{ \centering{\begin{minipage}{4.5in}\baselineskip=10pt\footnotesize \parindent=0pt #1\par \parindent=15pt #2\par \parindent=15pt #3 \end{minipage}}\par}
gr-qc0111002.tex
%--- updated in 29/9/93

\documentstyle[twoside,epsf]{article}

%--------------------NEW ADDITIONS TO EXISTING ARTICLE.STY---------------------
\catcode`\@=11
\long\def\@makefntext#1{
\protect\noindent \hbox to 3.2pt {\hskip-.9pt
$^{{\eightrm\@thefnmark}}$\hfil}#1\hfill}		%CAN BE USED

\def\thefootnote{\fnsymbol{footnote}}
\def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}}	%ORIGINAL

\def\ps@myheadings{\let\@mkboth\@gobbletwo
\def\@oddhead{\hbox{}
\rightmark\hfil\eightrm\thepage}
\def\@oddfoot{}\def\@evenhead{\eightrm\thepage\hfil
\leftmark\hbox{}}\def\@evenfoot{}
\def\sectionmark##1{}\def\subsectionmark##1{}}

%--------------------START OF FILE
%THIS STYLE FILE IS FOR REFERENCES FROM 1--9 ITEMS ONLY

%THIS STYLE FILE WILL PRODUCE
%$^1$ IN BODY TEXT AND 1. AT REFERENCE SECTION

\oddsidemargin=\evensidemargin
\addtolength{\oddsidemargin}{-30pt}
\addtolength{\evensidemargin}{-30pt}

%------------------------------------------------------------------------------
\newcommand{\symbolfootnote}{\renewcommand{\thefootnote}
	{\fnsymbol{footnote}}}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\newcommand{\alphfootnote}
	{\setcounter{footnote}{0}
	 \renewcommand{\thefootnote}{\sevenrm\alph{footnote}}}

%------------------------------------------------------------------------------
%NEWLY-DEFINED SECTION COMMANDS
\newcounter{sectionc}\newcounter{subsectionc}\newcounter{subsubsectionc}
\renewcommand{\section}[1] {\vspace{12pt}\addtocounter{sectionc}{1}
\setcounter{subsectionc}{0}\setcounter{subsubsectionc}{0}\noindent
	{\tenbf\thesectionc. #1}\par\vspace{5pt}}
\renewcommand{\subsection}[1] {\vspace{12pt}\addtocounter{subsectionc}{1}
	\setcounter{subsubsectionc}{0}\noindent
	{\bf\thesectionc.\thesubsectionc. {\kern1pt \bfit #1}}\par\vspace{5pt}}
\renewcommand{\subsubsection}[1] {\vspace{12pt}\addtocounter{subsubsectionc}{1}
	\noindent{\tenrm\thesectionc.\thesubsectionc.\thesubsubsectionc.
	{\kern1pt \tenit #1}}\par\vspace{5pt}}
\newcommand{\nonumsection}[1] {\vspace{12pt}\noindent{\tenbf #1}
	\par\vspace{5pt}}

%NEW MACRO TO HANDLE APPENDICES
%\newcounter{appendixc}
%\newcounter{subappendixc}[appendixc]
%\newcounter{subsubappendixc}[subappendixc]
%\renewcommand{\thesubappendixc}{\Alph{appendixc}.\arabic{subappendixc}}
%\renewcommand{\thesubsubappendixc}
%        {\Alph{appendixc}.\arabic{subappendixc}.\arabic{subsubappendixc}}

%\renewcommand{\appendix}[1] {\vspace{12pt}
%        \refstepcounter{appendixc}
%        \setcounter{figure}{0}
%        \setcounter{table}{0}
%        \setcounter{lemma}{0}
%        \setcounter{theorem}{0}
%        \setcounter{corollary}{0}
%        \setcounter{definition}{0}
%        \setcounter{equation}{0}
%        \renewcommand{\thefigure}{\Alph{appendixc}.\arabic{figure}}
%        \renewcommand{\thetable}{\Alph{appendixc}.\arabic{table}}
%        \renewcommand{\theappendixc}{\Alph{appendixc}}
%        \renewcommand{\thelemma}{\Alph{appendixc}.\arabic{lemma}}
%        \renewcommand{\thetheorem}{\Alph{appendixc}.\arabic{theorem}}
%        \renewcommand{\thedefinition}{\Alph{appendixc}.\arabic{definition}}
%        \renewcommand{\thecorollary}{\Alph{appendixc}.\arabic{corollary}}
%        \renewcommand{\theequation}{\Alph{appendixc}.\arabic{equation}}
%	\noindent{\tenbf Appendix \theappendixc. #1}\par\vspace{5pt}}
%        \noindent{\tenbf Appendix \theappendixc #1}\par\vspace{5pt}}
%\newcommand{\subappendix}[1] {\vspace{12pt}
%        \refstepcounter{subappendixc}
%        \noindent{\bf Appendix \thesubappendixc. {\kern1pt \bfit #1}}
%        \par\vspace{5pt}}
%\newcommand{\subsubappendix}[1] {\vspace{12pt}
%        \refstepcounter{subsubappendixc}
%        \noindent{\rm Appendix \thesubsubappendixc. {\kern1pt \tenit #1}}
%        \par\vspace{5pt}}

