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\markboth{\hfil A scaled characteristics method \hfil EJDE--1998/03}%
{EJDE--1998/03\hfil Chirakkal V. Easwaran \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~03, pp. 1--10. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
 \vspace{\bigskipamount} \\
   A scaled characteristics method for the asymptotic solution of 
weakly nonlinear wave equations 
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35C20, 35L20.
\hfil\break\indent
{\em Key words and phrases:} Multiple scale, perturbation, nonlinear, waves.
\hfil\break\indent
\copyright 1998 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted August 5, 1997. Published January 31, 1998.} }
\date{}
\author{Chirakkal V. Easwaran}
\maketitle

\begin{abstract} 
We formulate a multi-scale perturbation technique to asymptotically solve
weakly nonlinear hyperbolic equations. The method is based on a set of  
scaled characteristic coordinates. We show that  this technique leads 
to a simplified system of ordinary differential equations describing 
the weakly nonlinear interaction between left and right running waves. 
Using this method, a uniformly valid first order solution of a prototype
nonlinear equation is derived.  
\end{abstract}

\section{Introduction}

In this paper we consider initial-boundary value problems for weakly 
non-linear hyperbolic wave equations of the form
\begin{eqnarray}
&u_{tt}-u_{xx}+\epsilon h(u,u_t,u_x)=0,\quad t,x >0,\quad 0<\epsilon\ll 1&
\label{main} \\
&u(x,0)=a(x);\quad u_t(x,0)=b(x);\quad u(0,t)=\rho(t), \; t,x>0\,.&
\label{mainibc}
\end{eqnarray}
Such equations occur in many models in science and engineering
(See \cite{KC,Mye,Nay,Gbw}). 
We assume that $a(0)=\rho(0)$, $b(0)=\rho'(0)$, that $a(x)$, $\rho(t)$
are twice continuously differentiable, $b(x)$ is continuously 
differentiable, and that $h$ is analytic in its arguments. One can formally
write the solution of this system as two separate equations, one valid in
the region $0\le t \le x$ and the other valid for $t\ge x \ge 0$ \cite{TS},
\begin{eqnarray}
u(x,t) &=& {1\over 2}[ a(x+t)+a(x-t) ]+{1\over 2} \int_{x-t}^{x+t}
b(\lambda)d\lambda \\
 &&+ {\epsilon \over 2}\int_0^t\; d\tau \int_{x-(t-\tau)}^{x+(t-\tau)}
h(u, u_\tau(\lambda,\tau),u_\lambda(\lambda,\tau)) d\lambda, \quad 0\le t\le x,
\nonumber\\
u(x,t) &=&\rho(t-x)+{1\over 2}[ a(t+x)+a(t-x) ]+{1\over 2} 
\int_{t-x}^{t+x}b(\lambda)d\lambda  \\
       &&+ {\epsilon \over 2}\int_0^t\; d\tau \int_{|x-(t-\tau)|}^{x+(t-\tau)}
h(u, u_\tau(\lambda,\tau),u_\lambda(\lambda,\tau)) d\lambda, \quad t \ge x\ge 0.
\nonumber\end{eqnarray}
Using the above representation of solutions, and the assumptions on the
regularity of initial and boundary conditions, one can show that a twice
continuously differentiable solution $u(x,t;\epsilon)$ of the hyperbolic
system exists in a rectangle $0\le t,x \le O(\sqrt{\epsilon})$. This solution
depends continuously on the initial data, and formal perturbation
series expansions asymptotically converge to the solution in this rectangle.

Developing  uniformly valid perturbation solutions to hyperbolic systems 
of the above type has been a difficult task. 
When the boundary condition is zero, a multi-scale perturbation scheme 
based on a ``slow'' time scale $T=\epsilon t$ has been developed
\cite{KC}. For signaling problems in which the initial conditions are
zero and one wishes to study the propagation of boundary data, a perturbation
scheme based on a ``long'' distance scale $X=\epsilon x$ can be used. 
However, when the initial and boundary
conditions are both nonzero, the forward and backward going wave components 
(which are perturbations of the corresponding parts of the linear 
wave equation) interact, leading to considerable difficulties in finding
asymptotic solutions. 
In \cite{CE} we developed a perturbation scheme that could deal with this
situation. This scheme involved a coupled set of first order partial 
differential equations (PDEs) in terms of
two scaled variables $X=\epsilon x$ and $T=\epsilon t$, governing
the interaction between forward and backward going waves in the region 
$t > x$. In many cases we showed that these PDEs could be explicitly 
solved leading to
uniformly valid
$O(1)$ solutions of the system. Our purpose in this paper is to present
a modified perturbation scheme using a pair of scaled characteristic
variables, and certain assumptions on averaged quantities, leading to
coupled {\em ordinary differential equations} governing the interaction 
of left and
right running waves. This makes the perturbation analysis considerably
simpler. In section~3  we present comparisons of results of the present 
analysis with those in \cite{CE}.

