%%%%		-*- 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:      04/03/2025
%%%%
%%%% Purpose:   Nonlinear Schr\"{o}dinger Equation
%%%%

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\title{Nonlinear Schr\"{o}dinger Equation}
\subtitle{\textit{Partial Differential Equations}} % (optional)


\author % (optional, use only with lots of authors)
{J~Robert~Buchanan}
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%   affiliation.

\institute[Millersville University of Pennsylvania]
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{
  Department of Mathematics
}
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% - Keep it simple, no one is interested in your street address.

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

\subject{Talks}
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\DeclareMathOperator{\sech}{sech}
\begin{document}

\begin{frame}
  \titlepage
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%\begin{frame}
%  \frametitle{Outline}
%  \tableofcontents
  % You might wish to add the option [pausesections]
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% Since this a solution template for a generic talk, very little can
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% - Exactly two or three sections (other than the summary).
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% - 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 Schr\"{o}dinger wave equation, and 
\item find soliton solutions to the  Schr\"{o}dinger wave equation.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Standard Form of Nonlinear Schr\"{o}dinger
    Equation}

\[
%\label{eq:nlpde:schrodinger}
i \, \psi_{t} + \psi_{xx} + \gamma |\psi|^{2} \psi
	= 0	,
\]
where $i = \sqrt{-1}$ is the imaginary unit and $\psi$ is a complex-valued
function of $t$ and $x$.

\vspace*{0.10in}

Assume a plane wave solution of the form
\[
%\label{eq:nlpde:standingwave}
\psi(x, t)
	= A \, e^{i (k x - \omega t)}	,
\]
where $A$, $k$, and $\omega$ are constants to be determined.

\end{frame}
%%

\begin{frame}\frametitle{Plane Wave Solution}

\textbf{Recall}: the constant $k$ is
called the wave number and $\omega$ is the frequency.

\vspace*{0.10in}

Substitute the plane wave function in the nonlinear Schr\"{o}dinger
equation to produce
\[
i (-i \, \omega) + i^{2} k^{2} + \gamma \, A^{2}
	= 0	,
\]
which can be simplified as
\[
%\label{eq:nlpde:relation}
\omega
	= k^{2} - \gamma \, A^{2}	.
\]
Therefore, for any $k$
the function
\[
\psi(x, t)
	= A e^{i (k x - (k^{2} - \gamma A^{2}) t)}
\]
is a solution to the nonlinear
Schr\"{o}dinger equation.

\end{frame}
%%

\begin{frame}\frametitle{General Solution}

A more general solution to the nonlinear Schr\"{o}dinger equation
takes the form
\[
%\label{eq:nlpde:schtrav}
\psi(x, t)
	= f(x - c \, t) e^{i (k x - \omega t)}
\]
and the symbols $k$, $c$, and $\omega$ are
constants.
Note that this solution incorporates features of a traveling wave and
a plane wave.
Substitute this solution into the nonlinear
Schr\"{o}dinger equation,
\begin{align*}
0
	&= -c \, i f' + \omega f + f'' + 2 i \, k f' - k^{2} f + \gamma f^{3}	\\
	&= f'' + i (2 k - c) f' + (\omega - k^{2}) f + \gamma f^{3}	.
\end{align*}
Let $\xi = x - c \, t$ and assume $f$ is real-valued.
This implies $c = 2 k$ and
\[
f'' + (\omega - k^{2}) f + \gamma f^{3}
	= 0	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Solving for Function $f(\xi)$}

\[
f'' + (\omega - k^{2}) f + \gamma f^{3}
	= 0	.
\]
Multiply by $2 f'$ and antidifferente.
\[
%\label{eq:nlpde:feqn1}
(f')^{2}
	= A - (\omega - k^{2}) (f)^{2} - \frac{\gamma}{2} (f)^{4}	,
\]
where $A$ is an arbitrary constant of integration.

\vspace*{0.10in}

If the solution $\psi(x, t)$ and its derivative $\psi_{x}(x, t)$ vanish
as $x \to \pm\infty$, then $f \to 0$ and $f' \to 0$ as
$x \to \pm\infty$.
This implies $A = 0$ and function $f$ satisfies
\[
(f')^{2}
	= (k^{2} - \omega) (f)^{2} - \frac{\gamma}{2} (f)^{4}	.
\]
Solving for $f'$ produces
\[
\frac{df}{d\xi}
	= \pm |f| \sqrt{(k^{2} - \omega) - \frac{\gamma}{2} (f)^{2}}	.
\]

\end{frame}
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\begin{frame}\frametitle{Separate Variables}
\fontsize{9pt}{10.8}\selectfont

\[
\frac{df}{d\xi}
	= \pm |f| \sqrt{(k^{2} - \omega) - \frac{\gamma}{2} (f)^{2}}	.
\]
Separate variables in this ODE
and integrate both sides imply (taking the
negative root), 
\[
\xi + A
	= \int \frac{-1}{|f| \sqrt{(k^{2} - \omega) - \frac{\gamma}{2}
            (f)^{2}}} \, df
\]
where $A$ is an arbitrary constant of integration.

\vspace*{0.10in}

Make the change of variable,
\[
f
	= \sqrt{\frac{2 (k^{2} - \omega)}{\gamma}} \sech \zeta
\implies
df
	= -\sqrt{\frac{2 (k^{2} - \omega)}{\gamma}} \sech \zeta \tanh \zeta
\, d\zeta	.
\]
The indefinite integral is rewritten as
\begin{align*}
\int \frac{-1}{|f| \sqrt{(k^{2} - \omega) - \frac{\gamma}{2} (f)^{2}}} \,
  df
	&= \int \frac{1}{\sqrt{k^{2} - \omega}} \, d\zeta
	= \frac{1}{\sqrt{k^{2} - \omega}} \zeta	\\
	&= \frac{1}{\sqrt{k^{2} - \omega}}
\sech^{-1} \left( \sqrt{\frac{\gamma}{2 (k^{2} - \omega)}} f \right)	.
\end{align*}

\end{frame}
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\begin{frame}\frametitle{Solving for Function $\psi(x, t)$}

\[
\frac{1}{\sqrt{k^{2} - \omega}}
\sech^{-1} \left( \sqrt{\frac{\gamma}{2 (k^{2} - \omega)}} f \right)
	= \xi + A
\]
Solve for $f(\xi)$.
\[
%\label{eq:nlpde:schtravf}
f(\xi)
	= \sqrt\frac{2 (k^{2} - \omega)}{\gamma} \sech \left(
(k^{2} - \omega)^{1/2} (\xi + A) \right)
\]
Therefore, the following is a solution of the nonlinear
Schr\"{o}dinger equation,
\[
%\label{eq:nlpde:schtravsol}
\psi(x, t)
	= \sqrt{\frac{2 (k^{2} - \omega)}{\gamma}}
e^{i (k x - \omega t)} \sech \left(
(k^{2} - \omega)^{1/2} (x - 2 k \, t + A) \right)	.
\]
The solution found above is called an
\textbf{envelope soliton solution}.

\end{frame}
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\begin{frame}\frametitle{Illustration}

\begin{center}
\resizebox{0.75\textwidth}{!}{\includegraphics{schrodingersolns}}
\end{center}
The squared amplitude of two soliton solutions ($\psi_{1}(x,
  t) = e^{i t} \sech x$ and $\psi_{2}(x, t) = e^{i (-x/2 +3 t/4)}
  \sech (x + t)$) to the
  nonlinear Schr\"{o}dinger equation pass each other without
  constructive or destructive interference.

\end{frame}
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%\begin{frame}\frametitle{Homework}

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

%\end{frame}
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