%------------------------------------------------------------------------------
%FOLLOWING THREE COMMANDS ARE FOR `LIST' COMMAND.
\topsep=0in\parsep=0in\itemsep=0in
\parindent=15pt

%------------------------------------------------------------------------------
\newcommand{\textlineskip}{\baselineskip=13pt}
\newcommand{\smalllineskip}{\baselineskip=10pt}

%------------------------------------------------------------------------------
%MACRO FOR COPYRIGHT BLOCK
\def\eightcirc{
\begin{picture}(0,0)
\put(4.4,1.8){\circle{6.5}}
\end{picture}}
\def\eightcopyright{\eightcirc\kern2.7pt\hbox{\eightrm c}}

\newcommand{\copyrightheading}[1]
	{\vspace*{-2.5cm}\smalllineskip{\flushleft
      {\footnotesize Mod. Phys. Lett. A, submitted for publication   #1}\\
       {\footnotesize $\eightcopyright$\, World Scientific Publishing Company
      %{\footnotesize \LaTeX $~$ by H.C. Rosu (1999); physics/9908019 #1}\\
    %{\footnotesize Los Alamos electronic archives: gr-qc/01... v1 #1}\\
      % {\footnotesize $\eightcopyright$\\
        }\\
	 }}

%------------------------------------------------------------------------------
%MACRO FOR PUBLISHER INFORMATION SECTION
\newcommand{\pub}[1]{{\begin{center}\footnotesize\smalllineskip
	Received #1\\
	\end{center}
	}}

\newcommand{\publisher}[2]{{\begin{center}\footnotesize\smalllineskip
	Received #1\\
	Revised #2
	\end{center}
	}}

%------------------------------------------------------------------------------
%MARCO FOR ABSTRACT BLOCK
\def\abstracts#1#2#3{{
	\centering{\begin{minipage}{4.5in}\baselineskip=10pt\footnotesize
	\parindent=0pt #1\par
	\parindent=15pt #2\par
	\parindent=15pt #3
	\end{minipage}}\par}}

%------------------------------------------------------------------------------
%MARCO FOR KEYWORDS BLOCK
\def\keywords#1{{
	\centering{\begin{minipage}{4.5in}\baselineskip=10pt\footnotesize
	{\footnotesize\it Keywords}\/: #1
	 \end{minipage}}\par}}

%------------------------------------------------------------------------------
%NEW MACRO FOR BIBLIOGRAPHY
\newcommand{\bibit}{\nineit}
\newcommand{\bibbf}{\ninebf}
\renewenvironment{thebibliography}[1]
	{\frenchspacing
	 \ninerm\baselineskip=11pt
	 \begin{list}{\arabic{enumi}.}
        {\usecounter{enumi}\setlength{\parsep}{0pt}
	 \setlength{\leftmargin 12.7pt}{\rightmargin 0pt} %FOR 1--9 ITEMS
%	 \setlength{\leftmargin 17pt}{\rightmargin 0pt}   %FOR 10--99 ITEMS
%	 \setlength{\leftmargin 22pt}{\rightmargin 0pt}   %FOR 100+ABOVE ITEMS
         \setlength{\itemsep}{0pt} \settowidth
	{\labelwidth}{#1.}\sloppy}}{\end{list}}

%------------------------------------------------------------------------------
%LIST ENVIRONMENTS
\newcounter{itemlistc}
\newcounter{romanlistc}
\newcounter{alphlistc}
\newcounter{arabiclistc}
\newenvironment{itemlist}
    	{\setcounter{itemlistc}{0}
	 \begin{list}{$\bullet$}
	{\usecounter{itemlistc}
	 \setlength{\parsep}{0pt}
	 \setlength{\itemsep}{0pt}}}{\end{list}}

