center
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On the basis of a recently-proposed method to find solitary solutions of generalized nonlinear Schr\"{o}dinger equations \cite{1}-\cite{3}, the existence of an envelope solitonlike solutions of a nonlinear Schr\"{o}dinger equation containing an anti-cubic nonlinearity ($|\Psi|^{-4}\Psi $) plus a "regular" nonlinear part is investigated. In particular, in case the regular nonlinear part consists of a sum of a cubic and a quintic nonlinearities (i.e. $q_1|\Psi|^2\Psi+q_2|\Psi|^4\Psi$), an upper-shifted bright envelope solitonlike solution is explicitly found.
nlin0207054.tex
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%\title
\begin{center}
{\bf \Large Envelope Solitons of Nonlinear Schr\"{o}dinger
Equation with an Anti-cubic Nonlinearity\footnote{Work
supported by CRUI (Conferenza dei Rettori delle
Universit\'{a} Italiane) and DAAD (Deutscher Akademischer
Austauschdienst) within the research programm "Vigoni"
between University of Napoli Federico II and University
of Bayreuth  (Project {\it Nonlinear Phenomena,
Instability and Macroscopic Coherence in Charged-Particle
Beam Dynamics})}}
\end{center}

\medskip

%\author
\begin{center}{\bf \Large R. Fedele,
H. Schamel\footnote{Physikalisches
Institut, Universit\"{a}t Bayreuth, D-95440 Bayreuth,
Germany.\\ Email: hans.schamel@uni-bayreuth.de}, V.I.
Karpman\footnote{Racah Institute of Physics, Hebrew
University, Jerusalem 91904, Israel.\\ Email:
karpman@vms.huji.ac.il}, P.K. Shukla\footnote{Institut
f\"{u}r Theoretische Physik IV, Fakult\"{a}t f\"{u}r Physik und
Astronomie, Ruhr-Universit\"{a}t Bochum, D-44780 Bochum,
Germany and Department of Plasma Physics, Ume{\aa}
~University, SE-90187 Ume{\aa}, Sweden.\\ Email:
ps@tp4.ruhr-uni-bochum.de}}
\end{center}
%\address
\begin{center}{\it Dipartimento di Scienze Fisiche, Universit\`{a}
Federico II and INFN, \\ Complesso Universitario di M.S. Angelo,\\ Via
Cintia, I-80126 Napoli, Italy\\ Email: renato.fedele@na.infn.it}
\end{center}
%\maketitle

\bigskip

\begin{center}
{\bf Abstract}
\end{center}

%\abstracts
{On the basis of a recently-proposed method to find
solitary solutions of generalized nonlinear Schr\"{o}dinger
equations \cite{1}-\cite{3}, the existence of an envelope
solitonlike solutions of a nonlinear Schr\"{o}dinger equation
containing an anti-cubic nonlinearity ($|\Psi|^{-4}\Psi
$) plus a "regular" nonlinear part is investigated. In
particular, in case the regular nonlinear part consists
of a sum of a cubic and a quintic nonlinearities (i.e.
$q_1|\Psi|^2\Psi+q_2|\Psi|^4\Psi$), an upper-shifted
bright envelope solitonlike solution is explicitly
found.}

\bigskip\bigskip
\noindent
{\small PACS number(s): 52.35.Mw - Nonlinear waves;
05.45.Yv - Solitons;\\ 42.65 - Nonlinear optics; 67.57.Jj
- Collective modes in quantum fluids}