\section{The multi-scale perturbation procedure}

Equations (3)-(4) show that the solution of the hyperbolic system (1)-(2)
splits into two parts: the region $0\le t\le x$ where the boundary
conditions do not affect the solution, and the region $t \ge  x \ge 0$,
where boundary conditions come into play. See Figure 1.
Thus the analysis proceeds differently in these two regions. 

%\vspace {.4in}

\psfig{file=1.ps}

\subsection{The region $0\le t \le x$}

In $0\le t\le x$, only initial
conditions affect the solution. The asymptotics in this region is the same as
in \cite{CE},\cite{KC}.
The nonlinearity $h$ will have
a time-cumulative effect on the solution in this region.
To determine this effect,  a slow time  variable 
$T=\epsilon t$ is usually introduced, in addition 
to the ``fast'' 
variables $t$ and $x$. Strictly speaking, the fast variables should 
be relabeled, but for convenience we use $t$ and $x$ as the fast variables.
We then seek a formal perturbation series solution in
the form
\begin{equation}
u(x,t;\epsilon)=\sum_{n=0}^{N}\epsilon^n u_n(x,t,T)+ O(\epsilon^{n+1}).
\label{expn}
\end{equation}
The derivatives are replaced by
$\partial_t\longrightarrow\partial_t+\epsilon \partial_T$,
$\partial_{tt}\longrightarrow\partial_{tt}+2 \epsilon \partial_{tT}+
\epsilon^2\partial_{TT}$. 
We also introduce the characteristics $\sigma=x-t$ and $\xi=x+t$. 
After substituting into the original system and
equating like powers of $\epsilon$, a hierarchy of initial boundary
value problems are obtained:
\begin{eqnarray}
&-4u_{0_{\xi\sigma}} = 0&\\
&-4u_{1_{\xi\sigma}}+2 u_{0_{\xi T}}-2 u_{0_{\sigma T}}+h(u_0, u_{0_\xi}-
u_{0_\sigma},
u_{0_\xi}+u_{0_\sigma}) = 0&
\end{eqnarray}
The  solution of (6) is then sought in the form
\begin{equation}
u_0(x,t,T)=f(\sigma,T)+g(\xi,T)
\end{equation}
where $\sigma=x-t$ and $\xi=x+t$. When $\epsilon=0$ we should recover the
solution of the linear wave equation.

$f$ and $g$ are determined by requiring that the $O(\epsilon)$ 
solution,  given by (7), remains bounded. We eliminate secular terms
that would lead to non-uniformities at $O(\epsilon)$ level by 
substituting (8) into (7), and requiring that $({1\over\xi}) u_{1_\xi}$
and  $({1\over\sigma}) u_{1_\sigma}$ must go to $0$ as $\xi$ and $\sigma$ grow
large. Integrating (7) with respect to $\xi$ from $0$ to $M$, dividing by $M$
and taking the limit as $M\longrightarrow \infty$, we get 
\begin{equation}
-2 f_{\sigma T}+ \lim_{M \rightarrow \infty}   {1 \over M} \int_0^M 
h(f+g, g_\xi-f_\sigma, g_\xi+f_\sigma) d\xi =0. \\
\end{equation}
Repeating the process with respect to $\sigma$ yields
\begin{equation}
2 g_{\xi T}+ \lim_{M \rightarrow \infty}   {1 \over M} \int_0^M 
h(f+g, g_\xi-f_\sigma,  g_\xi+f_\sigma) d\sigma =0. \\
\end{equation}
These are a coupled set of ODEs for $f_\sigma$ and $g_\xi$ that determine the 
wave evolution in the region $0\le t \le x$.