\newenvironment{romanlist}
	{\setcounter{romanlistc}{0}
	 \begin{list}{$($\roman{romanlistc}$)$}
	{\usecounter{romanlistc}
	 \setlength{\parsep}{0pt}
	 \setlength{\itemsep}{0pt}}}{\end{list}}

\newenvironment{alphlist}
	{\setcounter{alphlistc}{0}
	 \begin{list}{$($\alph{alphlistc}$)$}
	{\usecounter{alphlistc}
	 \setlength{\parsep}{0pt}
	 \setlength{\itemsep}{0pt}}}{\end{list}}

\newenvironment{arabiclist}
	{\setcounter{arabiclistc}{0}
	 \begin{list}{\arabic{arabiclistc}}
	{\usecounter{arabiclistc}
	 \setlength{\parsep}{0pt}
	 \setlength{\itemsep}{0pt}}}{\end{list}}

%------------------------------------------------------------------------------
%FIGURE CAPTION
%\newcommand{\fcaption}[1]{
%        \refstepcounter{figure}
%        \setbox\@tempboxa = \hbox{\footnotesize Fig.~\thefigure. #1}
%        \ifdim \wd\@tempboxa > 5in
%           {\begin{center}
%        \parbox{5in}{\footnotesize\smalllineskip Fig.~\thefigure. #1}
%            \end{center}}
%        \else
%             {\begin{center}
%             {\footnotesize Fig.~\thefigure. #1}
%              \end{center}}
%        \fi}

%TABLE CAPTION
%\newcommand{\tcaption}[1]{
%        \refstepcounter{table}
%        \setbox\@tempboxa = \hbox{\footnotesize Table~\thetable. #1}
%        \ifdim \wd\@tempboxa > 5in
%           {\begin{center}
%        \parbox{5in}{\footnotesize\smalllineskip Table~\thetable. #1}
%            \end{center}}
%        \else
%             {\begin{center}
%             {\footnotesize Table~\thetable. #1}
%              \end{center}}
%        \fi}

%------------------------------------------------------------------------------
%MACROS FOR SETTING \cite{x} OR \citeup{x}
\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout
	{\string\citation{#2}}\fi
\def\@citea{}\@cite{\@for\@citeb:=#2\do
	{\@citea\def\@citea{,}\@ifundefined
	{b@\@citeb}{{\bf ?}\@warning
	{Citation `\@citeb' on page \thepage \space undefined}}
	{\csname b@\@citeb\endcsname}}}{#1}}

\newif\if@cghi
\def\cite{\@cghitrue\@ifnextchar [{\@tempswatrue
	\@citex}{\@tempswafalse\@citex[]}}
\def\citelow{\@cghifalse\@ifnextchar [{\@tempswatrue
	\@citex}{\@tempswafalse\@citex[]}}
\def\@cite#1#2{{$\null^{#1}$\if@tempswa\typeout
	{IJCGA warning: optional citation argument
	ignored: `#2'} \fi}}
\newcommand{\citeup}{\cite}

%________________________________________________________
%THIS IS FOR NORMAL-SIZED CITES LIKE (REF. 5).
%CALLED BY "\refcite{X}".
\def\@refcitex[#1]#2{\if@filesw\immediate\write\@auxout
	{\string\citation{#2}}\fi
\def\@citea{}\@refcite{\@for\@citeb:=#2\do
	{\@citea\def\@citea{, }\@ifundefined
	{b@\@citeb}{{\bf ?}\@warning
	{Citation `\@citeb' on page \thepage \space undefined}}
	\hbox{\csname b@\@citeb\endcsname}}}{#1}}

\def\@refcite#1#2{{#1\if@tempswa\typeout
        {IJCGA warning: optional citation argument
	ignored: `#2'} \fi}}

\def\refcite{\@ifnextchar[{\@tempswatrue
	\@refcitex}{\@tempswafalse\@refcitex[]}}

%------------------------------------------------------------------------------


%------------------------------------------------------------------------------
%FOR SUB/SUPERSCRIPT BOLDFACED + ITALICS
\def\pmb#1{\setbox0=\hbox{#1}
	\kern-.025em\copy0\kern-\wd0
	\kern.05em\copy0\kern-\wd0
	\kern-.025em\raise.0433em\box0}
\def\mbi#1{{\pmb{\mbox{\scriptsize ${#1}$}}}}
\def\mbr#1{{\pmb{\mbox{\scriptsize{#1}}}}}

