-invariant solution $u(x,t)=u(\xi)$ into \eqref{a1},
then \eqref{a1} reduces to an ordinary differential equation as follows
\begin{equation}\label{b1}
-cU'+UU'+(1+c)U'''=0,
\end{equation}
where $U'=dU/d\xi$. Integrating \eqref{b1} once with respect to $\xi$,
and taking integration constant equal to $h$, one obtains
\begin{equation}\label{b2}
-cU+\frac{U^2}{2}+(1+c)U''=h,
\end{equation}
or equivalent to the following two-dimensional dynamic system
\begin{equation}\label{b3}
\begin{gathered}
\frac{dU}{d\xi}=\frac{1}{1+c}V\\
\frac{dV}{d\xi}=h+cU-\frac{U^2}{2},
\end{gathered}
\end{equation}
which has the energy integral
\begin{equation}\label{b4}
E=\frac{1}{2(1+c)}V^2+P(U),
\end{equation}
where $P(U)=\frac{U^3}{6}-\frac{cU^2}{2}-hU$.
We select the integration constant $h$ as the bifurcation control parameter.
System \eqref{b3} has equilibrium points $(U_1,0)$ and $(U_2,0)$, and the
bifurcation point is $(U_b,-\frac{c^2}{2})$, where $U_1=c-\sqrt{c^2+2h}$,
$U_2=c+\sqrt{c^2+2h}$ and $U_b=c$ . Since the characteristic equation for
the equilibrium points $(U_i,0)(i=1,2)$ is $\lambda^2-\frac{1}{1+c}(c-U_i)=0$,
we know that the point $(U_1,0)$ is unstable saddle , and the points $(U_2,0)$
is stable center.
According to the above analysis, we see that the bifurcation diagram of
system \eqref{b3} is the parabolic shape curve in Figure \ref{fig1}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=7.5cm]{fig1} % fencha.eps
\end{center}
\caption{Steady bifurcation diagram of system \eqref{b3}}
\label{fig1}
\end{figure}
Next, we discuss the existence and solvability of solitary waves of \eqref{a1}.
The solitary wave of \eqref{a1} fact correspond to the homoclinic orbits of
two-dimensional dynamic system \eqref{b3} passing through the saddle $(U_1, 0)$.
The period wave of \eqref{a1} are in fact corresponding to the period orbits
of two-dimensional dynamic system \eqref{b3} passing through the center $(U_2, 0)$.
We should discuss two cases:
\subsection*{Case 1. $h>-c^2/2$}
When $E=P(U_1) $, from \eqref{b3} and \eqref{b4}, we obtain
\begin{equation}\label{b5}
\frac{dU}{d\xi}=\pm\sqrt{\frac{2}{1+c}G_1(U)},
\end{equation}
where $G_1(U)=P(U_1)-P(U)\geq0$ in some range of $U$, which depends on the
values of the bifurcation control parameter, and we have
\begin{equation}\label{b6}
G_1'(U_1)=-P'(U)=-\frac{1}{2}(U-U_1)(U-U_2),
\end{equation}
According to the existence theorem of zero points of the elementary continuous
function on the closed interval, it is obvious that the following results:
\begin{itemize}
\item[(i)] $G_1(U_1)=0$, $G_1'(U_1)=0$, i.e., $G_1(U)$ has the double root $U_1$;
\item[(ii)] $\lim_{U\to-\infty}G_1(U)=+\infty $ and
$\lim_{U\to+\infty}G_1(U)=-\infty $ i.e., $G_1(U)$ has a single root
$U_{1}^*>U_2$ .
\end{itemize}
Therefore, $G_1(U)=\frac{1}{6}(U-U_1)^2(U_{1}^*-U)$, and substituting it
into \eqref{b5}, we have
\begin{equation}\label{b7}
\frac{dU}{d\xi}=\pm|U-U_1|\sqrt{\frac{1}{3(1+c)}(U_{1}^*-U)}.
\end{equation}
Integrating \eqref{b7}, we obtain the solitary wave solution
\begin{equation}\label{b8}
u(x,t)=(U_1-U_1^*)\tanh\Big(\frac{1}{6}\sqrt{\frac{3(U_1^*-U_1)}{1+c}}
(x-ct-\xi_0)\Big)^2+U_1^*
\end{equation}
where $\xi_0$ is determined by the initial value $u_0=u(x_0,0)$.
It holds that $u\to U_{1}$ as $x\to\pm\infty $.
When $E=P(U_2) $, from \eqref{b3} and \eqref{b4}, we obtain
\begin{equation}\label{b9}
\frac{dU}{d\xi}=\pm\sqrt{\frac{2}{1+c}G_2(U)},
\end{equation}
where $G_2(U)=P(U_2)-P(U)\geq0$ in some range of $U$, which depends on
the values of the bifurcation control parameter, and we have
\begin{equation}\label{b10}
G_2'(U_2)=-P'(U)=-\frac{1}{2}(U-U_1)(U-U_2),
\end{equation}
According to the existence theorem of zero points of the elementary continuous
function on the closed interval, it is obvious that the following results:
\begin{itemize}
\item[(i)] $ G_2(U_2)=0$, $G_2'(U_2)=0$, i.e., $G_2(U)$ has the double root $U_2$;
\item[(ii)] $\lim_{U\to-\infty}G_2(U)=+\infty $ and $\lim_{U\to+\infty}G_2(U)=-\infty$,
i.e., $G_2(U)$ has a single root $U_{2}^*-\frac{c^2}{2}$, and there exist a breaking wave
solution if $h=-c^2/2$. It means that there is no solitary or period wave
if $ h<-c^2/2$.
We remark that the convergence of the power series solution \eqref{a16} and
\eqref{a21} can be easily proved\cite{c13}, thus power series solution is
an exact analytic solution of reduction equation.
Also we remark that integration constant influence the existence of solitary
or period wave, so we can control the phenomenon of solitary or period wave
by controlling the value of the integral constant.
\subsection*{Acknowledgments}
This research was supported by the Natural Science Foundation of Shanxi
(No. 2014021010-1).
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\end{document}