\subsection{The region $t\ge x \ge 0$}

In this region the boundary conditions also affect the solution. In \cite{CE},
we  introduced a ``slow'' time scale
$T=\epsilon t$ and a ``long'' space scale $X=\epsilon x$. The resulting 
analysis leads to a system of
coupled PDEs for the slow evolution of waves in this region. The left and 
right running waves interact with
each other through these stretched scales. 

In this paper we introduce a set of scaled  
characteristic variables, 
$\Delta= \epsilon (t+x) $ and $\Sigma= \epsilon (t-x) $, in addition to 
the regular ``fast'' characteristics $\mu=t-x$ and $\xi=t+x$ in this
region. Introduction of these scaled variables enables us to make 
certain assumptions about $\mu-$ and $\xi-$averaged quantities that
appear below. The resulting analysis, in 
many instances, is much simpler than previous methods.

With these new variables, 
the perturbation expansion becomes 
\begin{equation}
u(x,t;\epsilon)=\sum_{n=0}^{N}\epsilon^n u_n(\xi,\mu,\Delta,\Sigma)+ 
O(\epsilon^{n+1}).
\label{expn2}
\end{equation}

The $O(1)$ equation is $ u^0_{\xi \mu}=0$, for which we seek a solution in 
the form 
\begin{equation}
u^0=p(\mu,\Delta,\Sigma)+q(\xi, \Delta, \Sigma).
\end{equation}

$p$ and $q$ have to be determined subject to the conditions that 
when $t=x$, $q$ must equal $g$, the incoming wave in $t<x$, and when
$x=0$, the sum of $p$ and $q$ must equal $\rho(t)$. See Figure 1.
In addition, we must require uniform validity of the $O(1)$ and $O(\epsilon)$
solutions -- this would necessitate choosing
the $\Sigma$ and $ \Delta$ dependence of $p$ and $q$ to ensure 
that the  $O(\epsilon)$ solutions remain  bounded.  

 The $O(\epsilon)$ equation is
\begin{equation}
4 u^1_{\xi\mu}+4 q_{\xi\Sigma}+4 p_{\mu\Delta}+ h(p+q, p_\mu+q_\xi, q_\xi-p_\mu)
=0.
\end{equation}
Integrating (13) with respect to  $\mu$ from 0 to M, dividing by M and 
taking limit as $M \longrightarrow \infty$,
we get
\begin{equation}
4 q_{\xi\Sigma}+\lim_{M \rightarrow \infty}   {1 \over M} \int_0^M 
h(p+q, p_\mu+q_\xi, q_\xi-p_\mu)  d\mu =0,
\end{equation}
while the same process with respect to  $\xi$ gives 
\begin{equation}
4 p_{\mu\Delta}+\lim_{M \rightarrow \infty}   {1 \over M}  \int_0^M 
h(p+q, p_\mu+q_\xi, q_\xi-p_\mu)  d\xi =0.
\end{equation}

In calculating the $\xi$-average in (15), the $\Delta$ occurring in the 
argument of $q_\xi$ will be treated
as $\Delta=\epsilon \xi$. This is justified because for fixed $\epsilon$,
as $\xi\rightarrow \infty$, $\Delta \rightarrow \infty$, therefore the
averaging process should include $\Delta$. We leave $p$ unaffected because
we are interested only in the $O(1)$ contribution to the $\xi$-average, which
comes from $q$.  
Similarly in (14),  the $\Sigma$ that occurs in the 
argument  of $p_\mu$
will be treated as $\Sigma=\epsilon \mu$ when calculating the $\mu$ integral. 

The equations (14) and (15) are asymptotically equivalent to a pair
of partial differential equations derived in \cite{CE} to describe
the $O(1)$ solution in this region:
\begin{eqnarray}
&2p_{\mu T}+2 p_{\mu X}+ \lim_{M \rightarrow \infty}   {1 \over M} \int_0^M 
h(p+q, p_\mu+q_\xi, q_\xi-p_\mu)  d\xi =0&
\\
&2q_{\xi T}-2 q_{\xi X}+ \lim_{M \rightarrow \infty}   {1 \over M} \int_0^M 
h(p+q, p_\mu+q_\xi, q_\xi-p_\mu)  d\mu =0&
\end{eqnarray}
One can formally derive (14) and (15) from these two equations. However
the essential difference between (14)-(15) and (16)-(17) is 
in the way the averages are 
calculated in their last terms on the left side. The limiting process in
(14) and (15) affects the scaled variables $\Delta$ and $\Sigma$, while
the limiting process in (16) and (17) did not affect the scaled variables
$X$ and $T$. Up to $O(1)$, the two sets of equations are equivalent, since
the affected terms are $O(\epsilon)$. 