%------------------------------------------------------------------------------
%FOR FNSYMBOL FOOTNOTE AND ALPH{FOOTNOTE}
\def\fnm#1{$^{\mbox{\scriptsize #1}}$}
\def\fnt#1#2{\footnotetext{\kern-.3em
	{$^{\mbox{\scriptsize #1}}$}{#2}}}

%------------------------------------------------------------------------------
%FOR CREATING THE OPENING PAGE NUMBER
\def\fpage#1{\begingroup
\voffset=.3in
\thispagestyle{empty}\begin{table}[b]\centerline{\footnotesize #1}
	\end{table}\endgroup}

%------------------------------------------------------------------------------
%MACRO FOR RUNNINGHEAD
\def\runninghead#1#2{\pagestyle{myheadings}
\markboth{{\protect\footnotesize\it{\quad #1}}\hfill}
{\hfill{\protect\footnotesize\it{#2\quad}}}}
\headsep=15pt

%------------------------------------------------------------------------------
\font\tenrm=cmr10
\font\tenit=cmti10
\font\tenbf=cmbx10
\font\bfit=cmbxti10 at 10pt
\font\ninerm=cmr9
\font\nineit=cmti9
\font\ninebf=cmbx9
\font\eightrm=cmr8
\font\eightit=cmti8
\font\eightbf=cmbx8
\font\sevenrm=cmr7
\font\fiverm=cmr5

%------------------------------------------------------------------------------
\newtheorem{theorem}{\indent Theorem}
%OR \newtheorem{theorem}{Theorem}[sectionc] WHICH HAS SECTION NUMBER

\newtheorem{lemma}{Lemma}
%OR USE \newtheorem{lemma}{Lemma}[sectionc]

\newtheorem{definition}{Definition}
\newtheorem{corollary}{Corollary}
%USERS CAN ADD ON THEIR OWN NEW THEOREM-LIKE ENVIRONMENTS.

\newcommand{\proof}[1]{{\tenbf Proof.} #1 $\Box$.}

%--------------------END OF MPLA1.STY------------------------------------------



%--------------------START OF DATA FILE----------------------------------------
\textwidth=5truein
\textheight=7.8truein

%------------------------------------------------------------------------------
%DEFINITIONS
\def\qed{\hbox{${\vcenter{\vbox{			%HOLLOW SQUARE
   \hrule height 0.4pt\hbox{\vrule width 0.4pt height 6pt
   \kern5pt\vrule width 0.4pt}\hrule height 0.4pt}}}$}}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}	%USE SYMBOLIC FOOTNOTE

%------------------------------------------------------------------------------
\begin{document}

%\input crum1.tex
%
\newpage

\runninghead{Haret C. Rosu}
{KdV barotropic solitons}

\normalsize\textlineskip
\thispagestyle{empty}
\setcounter{page}{1}

\copyrightheading{}    %{Vol. 0, No.0 (1992) 000--000}

\vspace*{0.88truein}

%\fpage{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\centerline{Mod. Phys. Lett. A 16 (2001) xxx
%[gr-qc/0003108 v4]}
\bigskip
\centerline{\bf KdV ADIABATIC INDEX SOLITONS} %IN BAROTROPIC FRW COSMOLOGIES}
\centerline{\bf IN BAROTROPIC OPEN FRW COSMOLOGIES}
%INDICES\footnote{
%This essay received a ``honorable mention'' in the
%        A E Competition
%        F for the year 000 --- Ed.}}
\vspace*{0.035truein}
%\centerline{\bf MANUSCRIPTS USING COMPUTER SOFTWARE\footnote{For
%the title, try not to use more than 3 lines. Typeset the title
%in 10 pt Times Roman, uppercase and boldface.}}
\vspace*{0.37truein}
%\centerline{\footnotesize NAME}
%\footnote{Typeset names in
%10 pt Times Roman, uppercase. Use the footnote to indicate the
%present or permanent address of the author.}}
%\vspace*{0.015truein}
%\centerline{}
%\baselineskip=10pt
%\centerline{\footnotesize\it City, State ZIP/Zone,
%Country\footnote{State completely without abbreviations, the
%affiliation and mailing address, including country. Typeset in 8
%pt Times Italic.}}
\vspace*{10pt}
\centerline{\footnotesize HARET C. ROSU}
\vspace*{0.015truein}
%\centerline{\footnotesize [Received 17 August 1999] }
%\baselineskip=10pt
\centerline{\footnotesize  Instituto de F\'{\i}sica,
Universidad de Guanajuato, Apdo Postal E-143, 37150 Le\'on, Gto, Mexico}
\vspace*{0.225truein}
%\publisher{(May 15, 2000)}{}%{(May 14, 2000)}
%\centerline{\footnotesize Received (16 May 2000) }