%\newpage
\medskip\medskip

%\noindent
%{\small }

\newpage
%\section{Introduction}
Let us consider the following generalized Korteweg-de
Vries equation (GKdVE)
\begin{equation}
a~{\frac{\partial u}{\partial s }}~-~G\left[u\right]~
{\frac{\partial u}{\partial
x}}~+~\frac{\nu^2}{4}~{\frac{\partial^3 u}{\partial
x^3}}~=~0~~~,
\label{gkdve}
\end{equation}
where $a$ and $\nu$ are real constants, and
$G\left[u\right]$ is a real functional of $u$, and the
following generalized nonlinear Schr\"{o}dinger equation
(GNLSE)
\begin{equation}
i\alpha{\frac{\partial\Psi}{\partial
s}}~+~{\frac{\alpha^2}{2}}{\frac{
\partial^2\Psi}{\partial x^2}}~-
~U\left[|\Psi|^2\right]\Psi~=~0~~~,
\label{gnlse}
\end{equation}
where $U\left[|\Psi|^2\right]$ is a real functional of
$|\Psi|^2$ and $\alpha$ is a real constant. \newline
Recently, it has been shown that a correspondence between
(\ref{gkdve}) and (\ref{gnlse}) has been constructed in
such a way that, provided that the following equations
are satisfied, i.e.
\begin{equation}
\nu~=~\alpha
\label{nu-alpha}
\end{equation}
\begin{equation}
u_0 a~=~-u_{0}^{2}~+~2c_0
\label{}
\end{equation}
\begin{equation}
G\left[u\right]~=~u\frac{dU\left[u\right]}{du}~
+~2U\left[u\right]~~~,
\label{functional-G}
\end{equation}
if $u(x-u_0 s)\equiv u(\xi)$ ($u_0$ being a real
constant) is a non--negative stationary--profile solution
of (\ref{gkdve}) thus
\begin{equation}
\Psi (x,s)=\sqrt{u(\xi)}\exp\left\{\frac{i}{\alpha}
\left[\phi_0-\left(c_0 + u_{0}^{2}\right)s+u_0 x
+A_0\int\frac{d\xi}{u (\xi)}\right]\right\}~~~,
\label{Psi-solution}
\end{equation}
is a stationary--profile envelope solution of
(\ref{gnlse}), where $\phi_0$, $c_0$, and $A_0$ are real
constants (not all independent) \cite{2}. Note that
$u(\xi) = |\Psi (x,s)|^2$. This result has been
successfully applied to find new analytical solitary-wave
envelope solutions of some modified nonlinear Schr\"{o}dinger
equations (MNLSE) containing high--order nonlinearities
(with respect to the standard cubic nonlinearity)
\cite{2,3}.  In particular, analytical solutions in the
form of bright and dark/grey envelope solitonlike
solutions have been found for a MNLSE with cubic plus
quintic nonlinear terms, i.e.
\begin{equation}
{\cal U}\left(|\Psi|^2\right)
= q_1|\Psi|^2 + q_2|\Psi|^4~~~.
\label{cal-U}
\end{equation}
In fact, it has been shown that the following MNLSE
\begin{equation}
i\alpha{\frac{\partial\Psi}{\partial
s}}~+~{\frac{\alpha^2}{2}}{\frac{
\partial^2\Psi}{\partial x^2}}~-
~\left[q_1~|\Psi|^2~+~q_2~|\Psi|^4\right]\Psi~=~0~~~,
\label{mnlse-quartic-bis}
\end{equation}
has the following envelope solitonlike solutions
\cite{2}:
\begin{eqnarray}
\Psi (x,s)& = &\sqrt{\overline{u}~\left[1~+~\epsilon~\mbox{sech}
\left(\xi /\Delta\right)\right]}\exp\left\{\frac{i}{\alpha}
\left[\phi_0 - A s + u_0 x \right]\right\}{\times}\nonumber\\
& {\times} & \exp\left\{\frac{i B}{\alpha}
\left[\frac{\xi}{\Delta}+\frac{2\epsilon}{\sqrt{1-\epsilon^2}}
\mbox{arctan}\left(\frac{\left(\epsilon -1\right)\mbox{tanh}
\left(\xi /2\Delta\right)}{\sqrt{1-\epsilon^2}}\right)
\right]\right\}~~~,
\label{envelope-soliton-quartic}
\end{eqnarray}
where $\phi_0$ still plays the role of arbitrary
constant,
$$\epsilon ={\pm}\sqrt{1- 32|q_2|\left(u_0 -
V_0\right)^2/\left(3q_{1}^{2}\right)}~~~,$$
$$\Delta
=|\alpha |/\left(2\sqrt{2|E'_{0}|}\right)~~~,$$

