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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 07, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2004/07\hfil Asymptotic theory]
{Asymptotic theory for weakly non-linear wave equations in semi-infinite domains}
\author[Chirakkal V. Easwaran\hfil EJDE-2004/07\hfilneg]
{Chirakkal V. Easwaran}
\address{Department of Computer Science\\
State University of New York\\
New Paltz, NY 12561, USA}
\email{easwaran@newpaltz.edu}
\date{}
\thanks{Submitted November 19, 2003. Published January 2, 2004.}
\subjclass[2000]{35C20, 35L20}
\keywords{Multiple scale, perturbation, non-linear, waves, asymptotic}
\begin{abstract}
We prove the existence and uniqueness of solutions of a class of
weakly non-linear wave equations in a semi-infinite
region $0\le x$, $t< L/\sqrt{|\epsilon|}$ under arbitrary
initial and boundary conditions. We also establish the asymptotic
validity of formal perturbation approximations of the solutions in
this region.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\section{Introduction}
Many physical phenomena involve the action of weak non-linear
perturbations acting over long periods of space and time. Some of these
phenomena can be mathematically modeled by partial differential equations
containing small non-linear terms.
These non-linear effects, characterized
by a small parameter $\epsilon$, could accumulate over time and space to
significantly impact the space and time evolution of the systems.
Some examples of physical models
involving such weakly non-linear equations are:
\begin{itemize}
\item[(i)] The wave equation with a cubic non-linearity governing the slow oscillations
of overhead power lines \cite{Mye}.
\item[(ii)] The non-linear Schroedinger equation with slowly varying coefficients
with applications in water waves and non-linear optics \cite{Ablo,Gbw}.
\item[(iii)] The shallow water wave equations with small initial displacement, and the weakly
non-linear acoustics equations \cite{KC,Nay}.
\item[(iv)] The equations describing the motion of a slightly viscoelastic column with
viscous damping \cite{Lard}.
\end{itemize}
The traditional tool to study the effect of small non-linearities is perturbation expansion
in terms of the small parameter $\epsilon$.
Straight forward perturbation expansion of solutions usually become
unbounded at large times (or lengths) because of unbounded growth in
the wave amplitude. Averaging, matched asymptotic expansions and
multiple-scale techniques are the standard techniques used to develop
approximations of solutions of these
non-linear perturbation problems that remain bounded at large times and distances.
A review of these tools can be found in \cite{KC}, and a number of examples and related approaches can be found in \cite{Nay}.
Generally, multiple scale expansions of solutions of wave equations
anticipate in advance the dependence of solutions on different time scales in order
to construct solutions which are explicit functions of time.
Prior to 1992, most studies of weakly nonlinear wave equations dealt with
initial value problems on the infinite line $-\infty0$, a perturbation scheme based on a slow
spatial scale $X=\epsilon x$ can be used \cite{CVE2}.
In \cite{CE}, a multi-scale method was developed for weakly nonlinear wave
equations in the region $x>0$ with arbitrary initial and boundary data.
These problems
can not be transformed into initial value problems on the infinite interval
$-\infty0,\; 0<\epsilon\ll 1\\
u(x,0)=a(x);\quad u_t(x,0)=b(x);\quad u(0,t)=\rho(t), \quad t,x>0 %\label{mainibc}
\end{gathered}
\end{equation}
Let us assume that the initial and boundary data as well as the
non-linear function $h$ satisfy the following conditions:
\begin{itemize}
\item[(A1)] $a(x)$, $\rho(t)$ are twice continuously differentiable for
$x\ge 0$, $t\ge 0$.\\
$b(x)$is continuously differentiable for $x\ge 0$, the function
$h$ and its derivatives are analytic and uniformly bounded in its
arguments.\\
$a(0)=\rho(0)$, $b(0)=\rho'(0)$, $\rho(0)=0$, $a'(0)=0$, $\rho''(0)=0$, \\
$-a''(0)+\epsilon h(a(0),b(0),a'(0))=0$
\end{itemize} %(3)
\begin{theorem} \label{thm1}
Suppose $h$, $a(x)$, $b(x)$, $\rho(t)$ satisfy the conditions (A1).
Then for any $\epsilon$ with $0 < |\epsilon| \le \epsilon_0 \ll 1$,
the non-linear initial-boundary value problem \eqref{main}
has a unique, twice continuously differentiable solution in a region of the $x-t$
plane, $0\le x,t \le L/\sqrt{|\epsilon|}$, where $L$ is a
sufficiently small positive constant independent of $\epsilon$.
Moreover, this solution depends continuously on the initial-boundary data.
\end{theorem}
The conditions on the initial values of various
functions in (A1) ensure that the solution
$u$ and its first and second partial derivatives are continuous across $x=t$.