The advantage of our approach is that the resulting equations determining
the left and right running waves in the region $t\ge x$ are a set of 
coupled ODEs for $p_\mu$ and $q_\xi$, which can be solved in many cases. 
We next illustrate the procedure with examples.

\section{Examples}

We consider the prototype nonlinear equation  from \cite{CE}:
\begin{equation}
u_{tt}-u_{xx}+\epsilon( 2 u_t+u(u_t-u_x))=0
\end{equation}
with the initial boundary conditions (2).

In the region $t<x$, the analysis is the same as in \cite{CE},\cite{KC},  so 
we skip the details. The O(1) solution
is determined by  $u^0=f(\sigma,T)+g(\xi,T)$ , where 

$$2 f_{\sigma T}-2 f_{\sigma}-2 f f_\sigma-2 f_\sigma <g> =0$$ 
and
$$2 g_{\xi T}+g_\xi=0.$$
These can be integrated with respect to $\sigma$ and $\xi$ respectively. The
arbitrary function of $T$ resulting from the integration is identically 
zero because the solution has to depend continuously on the initial
data. Thus we get
$$2 f_T-2(1+<g>) -f^2=0$$ and
$$g_T+g=0,$$
where $<g>=\lim_{M \rightarrow \infty}   {1 \over M} \int_0^Mg(\xi,T) d\xi$.
The solution, subject to the appropriate initial conditions, is then given by
\begin{equation}
g(\xi,T)=G(\xi)e^{-T},
\end{equation}
where  
$$G(\xi)={1\over 2} \left( a(\xi)+\int_0^\xi b(\lambda) d\lambda\right)$$
and
\begin{equation}
f(\sigma,T)={F(\sigma)\lambda(T)\over
1+{1\over 2}F(\sigma)\int_0^T\lambda(\tau)d\tau}\,\, ,
\end{equation}
where 
\begin{eqnarray*}
&F(\sigma)={1\over 2} \left( a(\sigma)-\int_0^\sigma b(\lambda) 
d\lambda\right);& \\
&\lambda(T)=\exp\left[-\int_0^T(1+<g>)d\tau\right]\,.&
\end{eqnarray*}

For $t>x$, we follow the scaled characteristics approach outlined in 
section 2.2.
The O(1) problem has a solution in the form 
$$u^0=p(\mu, \Delta,\Sigma)+q(\xi,  \Delta, \Sigma).$$
The $O(\epsilon)$ equation is
$$ 4 u^1_{\mu\xi}+ 4 p_{\mu\Delta}+4 q_{\xi\Sigma}+2 (p_\mu+q_\xi)+
2 (p+q) p_\mu=0. $$

Integrating this with respect to  $\mu$ from $0$ to $M$, dividing by $M$
and taking limit as $M\rightarrow \infty$, we get
\begin{equation}
 q_{\xi\Sigma}+ q_\xi=0.
\end{equation}
A similar process with $\xi$ yields
\begin{equation}
2 p_{\mu\Delta} + p_\mu +  (p^2)_\mu+ <q> p_\mu=0,
\end{equation}
where 
$$<q>=\lim_{M \rightarrow \infty}   {1 \over M} \int_0^M 
q(\xi,\epsilon\xi,\Sigma) d\xi. $$

By integrating (20) with respect to $\xi$ and (21) with respect to 
$\mu$, we obtain a set of coupled ODEs:
\begin{eqnarray}
&2 q_{\Sigma}+ q = 0&\\
&2 p_{\Delta} + (1+<q>)p +  p^2 = 0&
\end{eqnarray}
The arbitrary functions of $\Delta$ and $\Sigma$ that result from these 
integrations are set to zero because the solution has to depend continuously
on initial data.