$$E'_0 =
-3q_{1}^{2}/\left(64|q_2|\right)~+~\left(u_0 -V_0\right)^2/2~~~,$$
provided that $E'_0 <0$ and $q_2 <0$ and
$$
-\sqrt{\frac{3q_{1}^{2}}{32|q_2|}}+V_0 <u_0 <
\sqrt{\frac{3q_{1}^{2}}{32|q_2|}}+V_0~~~,
$$
\begin{equation}
A~=~\frac{15 q_{1}^{2}}{64|q_2|}+\frac{\left(u_0 -
V_0\right)^2}{2} +\frac{u_{0}^{2}}{2}~~~,
\label{A-def}
\end{equation}
and
\begin{equation}
B~=-\frac{|\alpha|\left(u_0 -
V_0\right)}{2\sqrt{2|E'_0|}}~~~~.
\label{B-def}
\end{equation}

The following four cases have been discussed \cite{2}

\medskip
\noindent
(a). $0<\epsilon <1$ ($u_0 -V_0\neq 0$):
$$u(\xi=0)=(1+\epsilon)\overline{u}~,~~~\mbox{and}~~~
\lim_{\xi\rightarrow {\pm}\infty}u(\xi)=\overline{u}$$
which corresponds to a bright soliton of maximum
amplitude $(1+\epsilon)\overline{u}$ and up-shifted by
the quantity $\overline{u}$ ( {\it up-shifted bright
soliton}).

\bigskip
\noindent
(b). $-1<\epsilon <0$ ($u_0 -V_0\neq 0$):
$$u(\xi=0)=(1-\epsilon)\overline{u}~,~~~\mbox{and}~~~
\lim_{\xi\rightarrow {\pm}\infty}u(\xi)=\overline{u}$$
which is a dark soliton with minimum amplitude
$(1-\epsilon)\overline{u}$ and reaching asimptotically
the upper limit $\overline{u}$ (standard {\it gray
soliton}).

\bigskip
\noindent
(c). $\epsilon =1$ ($u_0 -V_0 = 0$):
$$u(\xi=0)=2\overline{u}~,~~~\mbox{and}~~~
\lim_{\xi\rightarrow {\pm}\infty}u(\xi)=\overline{u}$$
which corresponds to a bright soliton of maximum
amplitude $2\overline{u}$ and up-shifted by the maximum
quantity $\overline{u}$ ({\it upper-shifted bright
soliton}).

\bigskip
\noindent
(d). $\epsilon =-1$ ($u_0 -V_0 = 0$):
$$u(\xi=0)= 0~,~~~\mbox{and}~~~
\lim_{\xi\rightarrow {\pm}\infty}u(\xi)=\overline{u}$$
which is a dark soliton (zero minimum amplitude),
reaching asimptotically the upper limit $\overline{u}$
(standard {\it dark soliton}).