The proof of this theorem depends crucially on the integral representation of
the solution to the hyperbolic system \eqref{main}-\label{mainibc} (see \cite{TS}):
\begin{equation} \label{e4}
u(x,t) = \begin{cases}
\frac 12 [ a(x+t)+a(x-t) ]+\frac 12 \int_{x-t}^{x+t} b(\lambda)\,d\lambda \\
+ \frac{\epsilon}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,
&\mbox{if } 0\le t\le x, \\[5pt]
\rho(t-x)+\frac 12 [ a(t+x)+a(t-x) ]+\frac 12
\int_{t-x}^{t+x}b(\lambda)d\lambda \\
+ \frac{\epsilon}{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,
&\mbox{if } t \ge x\ge 0,
\end{cases}
\end{equation}
By direct differentiation, one can show the equivalence of the system
\eqref{e4} to \eqref{main} whenever $C^2$ solutions
of \eqref{main} exist.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1.eps}
\end{center}
\caption{Region of integration for the first case in \eqref{e4}. This region has
area $t^2$}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2.eps}
\end{center}
\caption{Region of integration for the second case in \eqref{e4}.
This region has area $2tx-x^2$}
\end{figure}
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 $0\le t,x \le O(1/\sqrt{\epsilon})$.
This solution
depends continuously on the initial-boundary data, and formal perturbation
series expansions asymptotically converge to the solution in this rectangle.
Let $T:S\to S$ denote the integral operator
defined by \eqref{e4}, where
\[
S=\{ (x,t)|0\le x,t \le {L\over \sqrt{|\epsilon|}}\}
\]
We represent the integral equations \eqref{e4} in the form $u=Tu$.
The proof consists of showing that
$T$ is a contraction on a space of twice continuously differentiable functions
defined on $S$, and therefore by Banach's
Fixed Point Theorem, a unique solution $u$ exists.
Let $C^2_M(S)$ be the space of twice continuously differentiable
functions on $S$ with norm
\[ %7
\|f\|=\sum_{i,j=0\; i+j\le 2}^2 \max_{(x,t)\in S}
\big|{\partial^{i+j}f(x,t)\over \partial x^i \partial t^j}\big| \le M
\]
Let us write
\begin{equation} \label{e8}
Tu=u_I+T_\epsilon u
\end{equation}
where
\begin{equation} \label{e9}
u_I(x,t) =\begin{cases}
\frac 12 [ a(x+t)+a(x-t) ]+ \int_{x-t}^{x+t}
b(\lambda)d\lambda, &\mbox{if } 0\le t\le x \\[3pt]
\rho(t-x)+\frac 12 [ a(t+x)+a(t-x) ] + \frac 12
\int_{t-x}^{t+x}b(\lambda)d\lambda,
&\mbox{if } t \ge x\ge 0
\end{cases}
\end{equation}
and
\begin{equation} \label{e11}
T_\epsilon u= \begin{cases}
\frac{\epsilon}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,
&\mbox{if } 0\le t\le x \\[3pt]
= \frac{\epsilon}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,
&\mbox{if } t \ge x\ge 0
\end{cases}
\end{equation}
From \eqref{e8} we get
\begin{equation} \label{e13}
\|Tu\|\le \|u_I\|+\|T_\epsilon u\|
\end{equation}
Because of the boundedness conditions on $\rho$, $a$ and $b$, there
exists a nonnegative constant $M_1$, independent of $\epsilon$,
such that
\begin{equation} \label{e14}
\|u_I\| \le {M_1\over 2}
\end{equation}
From Figures 2 and 3, it follows that there exist constants
$M_2^1$ and $M_2^2$ such that
\[
\|T_\epsilon u\| \le
\begin{cases} M_2^1 \epsilon t^2, &\mbox{if } t \le x\\
M_2^2 \epsilon (2tx-x^2), &\mbox{if } t \ge x %16
\end{cases}
\]
Thus for $t$ and $x$ in the region $S$, one can find an
$\epsilon$-independent constant $M_2$ such that
\[ %17
\|T_\epsilon u\|\le M_2 L^2
\]
Combining this inequality with \eqref{e14} and \eqref{e13},
\[ %18
\|Tu\|\le {M_1\over 2} +M_2 L^2 \le M_1
\]
for sufficiently small $L$.
This shows that $T: C_{M_1}^2(S) \to C_{M_1}^2(S)$.
We next show that $T$ is a contraction on $ C_{M_1}^2(S)$.