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{0.21truein}
\abstracts{Applying standard mathematical methods,
it is explicitly shown how the Riccati equation
for the Hubble parameter $H(\eta)$ of
barotropic open FRW cosmologies is connected with a
Korteweg-de Vries equation for adiabatic index solitons.
It is also shown how one can embed a discrete sequence of adiabatic indices
of the type $n^2(\frac{3}{2}\gamma -1)^2$ ($\gamma \neq 2/3$)
in the ${\rm sech}$ FRW adiabatic index soliton.   }{}{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\vspace*{10pt}
%\keywords{The contents of the keywords}

\textlineskip                  %) USE THIS MEASUREMENT WHEN THERE IS
\vspace*{12pt}                 %) NO SECTION HEADING

\vspace*{1pt}\textlineskip	%) USE THIS MEASUREMENT WHEN THERE IS
%\section{General Appearance}    %) A SECTION HEADING
\vspace*{-0.5pt}
\noindent

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%PACS number(s):  98.80.Hw, 11.30.Pb

\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newpage

%\pagebreak

%\textheight=7.8truein
%\setcounter{footnote}{0}
%\renewcommand{\thefootnote}{\alph{footnote}}


\bigskip
\bigskip

%\section{The Main Text}

{\bf 1} - {\bf Riccati equation for barotropic FRW cosmologies}

\bigskip

\noindent
Recently, Faraoni,\cite{Far} showed that the equations
describing barotropic FRW cosmologies can be combined in a simple
Riccati equation leading in a much easier way
to the textbook solutions for the FRW scale factors.
Faraoni obtained the following cosmological Riccati equation
%(${\rm c > 0}$ henceforth)
$$
 \frac{dH}{d\eta}=-cH^2-\kappa c~,
\eqno(1a)
$$
for the log derivative of the FRW scale factor, the famous Hubble
parameter $H(\eta)=\frac{da/d\eta}{a}$.
The independent variable is the conformal time $\eta$, $c$ is simply related
to the adiabatic index of the cosmological fluid,
$c=\frac{3}{2}\gamma -1$ which was assumed constant by Faraoni,
and $\kappa=0,\pm1$ is the curvature index of the flat, closed, open
FRW universe, respectively. For the mathematical considerations of
this work we are more interested in the open cosmologies
$\kappa =- 1$. The flat case is special and has been discussed in a
previous work,\cite{flat} whereas the case $\kappa =+ 1$ will be commented in
subsection 3.2 herein.
%The adiabatic index $\gamma$ is defined through
%the barotropic equation of state $P=(\gamma -1)\rho$, where $P$ and $\rho$
%are, respectively, the pressure and energy density of the cosmological fluid
%under consideration. It is a constant quantity in the approach of Faraoni.
%It is a simple matter to write
%the first Friedmann dynamical equation $\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}
%(\rho +P)$ in the form $\frac{\ddot{a}}{a}=-\frac{4\pi G\rho}{3}c$ for the
%barotropic case. Thus, the common lore is that
%a negative $c$ ($\gamma < \frac{2}{3}$)
%implies an accelerating universe as
%claimed in the astrophysics of supernovae.

We notice that from the mathematical point of view, Faraoni's FRW
Riccati equation being of constant coefficients is directly integrable.
The solutions are
$$
H^+=-\tan (c\eta -\phi _+) ~, \qquad H^-={\rm tanh}(c\eta - \phi _-)~,
\eqno(1b)
$$
for the closed and open FRW universes, respectively. Of course, depending on
the initial conditions, the solutions can be written in terms of the
corresponding cofunctions as well. The minus sign in front of the phases has
been chosen for later convenience.