In this paper, we use the method mentioned above to find
envelope solitonlike solutions of the following MNLSE
containing, besides the cubic and quintic nolinearities,
an "anti-cubic" nonlinearity (i.e. $|\Psi|^{-4}\Psi$),
namely
\begin{equation}
i\alpha{\frac{\partial\Psi}{\partial
s}}~+~{\frac{\alpha^2}{2}}{\frac{
\partial^2\Psi}{\partial x^2}}~-
~\left[Q_0~|\Psi|^{-4}~+~q_1~|\Psi|^2~+~q_2~|\Psi|^4\right]\Psi~=~0~~~,
\label{mnlse-anti-cubic}
\end{equation}
where $Q_0$ is a real constant. \newline It is easily
seen that Eq. (\ref{functional-G}) has the following
general solution
\begin{equation}
U\left[u\right]~=~\frac{1}{u^2}\left[K_0~+~
\int~G\left[u\right]~u~du\right]\,,
\label{functional-solution}
\end{equation}
where $K_0$ is an arbitrary real constant. However, once
the function $u(\xi)$ is a stationary profile of
(\ref{gkdve}) with
\begin{equation}
G\left[u\right]~=~3q_1~u~+~4q_2~u^2\,,
\label{functional-G-1}
\end{equation}
it follows that, provided that $Q_0 = K_0$,
\begin{equation}
{\cal U}\left[u\right]~= ~{1\over u^2
}\int~G\left[u\right]~u~du\,,
\label{functional-solution-1}
\end{equation}
where ${\cal U}$ is defined by (\ref{cal-U}).
Consequently, it seems that the mapping
(\ref{Psi-solution}) allows us to construct solutions of
the (\ref{mnlse-anti-cubic}) which have the functional
form of the ones satisfying the
(\ref{mnlse-quartic-bis}). The only restriction for
(\ref{mnlse-anti-cubic}) is that $u(\xi)$ must not vanish
somewhere, which implies that we have to impose that
${u(\xi)}$ be a positive solitonlike solution of the
MKdVE of the type (\ref{gkdve}) with the nonlinearity
given by (\ref{functional-G-1}). In fact, the vanishing
of $u$ corresponds to a divergence of the nonlinear
potential term $|\Psi|^{-4}$ in (\ref{mnlse-anti-cubic}).
This circumstance excludes the standard dark solitonlike
solutions.
\newline On the other hand, it can be shown that \cite{2}
\begin{equation}
-\frac{\nu^2}{2}\frac{d^2 u^{1/2}}{d\xi^2
}~+~\frac{Q_0}{u^{3/2}}~+~\frac{1}{u^{3/2}}~
\int~G\left[u\right]~u~du~=~\left(c_0~+~
\frac{u_{0}^{2}}{2}\right)u^{1/2}~-~\frac{A_{0}^{2}}{2u^{3/2}}~~~.
\label{u-equation}
\end{equation}
It is then clear from (\ref{u-equation}) that a family of
solitary wave solutions of (\ref{mnlse-anti-cubic}) can
be obtained by imposing the following condition
\begin{equation}
A_0~=~{\pm}\sqrt{-2Q_0}\,,
\label{A-zero-condition}
\end{equation}
which implies that such a kind of family of solution
exists for negative values of $Q_0$. Consequently, Eq.
(\ref{u-equation}) becomes the following NLSE for
stationary states
\begin{equation}
-\frac{\alpha^2}{2}\frac{d^2 u^{1/2}}{d\xi^2
}~+~{\cal U}\left[u\right]u^{1/2}~=~E_0 u^{1/2}~~~,
\label{stationary-gnlse}
\end{equation}
where $E_0=c_0+u_{0}^{2}/2$. This equation admits
solitary solutions whose form is the squared modulus of
(\ref{envelope-soliton-quartic}). It follows that for any
\begin{equation}
Q_0~<~0~~~,
\label{Q-zero-condition}
\end{equation}
under condition (\ref{A-zero-condition}), solitary-wave solutions of
(\ref{mnlse-anti-cubic}) can be directly constructed from both
(\ref{Psi-solution}) and (\ref{envelope-soliton-quartic}) as
\begin{eqnarray}