Using the Lipschitz property of $h$, there exists a constant
$M'$:
\[
\|h(u,u_t,u_x)-h(v,v_t,v_x)\|\le M' \|u-v\|
\]
for all $(x,t)\in S$. It follows from \eqref{e11} that one can
find constants $K_1$ and $K_2$ satisfying
\[
\|T_\epsilon u-T_\epsilon v\|
\le \begin{cases} \epsilon t^2 K_1 \|u-v\|, &\mbox{if } t\le x \\ %20
\epsilon (2tx-x^2) K_2 \|u-v\|, &\mbox{if } t\ge x %21
\end{cases}
\]
Thus there is a nonnegative constant $K$ such that
\[
\|Tu-Tv\|=\|T_\epsilon u-T_\epsilon v\|\le K L^2\|u-v\|
\]
for all $(x,t)\in S$. Then for sufficiently
small $L$, independent of $\epsilon$,
\begin{equation} \label{e23}
\|Tu-Tv\|\le k \|u-v\|
\end{equation}
for all $u, v \in C^2_{M_1}(S)$ and $0\le k< 1$. This shows
that $T$ is a contraction of $C^2_{M_1}(S)$ into itself for
sufficiently small $L$. By applying Banach fixed point theorem, it
follows that $T$ has a unique fixed point in $C^2_{M_1}(S)$.
Since solutions of the integral equation \eqref{e4} are also solutions
of the hyperbolic system \eqref{main}, we have proved that \eqref{main} has a unique
solution in the space $S$.
We next prove that the solutions of the hyperbolic system \eqref{main}
depends continuously on the initial-boundary values, in the sense
that small changes in these values result in small changes in the
solution within the region $S$ in which existence-uniqueness
of solutions have been proved.
Let $\tilde{u}$ be the solution of the hyperbolic system \eqref{main}
corresponding to the initial boundary conditions
\begin{equation}
u(x,0)=\tilde{a}(x);\quad u_t(x,0)=\tilde{b}(x);\quad
u(0,t)=\tilde{\rho}(t), \quad t,x>0
\label{ibc2}
\end{equation}
Assume that the initial-boundary data in (24) satisfy conditions
similar to (A1). Following \eqref{e8}, we let
\[ %25
T\tilde{u} = \tilde{u}_I+T_\epsilon \tilde{u}.
\]
Then the following estimate can be made:
\[ %26
\|u-\tilde{u}\|\le \|u_I-\tilde{u}_I\|+\|T_\epsilon u-T_\epsilon\tilde{u}\|
\]
From an argument similar to \eqref{e23},
there exists a $k$, $0\le k< 1$, such that
\[ % 27
\|T_\epsilon u -T_\epsilon\tilde{u}\|\le k \|u-\tilde{u}\|
\]
so that
\[ %28
\|u-\tilde{u}\|\le \|u_I-\tilde{u}_I\|+k \|u-\tilde{u}\|
\]
and
\[
\|u-\tilde{u}\| \le {1\over 1-k}\|u_I-\tilde{u}_I\|
\]
whenever $u$, $\tilde{u}$ are in $C^2_{M_1}(S)$.
This proves that small changes in initial data lead to small
changes in the solutions.
\section{Asymptotic validity of formal expansions}
Next we prove that perturbation series expansions of
solutions of \eqref{main} are asymptotically
convergent to the exact solutions.
Let $v(x,t)$ be defined on $S$ satisfying
\begin{equation} \label{e30-33}
\begin{gathered}
v_{tt}-v_{xx}+\epsilon h(v,v_t,v_x)=\epsilon^n r_1(x,t) \\
v(x,0)=a(x)+\epsilon^{n-1} r_2(x) \\
v_t(x,0)=b(x)+\epsilon^{n-1} r_3(x) \\
v(0,t)= \rho(t)+\epsilon^{n-1} r_4(t)
\end{gathered}
\end{equation}
The motivation to study this system is that they satisfy the
original partial differential equation and initial/boundary conditions to
$O(\epsilon^{n})$ and $O(\epsilon^{n-1})$
respectively, where usually $n=2$.
\begin{itemize}
\item[(A2)]
Assume that $h, r_1, r_2, r_3, r_4$ satisfy boundary conditions as in
(A1), and that $r_3, r_4$ are in $C^1$ %34
\end{itemize}
\begin{theorem} \label{thm2}
Let $v$ satisfy \eqref{e30-33} and the $r_{i(1\le i\le 4)}$
satisfy (A2). Then for $n>1$, $\|u-v\|= O(\epsilon^{n-1})$
for all $t,x \in S$. Thus in the limit of small $\epsilon$, the
formal approximation $v$ converges to the solution $u$.
\end{theorem}
\begin{proof} Note that a formal integral
representation of \eqref{e30-33} can be written in the form
\begin{equation} \label{e35}
v=\tilde{v}_I+T_\epsilon v + \tilde{T}_\epsilon r_1
\end{equation}
where
\[ %36-37
\tilde{T}_\epsilon r_1 = \begin{cases}
{\epsilon^n\over 2} \int_0^t\,d\tau \int_{x-(t-\tau)}^{x+(t-\tau)}
r_1(\lambda,\tau) d\lambda, &\mbox{if } 0\le t\le x\\[3pt]
{\epsilon^n \over 2}\int_0^t\,d\tau \int_{|x-(t-\tau)|}^{x+(t-\tau)}
r_1(\lambda,\tau) d\lambda, &\mbox{if } t \ge x\ge 0
\end{cases}
\]
and $v_I$ is analogous to \eqref{e9}.