The coupled ODEs (23) and (24) have to be solved subject to the conditions
that when $t=x$, $q$ equals $g$, the backward-going component in $t<x$, and
when $x=0$, the sum of $p$ and $q$ should equal $\rho(t)$, the boundary
condition. One can compute these solutions readily:
\begin{eqnarray}
&q(\xi, \Delta, \Sigma) = g(\xi, {\Delta\over 2}) e^{-{\Sigma /2}}&\\
&p(\mu, \Delta,\Sigma) = \left( ({1\over 2k}+
{1 \over \rho(\mu)-q(\mu,\Sigma,\Sigma)} )e^{-k(\Sigma-\Delta)/2}
-{1\over 2k} \right)^{-1}&
\end{eqnarray}
where $k(\Sigma;\epsilon)=1+<q>$. 
We now numerically compare the solutions  obtained in this paper,
(25)-(26),  with those obtained in \cite{CE}.

Figure 2 plots the $O(1)$ solution at $t=5$, using equations (25)-(26)
and the solution derived in \cite{CE} by solving the PDEs resulting
from (16)-(17). We used 
the set of initial-boundary conditions
\begin{eqnarray}
&u(x,0) = \sin (3 x)&\\
&u_t(x,0) = 0&\\
&u(0,t) = {2t\over 1+t^2} \sin t\,.&
\end{eqnarray}
In this case $<q>=0$ in equation (26). The two plots are 
indistinguishable. Figure 3. plots the same solutions at $x=8$.

\vspace{.2in}
 
\psfig{file=2.ps}

\vspace{.3in}

\psfig{file=3.ps}

Figure 4 
plots the solutions at $t=5$ for $u(x,0)=\sin^2(3x)$, and the other
conditions as in (28)-(29). In this case $<q>\ne 0$. We plotted the
solution in the region $x>5$ where interaction between left and
right going waves take place. In Figure 5
are solutions at $x=8$. Note that each of figures 2-5 contains two plots
which are indistinguishable except in figure 5. 

\vspace{.2in}

\psfig{file=4.ps}

\vspace {.3in}

\psfig{file=5.ps}

\section{Conclusion}

We presented a method to develop uniformly valid asymptotic approximations
to weakly nonlinear wave equations that is considerably simpler than 
previously available methods. In principle, the procedure could be carried
out to higher order, although we presented only first order approximations.
The new method offers a way to study the nonlinear interaction of left and
right running waves  through a set of coupled ordinary differential
equations. This should be compared with previous methods that required
solving coupled sets of partial differential equations. Our method
used a pair of ``slow'' characteristic variables 
$\Delta=\epsilon (t+x)$ and $\Sigma=\epsilon (t-x)$ and certain assumptions
on the averages of forward and backward going wave
amplitudes to simplify the analysis. This simplification will hopefully
help analyze a variety of weakly nonlinear wave equations that
occur in applications such as gas dynamics and acoustics 
(\cite{KC},\cite{Nay}),
in which the boundary and initial data are non-zero. 

%\newpage

\begin{thebibliography}{1000}

\bibitem{CE} S. C. Chikwendu and C. V. Easwaran, {\em Multiple-scale
solution of initial-boundary value problems for weakly nonlinear wave
equations on the semi-infinite line}. SIAM J. Appl. Math. {\bf 52}(1992),
pp. 964 -- 958.

\bibitem{KC} J. Kevorkian and J. D. Cole, {\em 
Multiple Scale and Singular Perturbation Methods}.
Springer Verlag, New York, 1996.

\bibitem{Mye} C. J. Myerscough, {\em A simple model for the growth of
wind-induced oscillations in overhead lines}. J. Sound and
Vibration {\bf 39}(1975), pp. 503 -- 517.

\bibitem{Nay} A. H. Nayfeh, {\em A comparison of perturbation methods 
for nonlinear hyperbolic waves.} In {\em Singular Perturbations and 
Asymptotics},
R.E. Mayer and S.U.Parter, Eds, Academic Press, NY 1980, pp 223-276.

\bibitem{TS} A. N. Tychonov and A. A. Samarski, {\em Partial Differential
Equations of Mathematical Physics}. Holden-Day, San Francisco, 1964.

\bibitem{Gbw} G.B. Whitham, {\em Linear and Non-Linear Waves}. 
Wiley-Interscience, New York, 1974.

\end{thebibliography}
\bigskip


{\sc Chirakkal V. Easwaran}\\
Department of Mathematics and Computer Science\\
State University of New York\\
New Paltz, NY 12561 USA\\
E-mail address: easwaran@mcs.newpaltz.edu\\
http://www.mcs.newpaltz.edu/$\sim$easwaran

\end{document}