& &\Psi_{{\pm}} (x,s)=\sqrt{\overline{u}~\left[1~+~\epsilon~\mbox{sech}
\left({\xi\over\Delta}\right)\right]}~{\times} \nonumber \\ & {\times} &
\exp\left\{\frac{i}{\alpha}
\left[\phi_0 -\left(E_0 + {u_{0}^{2}\over 2}\right)s+u_0 x
~{\pm}\sqrt{2|Q_0|}~
\int {d\xi \over\sqrt{\overline{u}~\left[1~+
~\epsilon~\mbox{sech}
\left(\xi /\Delta\right)\right]}}\right]\right\},
\label{Psi-solution-1}
\end{eqnarray}
where, in principle, according to Ref. \cite{2},
$\epsilon$ should be taken in the following range
\begin{equation}
-1~<\epsilon~\leq 1~~~,
\label{epsilon-condition}
\end{equation}
which excludes the standard "dark" solitary waves
($\epsilon
=-1$), namely the condition for which the modulus of $\Psi$
vanishes at $\xi = 0$. \newline Actually, the direct
substitution of $u=|\Psi|^2$ given by
(\ref{Psi-solution-1}) into the eigenvalue equation
(\ref{stationary-gnlse}) allows us to find
\begin{equation}
\epsilon~=~1~~~,
\label{epsilon-value}
\end{equation}
\begin{equation}
\overline{u}~=~-{3 q_1\over 8 q_2}~~~,
\label{u-overline}
\end{equation}
\begin{equation}
q_1~>~0~~~,~~~~~~q_2~<~0~~~,
\label{q1-q2-conditions}
\end{equation}
\begin{equation}
E_0~=~-{15 q_{1}^{2}\over 64 q_2}~~~,
\label{E-zero}
\end{equation}
\begin{equation}
\Delta=~{2|\alpha|\over q_1}\sqrt{2|q_2|\over 3}~~~.
\label{Delta}
\end{equation}
Consequently, solution (\ref{Psi-solution-1}) can be cast as
\begin{eqnarray}
&&\Psi_{{\pm}} (x,s)=\sqrt{{3 q_1\over 8 |q_2|}~\left[1~+~\mbox{sech}
\left({\xi\over\Delta}\right)\right]}~{\times} \nonumber \\ &{\times}&\exp\left\{\frac{i}{\alpha}
\left[\phi_0-\left({15 q_{1}^{2}\over 64 |q_2|} + {u_{0}^{2}\over 2}\right)s+u_0 x
~{\pm}\sqrt{2|Q_0|}~{16 |\alpha||q_2|\over 3q_{1}^{2}}\sqrt{2|q_2|\over
3}\left({\xi\over\Delta}-
\mbox{tanh}\left({\xi\over 2\Delta}\right)\right)\right]\right\}~~~,\nonumber \\
\label{Psi-solution-2}
\end{eqnarray}
where $u_0$ is a fully arbitrary soliton velocity.
According to the classification of the solitary waves
given in Ref. \cite{2}, the (\ref{Psi-solution-2})
represents an upper-shifted bright envelope solitonlike
solution of Eq. (\ref{mnlse-anti-cubic}), provided that
the coefficients $Q_0$, $q_1$ and $q_2$ satisfy the
conditions (\ref{Q-zero-condition}) and
(\ref{q1-q2-conditions}), respectively. It is clear that
Eq. (\ref{epsilon-value}), which does not contradict
condition (\ref{epsilon-condition}), implies that also
gray solitary solutions do not exist in the solution form
(\ref{Psi-solution-1}).\newline In conclusion, in this
paper, by using a recently developed method for solving a
wide family of MNLSE with high-order nonlinearities
\cite{1}-\cite{3} on the basis of the knowledge of the
solution of the associated MKdVE, an upper-shifted bright
envelope solitonlike solution of a MNLSE, containing,
besides the standard cubic and quintitic nonlinearities,
an "anti-cubic" nonlinear term (see Eq.
(\ref{mnlse-anti-cubic})~). Our analysis has shown that,
once the form of the envelope solution of
(\ref{mnlse-anti-cubic}) is taken according to
(\ref{Psi-solution-1}), dark and gray envelope solitary
solutions do not exist that, in contrast, exist in case
the "anti-cubic" term is missing.

