%%%%		-*- Mode: LaTeX -*-
%%%% Author:    J. Robert Buchanan
%%%% Address:   Department of Mathematics
%%%%		Millersville University
%%%%            P.O. Box 1002
%%%%            Millersville, PA 17551-0302
%%%% Phone:     717-871-7305
%%%% FAX:	717-871-7948
%%%%
%%%% Email:     Robert.Buchanan@millersville.edu
%%%%
%%%% Date:      04/01/2025
%%%%
%%%% Purpose:   Traveling Waves
%%%%

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\title{Traveling Waves}
\subtitle{\textit{Partial Differential Equations}} % (optional)


\author % (optional, use only with lots of authors)
{J~Robert~Buchanan}
% - Use the \inst{?} command only if the authors have different
%   affiliation.

\institute[Millersville University of Pennsylvania]
% (optional, but mostly needed)
{
  Department of Mathematics
}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date{ % (optional)
Fall
%Spring
%Summer
\the\year}

\subject{Talks}
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% the beginning of each subsection:
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{
  \begin{frame}<beamer>
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\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%\begin{frame}
%  \frametitle{Outline}
%  \tableofcontents
  % You might wish to add the option [pausesections]
%\end{frame}


% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:  

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

%\section{Introduction}

\begin{frame}\frametitle{Objectives}

In this lesson we will:
\begin{itemize}
\item introduce the traveling wave form of a solution to a PDE, and
\item use the traveling wave to solve some PDEs.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Traveling Wave}

If $u(x, t)$ represents a disturbance or signal strength
at location $x$ at time $t$, it is a
\textbf{traveling wave} if $u$ has the
form,
\[
%\label{eq:nlpde:travsol}
u(x, t)
	= U(x - c \, t)
\]
for some single-variable function $U$ and a fixed real constant $c$.

\vspace*{0.10in}

\uncover<2->{%
The solution of the standard wave
equation on the real number line with the initial conditions
\[
u(x, 0)
	= f(x)
\mbox{ and }
u_{t}(x, 0)
	= 0	,
\]
(the plucked string) is given by
\[
u(x, t)
	= \frac{1}{2} ( f(x - c \, t) + f(x + c \, t) )	.
\]
\begin{itemize}
\item $u(x, t)$ is the sum of two simple traveling waves, the initial
  disturbance $f(x)$ of the string is propagated to the right and left
  with speed $|c|$, each at half of the magnitude.
\item Strictly speaking, $u(x, t)$ is not a traveling wave
itself, it is the sum of two simple traveling waves.
\end{itemize}
}

\end{frame}
%%

\begin{frame}\frametitle{Example}

Find all bounded, traveling wave solutions of the (linear) partial
differential equation,
\[
%\label{eq:nlpde:example1}
u_{t} - u_{xxx}
	= 0	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Solution (1 of 2)}

Set $\xi = x - c \, t$ and substitute $u(x, t) = U(\xi)$ in the equation,
\[
-c \, U'(\xi) - U'''(\xi)
	= 0	.
\]
Integrating the above equation produces
\[
\label{eq:nlpde:example1ode}
U''(\xi) + c \, U(\xi)
	= A	,
\]
where $A$ is an arbitrary constant of integration.
If $c < 0$, the general solution is
\[
U(\xi)
	= A_{1} e^{\sqrt{-c} \, \xi} + A_{2} e^{-\sqrt{-c} \, \xi} + \frac{A}{c}
\]
Where $A_{1}$ and $A_{2}$ are arbitrary constants.

\vspace*{0.10in}

\textbf{Remark}: unless $A_{1}$ and $A_{2}$
are both zero, $U(\xi)$ is unbounded.

\end{frame}
%%

\begin{frame}\frametitle{Solution (2 of 2)}

If $c = 0$, there is no bounded solution
other than constant solutions.

\vspace*{0.10in}

If $c > 0$, the general solution is
\[
U(\xi)
	= A_{1} \cos(\sqrt{c} \, \xi) + A_{2} \sin(\sqrt{c} \, \xi) + \frac{A}{c}
\]
where $A_{1}$ and $A_{2}$ are again arbitrary constants.

\vspace*{0.1in}

The function $U(\xi)$ is bounded and leads to the following bounded
traveling wave solution,
\[
%\label{eq:nlpde:example1sol}
u(x - c \, t)
	= A_{1} \cos(\sqrt{c} (x - c \, t)) + A_{2} \sin(\sqrt{c} (x - c \, t)) + \frac{A}{c}
\]
for any choices of constants $A_{1}$, $A_{2}$, and $A$. 

\end{frame}
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\begin{frame}\frametitle{Remarks}

\begin{itemize}
\item The PDE of the previous example is a linear, dissipative partial differential
equation.
\item The nonconstant, bounded traveling wave solutions occur
only for $c > 0$.
This implies that all traveling wave solutions travel to the right
only.
\item Disregarding the constant $A/c$, all nonconstant traveling
wave solutions are linear combinations of
\[
u_{1}(x, t) = \cos(\sqrt{c} (x - c \, t))
\mbox{ and }
u_{2}(x, t) = \sin(\sqrt{c} (x - c \, t))	.
\]
\item Setting $k = \sqrt{c}$ and $\omega = k^{3}$, these two solutions can
be written as
\[
u_{1}(x, t) = \cos(k x - \omega t)
\mbox{ and }
u_{2}(x, t) = \sin(k x - \omega t)	.
\]
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Remarks}

\begin{itemize}
\item Such traveling wave solutions are often called one-dimensional
\textbf{plane wave solutions}
and are often conveniently represented in the form of complex
exponentials as $u(x, t) = e^{i (k x - \omega t)}$
and its complex conjugate.
\item The constant $k$ is called the
\textbf{wave number}
and $\omega$ is the \textbf{the angular frequency}.
\item The equality $\omega = k^{3}$ describes the relationship between the
angular frequency and the
wave number and is often referred to as the
\textbf{dispersion relation}.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Free Schr\"{o}dinger Equation}

Consider the partial differential equation 
\[
%\label{eq:nlpde:freeschrodinger}
\psi_{t}
	= i \, \psi_{xx}
\]
referred to as the free
Schr\"odinger equation for free particles.

\vspace*{0.10in}

If $\psi(x, t) = e^{i (k x - \omega t)}$ is a solution,
the following
dispersion relation holds, 
\[
-i \, \omega
	= -i \, k^{2}
\iff
\omega
	= k^{2}	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Solitons}

\begin{itemize}
\item A class of traveling wave solution, of particular
importance for nonlinear partial differential equations, is
the
\textbf{solitary wave} or
\textbf{soliton}, solution which is a special type of
traveling wave solution.
\item A solitary wave solution has two important
features.
\begin{itemize}
\item It must have a permanent form, meaning that its
profile will not change as it propagates with time.
\item It is localized, meaning that it decays to 0 or approaches a
constant at infinity.
\end{itemize}
\item A soliton solution is a solitary wave
solution that is stable in the sense that it maintains its wave
profile even after colliding with another soliton.
\end{itemize}

\end{frame}
%%

%\begin{frame}\frametitle{Homework}

%\begin{itemize}
%\item Read Section $x.y$
%\item Exercises: 
%\end{itemize}

%\end{frame}
%%

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\end{document}

%%%%
%%%% EOF
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