\bigskip
\newpage

{\bf 2 -} {\bf Cosmological Korteweg de Vries equation}

\bigskip


\noindent
We now pay special attention to the $\kappa =-1$ case for which a connection
with the KdV equation is possible.
Employing the well-known change of function
$H^{-}=\frac{1}{c}\frac{w^{'}}{w}$ one can pass from the nonlinear Riccati
equation to the linear second-order differential equation
$$
w^{''}- c^2w=0~,
\eqno(2)
$$
where the prime means derivation with respect to $\eta$.
This equation can be factorized
$$
A_1A_2w=0
\eqno(3)
$$
by means of the operators
$$
A_1=\left(\frac{d}{d\eta}+ W\right)~,
\qquad \qquad
A_2=\left(\frac{d}{d\eta}-W\right)~,
\eqno(4)
$$
where $W=cH^{-}$.
The general linear solution (the general zero mode) is of the form
$$
w(\eta, T)=\alpha (T) e^{c\eta}+\beta (T) e^{-c\eta}~,
\eqno(5)
$$
where we have assumed that the superposition constants are functions of a
new coordinate $T$ that is independent of $\eta$.
It is easy to show the relationship between the two superposition constants
and the phase $\phi _-$
$$
\phi _{-} (T)=\frac{1}{2}\ln \beta /\alpha~.
\eqno(6)
$$
Inverting as in SUSY QM the order of application of the factorization
operators, one gets an equation of the type
$$
A_2A_1u=0 \qquad \qquad {\rm or} \qquad \qquad u^{''}+c_1^2(\eta , T)u=0~,
\eqno(7)
$$
where $c_1(\eta ,T)$ is the ${\rm sech}$ soliton with the asymptotic wings going
to $c^2$
%From here, one can get the Bargmann potential
$$
c_1^2(\eta ,T)=c^2-2c^2{\rm sech} ^2(c\eta -\phi _{-} (T))~.
\eqno(8)
$$
It is now easy to apply basic results from the
KdV mathematics allowing to interpret
$c_1^2$ as a single ${\rm sech}$ soliton of the following KdV
equation,\cite{KdV}
$$
\frac{dC}{dT}-6C\frac{dC}{d\eta}+\frac{d^3C}{d\eta ^3}=0~,
\eqno(9)
$$
if the condition $\frac{d\phi _-}{dT}=4c^3$ is imposed. Moreover, by choosing
initial conditions at $T=0$, one can see that $T$ is the evolution variable
whereas the conformal time plays a role similar to a spatial variable.\cite{fmp}
\bigskip


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip
\bigskip

\newpage

{\bf 3 -} {\bf SUSY recursive procedure}

\bigskip

\noindent
{\bf 3.1 -}
The scheme presented in the previous section can be iterated in a simple and
well-known manner.\cite{r,hr} We put $T={\rm const}$, i.e., we consider an
instantaneous soliton profile
and take $\phi _- =0$ for simplicity. Thus, we start with a
`quantum' mechanical system
in a constant `potential' $c^2$ and relate it
to a Schr\"odinger-type equation in the conformal time domain which has a
fundamental `frequency' at $-c^2$.
We solve the ``fermionic" Riccati equation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%$$
%W^2_{1}-W^{'}_{1}+\omega _{d}^{2}=0~,
%\eqno(3)
%$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
i.e.,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
W^2_{1}-W^{'}_{1}-c^2=0~,
\eqno(10)
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
to find Witten's superpotential $W_1(\eta)=-c\tanh [c\eta]$
and next go to the ``bosonic" Riccati equation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
W^2_{1}+W^{'}_{1}+c _{1}^{2}(\eta)-c^{2}=0~,
\eqno(11)
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
in order to get $c _{1}^{2}(\eta)=-2c^{2}
{\rm sech}^2[c\eta]$. Moreover, one can write the Schr\"odinger
equation corresponding to the ``bosonic" Riccati equation as follows
%%%%%%%%%%%%%%%%%%%%%%%
$$
-\tilde{y}^{''}+c _{1}^{2}(\eta)\tilde{y}=-c^{2}\tilde{y}~,
\eqno(12)
$$
%%%%%%%%%%%%%%%%%%%%%%%
with the solution
$\tilde{y}\propto c\,{\rm sech}(c\eta)$.
The physical picture is that of a ${\rm sech}$ soliton profile
containing a
single mode self-trapped at $-c^2$
within the frequency pulse. One can employ the scheme recursively to get
several localized modes embedded in the soliton
profile.  %${\rm sech}$ chirp pulse.
Indeed, suppose we would like to introduce $N$
adiabatic indices of the type $c _{n}^{2}=-n^2c^{2}$, $n=1,...N$ in the
${\rm sech}$ pulse. Then, one has to solve the sequence of equations
$$
W_{n}^{2}-W_{n}^{'}=c _{n-1}^2+n^{2}c^{2}~,
\eqno(13)
$$
$$
W_{n}^{2}+W_{n}^{'}=c _{n}^2+n^{2}c^{2}~,
\eqno(14)
$$
inductively for $n=1...N$.\cite{r,hr} The soliton
containing $N$ adiabatic indices $n^2c^{2}$, $n=1...N$ is of the form
$c _{N}^{2}(\eta)=-N(N+1)c^{2}{\rm sech}^2(c\eta)$. The
corresponding modes can be written in a compact form as follows
%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
\tilde{y}_{n}(\eta ;N)\approx A^{\dagger}(\eta ;N)A^{\dagger}(\eta ;N-1)
A^{\dagger}(\eta ;N-2)...A^{\dagger}(\eta ;N-n+2){\rm sech} ^{N-n+1}
c\eta~,
\eqno(15)
$$
%%%%%%%%%%%%%%%%%%%%%%%
i.e., by applying the first-order operators
$A^{\dagger}(\eta;a_{n})=-\frac{d}{d\eta}-a_{n}c\tanh (c\eta)$,
where $a_{n}=N-n$, onto the ``ground state" sech mode.
%This scheme can be easily generalized to embedding `frequencies' of the
%type $-c _{i}^2=f _{i}$ and moreover, to the
%construction of chirp profiles having a given continuous spectrum of
%localized modes but we shall not pursue this task here.