\begin{thebibliography}{99}

\bibitem{1} R. Fedele and H. Schamel, to be published in
{\it Eur. Phys. J. B} {\bf 27}, issue n.3 (June 2002).

\bibitem{2} R. Fedele, {\it Physica Scripta} {\bf 65}, 506 (2002).

\bibitem{3} R. Fedele, H. Schamel and P. K. Shukla,
{\it Physica Scripta} {\bf T98}, 18 (2002)

\end{thebibliography}

\end{document}

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\oddsidemargin=0pt
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\begin{document}

%\title
\begin{center}
{\bf \Large Envelope Solitons of Nonlinear Schr\"{o}dinger
Equation with an Anti-cubic Nonlinearity\footnote{Work
supported by CRUI (Conferenza dei Rettori delle
Universit\'{a} Italiane) and DAAD (Deutscher Akademischer
Austauschdienst) within the research programm "Vigoni"
between University of Napoli Federico II and University
of Bayreuth  (Project {\it Nonlinear Phenomena,
Instability and Macroscopic Coherence in Charged-Particle
Beam Dynamics})}}
\end{center}

\medskip

%\author
\begin{center}{\bf \Large R. Fedele, H. Schamel\footnote{Physikalisches
Institut, Universit\"{a}t Bayreuth, D-95440 Bayreuth,
Germany.\\ Email: hans.schamel@uni-bayreuth.de}, V.I.
Karpman\footnote{Racah Institute of Physics, Hebrew
University, Jerusalem 91904, Israel.\\ Email:
karpman@vms.huji.ac.il}, P.K. Shukla\footnote{Institut
f\"{u}r Theoretische Physik IV, Fakult\"{a}t f\"{u}r Physik und
Astronomie, Ruhr-Universit\"{a}t Bochum, D-44780 Bochum,
Germany and Department of Plasma Physics, Ume{\aa}
~University, SE-90187 Ume{\aa}, Sweden.\\ Email:
ps@tp4.ruhr-uni-bochum.de}}
\end{center}
%\address
\begin{center}{\it Dipartimento di Scienze Fisiche, Universit\`{a}
Federico II and INFN, \\ Complesso Universitario di M.S. Angelo,\\ Via
Cintia, I-80126 Napoli, Italy\\ Email: renato.fedele@na.infn.it}
\end{center}
%\maketitle