From the geometry of Figures 1 and 2, it follows that there exists constants
$M_3^1$ and $M_3^2$ such that
\[ %38-39
\|\tilde{T}_\epsilon r_1\| \le \begin{cases}
M_3^1 \epsilon^n t^2, &\mbox{if } t \le x\\
M_3^2 \epsilon^n (2tx-x^2),&\mbox{if }t\ge x.
\end{cases}
\]
Thus for $t$ and $x$ in the region $S$, one can find a constant $M_3$ such
that
\[ %40
\|\tilde{T}_\epsilon r_1\|\le \epsilon^{n-1}M_3 L^2.
\]
In addition, there exists a non-negative constant $M_4$ such that,
\[ %41
\|u_I-\tilde{v}_I\|\le \epsilon^{n-1} M_4.
\]
From \eqref{e8} and \eqref{e35} it follows that
\[ %42-43
\|u-v\| \le \|T_\epsilon u-T_\epsilon v\|+\|u_I-\tilde{v}_I\|
+\|\tilde{T}_\epsilon r_1\|
\le k \|u-v\|+ \epsilon^{n-1}(M_3 L^2+M_4)
\]
so that
\[
\|u-v\|\le {\epsilon^{n-1}\over 1-k}(M_3 L^2+M_4)
\]
showing that $\|u-v\| = O(\epsilon^{n-1})$ for $x,t\in S$.
This establishes Theorem \ref{thm2}.
\end{proof}
\section{Applications of the asymptotic theory}
The results obtained above give formal theoretical support to a
multiple-scale solution technique for weakly non-linear wave equations
developed in \cite{CE}. It is shown there
that the solution of \eqref{main} involves two scaled (or slow) variables,
$X=\epsilon x$ and $T=\epsilon t$, in addition to the regular (or fast) variables
$x$ and $t$. The perturbation solution then defines two regions, $t>x$ and
$t \le x$. For $t\le x$, the first order solution has the form
\[ %45
u_0(x,t,T)=f(\sigma,T)+g(\xi,T)
\]
where $\sigma=x-t$ and $\xi=x+t$ are the forward and backward going characteristics of
the wave equation. It is shown in \cite{CE} that $f$ and $g$ are governed by:
\begin{gather*} %46-47
2 f_{\sigma T}-\lim_{M\to \infty}
{1\over M}\int_0^M h(f+g, g_\xi - f_{\sigma},g_\xi + f_{\sigma})\,d\xi =0,\\
2 g_{\xi T}+\lim_{M\to \infty}
{1\over M}\int_0^M h(f+g, g_\xi - f_{\sigma},g_\xi + f_{\sigma})\,d\sigma =0
\end{gather*}
with appropriate initial-boundary conditions.
For the region $t>x$, the first order solution takes the form
\[ %48
u_0(x,t,X,T)=p(\mu,X,T)+q(\xi,X,T)
\]
where $\mu=t-x$ and $\xi=t+x$. In general the PDEs governing $p$ and $q$ are complicated
coupled nonlinear equations; but for special cases, they are considerably simplified. For example, when $h$ involves only the first derivatives of $u$, $h=h(u_t, u_x)$, it can be
shown that
\begin{gather*} %49-50
2 p_{\mu T}+2 p_{\mu X}+\lim_{M\to \infty}
\frac 1M \int_0^M h(p_\mu+q_\xi, q_\xi-p_\mu)d\xi =0\,,\\
2 q_{\xi T}-2 q_{\xi X}+\lim_{M\to \infty}
\frac 1M \int_0^M h(p_\mu+q_\xi, q_\xi-p_\mu)d\mu =0\,.
\end{gather*}
The important point here is that the two equations governing the interaction of
backward and forward propagating waves form a pair of coupled, nonlinear
first-order PDEs whose theory is well-developed. This is a considerable
simplification from the original second order nonlinear equation.
We refer to \cite{CE} for details of solutions of these equations as well as
explicit solutions for the cases $h=2u_t+u (u_t-u_x)$ and $h=2u_t$.
\subsection*{Concluding remarks}
For weakly non-linear hyperbolic partial differential equations in the region
$t>0$, $x>0$, we established the existence, uniqueness and continuous
dependence of solutions on initial data within a rectangle
$0\le x,t< L/\sqrt{|\epsilon|}$. Although it appears from numerical evidence
that the existence-uniqueness theorems could hold in much longer
space and time intervals, it is not clear how to establish that. We also
proved the asymptotic validity of formal perturbation expansions
of solutions within this
rectangle.
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\end{document}