%\bigskip

%{\bf 3.2 -} We notice
%that an interesting, polar analysis of the chirp `frequency' profile can
%be performed by means of the change of variable
%$\eta={\rm ln}({\rm tan}\frac{\theta}{2})$ leading to an associated Legendre
%equation in the spherical polar coordinate $\theta$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%$$
%\frac{d^2\tilde{y}}{d\theta ^2}+\cot \theta\frac{d\tilde{y}}{d\theta}+
%\Big[N(N+1)-\frac{n^{2}}{\sin ^{2}\theta}\Big]\tilde{y}=0~.
%\eqno(12)
%$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip

\noindent
{\bf 3.2 -} For the case $\kappa =1$ the ``fermionic"
Riccati equation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
W^2_{1}-W^{'}_{1}+c^{2}=0
\eqno(16)
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
leads to the solution $W_{1}=c\,{\rm tan}(c\eta)$ and from
the ``bosonic" Riccati equation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
W^2_{1}+W^{'}_{1}+c _{1}^{2}+c^{2}=0~,
\eqno(17)
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
one will find $c _{1}^{2}(\eta)
=2c^{2}{\rm sec}^{2}(c\eta)$. Consequently, the Schr\"odinger-like equation
%%%%%%%%%%%%%%%%%%%%%%%%
$$
-\tilde{y}^{''}+c _{1}^{2}(\eta)\tilde{y}=c^{2}\tilde{y}
\eqno(18)
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
has solutions of the type
$\tilde{y}\propto c\,{\rm sec}(c\eta)$, which is not of the localized soliton
type, and therefore the approach leads to unphysical results.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%$$
%{\rm c_{\kappa,susy}(\eta)=-u_{p}^{'}+cu_{p}^2=
%\left\{ \begin{array}{ll}
%c(1+2{\rm tan}^2 c\eta) & \mbox{if $\kappa >1$}\\
%c(-1+2{\rm tanh}^2 c\eta) & \mbox{if $\kappa <1$}
%\end{array} \right.}
%\eqno(7d)
%$$
%\end{equation}




\bigskip


{\bf 4 - Conclusion}

\bigskip

\noindent
A cosmological KdV equation for adiabatic index solitons
in the realm of barotropic FRW models has been
introduced in this work. A simple recursive nonrelativistic SUSY scheme has
been also applied
to show that in the barotropic soliton profile one can embed a discrete
sequence of adiabatic indices. Each eigenmode of the soliton corresponds
to one of the embedded adiabatic indices.


\bigskip
\bigskip
\bigskip
\bigskip

KdVc.tex




%\nonumsection{Acknowledgements}

%\noindent
%The authors thank
%This work was supported by a Project from CONACyT (No.~x).

\newpage
\bigskip
\noindent
{\bf References}

\begin{thebibliography} {000}
\bibitem{Far} V. Faraoni,
%``Solving for the dynamics of the universe",
Am. J. Phys. {\bf 67}, 732 (1999) [physics/9901006].
See also, J.A.S. Lima,  Am. J. Phys., to appear, [astro-ph/0109215].
For supersymmetric approach with application to
cosmological acceleration, see H.C. Rosu, Mod. Phys. Lett. A {\bf 15},
979 (2000) [gr-qc/0003108].