a correspondence between a wide family of generalized
nonlinear Schr\"{o}dinger equations (GNLSEs) and a wide
family of generalized Korteweg-de Vries equations
(GKdVEs) is constructed in such a way that
stationary-profile solutions of the latter are the
squared modulus of stationary-profile envelope solutions
of the former. This is done by extending a recent work
\cite{1} that has considered a similar correspondence
involving the cubic nonlinear Schr\"{o}dinge equation (NLSE)
and the standard Korteweg-de Vries equation (KdVE).
\newline From one side, the investigation is carried out
considering the following GNLSE for a complex
wavefunction $\Psi$:
\begin{equation}
i\alpha{\frac{\partial\Psi}{\partial
s}}~+~{\frac{\alpha^2}{2}}{\frac{
\partial^2\Psi}{\partial x^2}}~-
~U\left[|\Psi|^2\right]\Psi~=~0~~~,
\label{gnlse}
\end{equation}
where the real constant $\alpha$ accounts for the
dispersive/diffractive effects and
$U\left[|\Psi|^2\right]$ is a real functional of
$|\Psi|^2$; from the other side, the following GKdVE for
the real function $u$ is considered:
\begin{equation}
a~{\frac{\partial u}{\partial s }}~-~G\left[u\right]~
{\frac{\partial u}{\partial
x}}~+~\frac{\alpha^2}{4}~{\frac{\partial^3 u}{\partial
x^3}}~=~0~~~,
\label{gkdve}
\end{equation}
where $a$ and $\alpha$ are real constants, and
$G\left[u\right]$ is a real functional of $u$
\footnote{It is easy to see that, without loss of
generality, we have chosen , in Eq. (\ref{gkdve}), a
positive coefficient of the third-derivative term.
Actually, the coefficient $a$ in principle can be
positive or negative in such a way that the dispersion
coefficient of (\ref{gkdve}), i.e. $\mu\equiv \nu^2
/4a$, can be positive or negative, as well}. In both equations,
$x$ is the 1-D configurational space coordinate and $s$ is the timelike
coordinate. In particular, in this paper, special attention will be devoted
to a correspondence between the special case of (\ref{gnlse}) when
$U\left[|\Psi |^2\right]=q_0|\Psi |^{2\beta}$, $q_0$ and $\beta$ being real
and positive real numbers, respectively, namely
\begin{equation}
i\alpha{\frac{\partial\Psi}{\partial
s}}~+~{\frac{\alpha^2}{2}}{\frac{
\partial^2\Psi}{\partial x^2}}~-
~q_0~|\Psi |^{2\beta}~\Psi~=~0~~~,
\label{mnlse}
\end{equation}
and the special case of (\ref{gkdve}) when
$G\left[u\right]=p_0 u^{\gamma}$, $p_0$ and $\gamma$
being real and positive real numbers, respectively,
namely
\begin{equation}
a~{\frac{\partial u}{\partial s }}~-~p_0~u^{\gamma}
{\frac{\partial u}{\partial
x}}~+~\frac{\nu^2}{4}~{\frac{\partial^3 u}{\partial
x^3}}~=~0~~~.
\label{mkdve}
\end{equation}
\newline  According to
other authors \cite{2}-\cite{5}, let us define
(\ref{mnlse}) as {\it modified nonlinear Schr\"{o}dinger
equation} (MNLSE) and (\ref{mkdve}) as {\it modified
Korteweg-de Vries equation} (MKdVE). \newline Our goals
are: (i) to construct a correspondence between
(\ref{gnlse}) and (\ref{gkdve}) in such a way that each
positive stationary-profile solution of (\ref{gkdve}),
namely $u(x,s)=u(\xi)$ with $\xi = x-u_0 s$ ($u_0$ being
a real dimensionless constant velocity), is the squared
modulus of a stationary-profile envelope solution of
(\ref{gnlse}) having the form
\begin{equation}
\Psi(x,s)~=~\sqrt{\rho (\xi )}~\exp \left[i\Theta (x,s)\right] ~~~,
\label{envelope-solution}
\end{equation}
with $\Theta$ real function and $\rho$ real and positive
function such that $u(\xi )=|\Psi (x,s)|^2=\rho (\xi )$,
and consider explicitly the conditions that guarantee the
existence of solitonlike solutions; (ii) to apply the
above correspondence to the case of (\ref{mnlse}) and
(\ref{mkdve}) and find for them solitonlike solutions,
generalizing the set of solitonlike solutions given
already in literature \cite{6,7}; (iii) to apply, in
turn, the results of (ii) to find envelope solitons of
(1) in the case
\begin{equation}
U(|\Psi |^2 )~=~q_1~|\Psi|^2~+~q_2~|\Psi|^4~~~.
\label{sum}
\end{equation}
\newline The framework in which the above goals will be
reached is the same as in Ref. \cite{1}, namely the
Madelung's fluid \cite{8} picture of the GNLSE
(\ref{gnlse}).
\newline
As it is well known, once $\Psi (x,s)=
\sqrt{\rho (x,s)}~\exp\left[i\Theta (x,s)/\alpha\right]$
is assumed, the (\ref{gnlse}) can be cast as a closed
system of nonlinear coupled equations (Madelung's fluid
\cite{8}): continuity + motion equation. However,
according to Ref.\cite{1}, this system can be, in turn,
reduced to the following closed system of coupled
equations:
\begin{equation}
{\frac{\partial \rho }{\partial s}} + {\frac{\partial
}{\partial x }}
\left(\rho V\right) =0\,,
\label{madelung-fluid-continuity}
\end{equation}
\begin{equation}
-\left({\frac{\partial V}{\partial s}}\right)\rho+V{\frac{\partial\rho}{
\partial s}}+2\left[c_0 (s)-\int\left({\frac{\partial V}{\partial s}}
\right)~dx\right]{\frac{\partial\rho}{\partial x}}-\left({\frac{\partial U}{
\partial x}}\rho+2~U{\frac{\partial\rho}{\partial x}}\right)~+~{\frac{
\alpha^2}{4}}{\frac{\partial^3\rho}{\partial x^3}}~=~0~~~,  \label{c}
\end{equation}
where $V$ is the current velocity, given by
\begin{equation}
V(x,s)={\frac{\partial \Theta (x,s)}{\partial x}}\,,
\label{V-definition}
\end{equation}
and $c_0 (s)$ is an arbitrary real function of $s$.