\bibitem{flat} H.C. Rosu,  Mod. Phys. Lett. A {\bf 16}, 1147 (2001)
[gr-qc/0003108].

\bibitem{KdV}
It seems that a water KdV sech soliton has been first observed in a
shallow drainage channel during the month of August 1834 by J.S. Russell, who
followed it on horseback in a chase of one or two miles. The KdV equation has
been first obtained in 1895.

\bibitem{fmp}
Using the direction of propagation as an evolution coordinate is a well-known
procedure for pulse propagation in nonlinear materials and in waveguide physics,
see, e.g., G. Fibich, V.M. Malkin, G.C. Papanicolau,
Phys. Rev. A {\bf 52}, 4218 (1995).

\bibitem{r}
J.L. Rosner,
Ann. Phys. {\bf 200}, 101 (1990); W. Kwong and J.L. Rosner, Prog.
Theor. Phys. Suppl. {\bf 86}, 366 (1986).

\bibitem{hr}
H. Rosu, J.L. Romero, J. Socorro, Nuovo Cimento B {\bf 113}, 549 (1998)
[physics/9707012].

\end{thebibliography}


%\bibitem{das} A. Das, {\em Integrable Models}, World Scientific, 1989;
%For recent work, see e.g., A. Das and Z. Popowicz, Phys. Lett. A {\bf 274},
%30 (2000) and references therein.


%\newpage

%%%%%%%%%%%%%%%%%
%\vskip 2ex
%\centerline{
%\epsfxsize=180pt
%\epsfbox{hubble1.eps}}
%\vskip 12ex
%\begin{center}
%{\small{Fig. 1}:$\quad$%\\
%The function $dH_{11}^{0}/dt+
%(H_{11}^{0})^{2}=\frac{d^2a_{11}/dt^2}{a_{11}}$
%showing that for this
%superfield component there is
%cosmological acceleration (of different strength) for all values of $c$
%(here $c\in [-0.1, 1]$, $t\in[0.15, 2.5])$. At small times,
%one can see a deflationary phase.}
%\end{center}
%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%
%\vskip 2ex
%\centerline{
%\epsfxsize=180pt
%\epsfbox{hubble2.eps}}
%\vskip 12ex
%\begin{center}
%{\small{Fig. 2}:$\quad$%\\
%The same plot as in Fig. 1 for the same time interval and
%$c\in [-0.9, -0.1]$ . One can see the strong
%acceleration effect corresponding to fluids in the neighborhood of the
%cosmological vacuum and also traces of the deflationary phase.}
%\end{center}
%%%%%%%%%%%%%%%%


\end{document}




gr-qc0111002.extracted_bib
\begin{thebibliography} {000}
\bibitem{Far} V. Faraoni,
%``Solving for the dynamics of the universe",
Am. J. Phys. {\bf 67}, 732 (1999) [physics/9901006].
See also, J.A.S. Lima,  Am. J. Phys., to appear, [astro-ph/0109215].
For supersymmetric approach with application to
cosmological acceleration, see H.C. Rosu, Mod. Phys. Lett. A {\bf 15},
979 (2000) [gr-qc/0003108].


\bibitem{flat} H.C. Rosu,  Mod. Phys. Lett. A {\bf 16}, 1147 (2001)
[gr-qc/0003108].

\bibitem{KdV}
It seems that a water KdV sech soliton has been first observed in a
shallow drainage channel during the month of August 1834 by J.S. Russell, who
followed it on horseback in a chase of one or two miles. The KdV equation has
been first obtained in 1895.

\bibitem{fmp}
Using the direction of propagation as an evolution coordinate is a well-known
procedure for pulse propagation in nonlinear materials and in waveguide physics,
see, e.g., G. Fibich, V.M. Malkin, G.C. Papanicolau,
Phys. Rev. A {\bf 52}, 4218 (1995).

\bibitem{r}
J.L. Rosner,
Ann. Phys. {\bf 200}, 101 (1990); W. Kwong and J.L. Rosner, Prog.
Theor. Phys. Suppl. {\bf 86}, 366 (1986).

\bibitem{hr}
H. Rosu, J.L. Romero, J. Socorro, Nuovo Cimento B {\bf 113}, 549 (1998)
[physics/9707012].

\end{thebibliography}