\section{Correspondence between solitons and envelope solitons}
Let us denote with ${\cal E}= \left\{\Psi\right\}$
the set of all the envelope
solutions (\ref{envelope-solution}) of (\ref{gnlse}),
i.e. the set of all the stationary-profile envelope
solutions of the GNLSE. Let us also denote with ${\cal
S}=
\left\{u(\xi)\geq 0\right\}$ the set of all non-negative
stationary-profile solutions of the GKdVE
(\ref{gkdve}).\newline In order to construct the above
correspondence between ${\cal E}$ and ${\cal S}$, we
observe that if $\Psi\in {\cal E}$, thus $\rho$ and $V$
have the form $\rho
=\rho (\xi)$ and $V= V(\xi)$, respectively.\newline Under
the above hypothesis, it is easy to see that: (a). $c_{0}
(s)$ becomes constant (so that, let us put $c_{0} (s)
\equiv c_0$); (b). continuity equation
(\ref{madelung-fluid-continuity}) becomes:
\begin{equation}
u_0\frac{d\rho}{d\xi} = \frac{d}{d\xi}\left(\rho
V\right)~~~,
\label{continuity-bis}
\end{equation}
which integrated gives:
\begin{equation}
V(\xi) = u_0 + \frac{A_0}{\rho (\xi)}~~~,
\label{solution-continuity}
\end{equation}
where $A_0$ is an arbitrary constant. By combining
(\ref{c}) and (\ref{continuity-bis}), we easily have:
\begin{equation}
\left(u_{0}^{2}+2c_0\right)\frac{d\rho}{d\xi}~-~
{\cal
I}\left[\rho\right]\frac{d\rho}{d\xi}~+~\frac{\alpha^2}{4}
\frac{d^3\rho}{d\xi^3}~=~0~~~,
\label{stationary-gkdve}
\end{equation}
where the functional ${\cal I}\left[\rho\right]$ is
defined as:
\begin{equation}
{\cal
I}\left[\rho\right]~=~\rho\frac{dU\left[\rho\right]}{d\rho}~
+~2U\left[\rho\right]~~~.
\label{functional-I}
\end{equation}
On the other hand, for stationary-profile solution
$u=u(\xi)$, Eq. (\ref{gkdve}) becomes:
\begin{equation}
-u_{0}a~\frac{du}{d\xi}~-~
G\left[u\right]\frac{du}{d\xi}~+~\frac{\nu^2}{4}
\frac{d^3 u}{d\xi^3}~=~0~~~.
\label{stationary-gkdve-bis}
\end{equation}
Consequently, (\ref{stationary-gkdve}) and
(\ref{stationary-gkdve-bis}) have the same solutions, if
the same boundary conditions are taken for them and
provided that their coefficients are respectively
proportional. In particular, it follows that $u(\xi)$ is
a non-negative stationary-profile solution of the
following GKdVE ($u_0\neq 0$):
\begin{equation}
-\frac{u_{0}^{2}+2c_0}{u_0}~{\frac{\partial \rho}{\partial s }}~
-~{\cal I}\left[\rho\right]~
{\frac{\partial \rho}{\partial
x}}~+~\frac{\alpha^2}{4}~{\frac{\partial^3 \rho}{\partial
x^3}}~=~0~~~,
\label{gkdve-bis}
\end{equation}
where the following identies have been used:
\begin{equation}
a\equiv
-\left(u_{0}^{2}+2c_0\right)/u_0~,~~~~~G\left[u\right]\equiv
{\cal I}\left[u\right]~,~~~~~\nu\equiv\alpha~~~.
\label{identities}
\end{equation}
Thus, it results that, starting from Eq. (\ref{gnlse}),
we have constructed the following correspondence:
\begin{eqnarray}
{\cal F}~&:&~\Psi\in {\cal E}\rightarrow u\in {\cal
S}~~~,\nonumber \\ u~&=&~{\cal
F}\left[\Psi\right]~=~|\Psi|^2~=~\rho (\xi)~~~.
\label{correspondence-F}
\end{eqnarray}
${\cal F}$ associates a stationary-profile envelope
solution of (\ref{gnlse}) to a stationary-profile
solution of the associated GKdVE (\ref{gkdve-bis}). In
particular, it may associate an envelope solitonlike
solution of (\ref{gnlse}) with a solitonlike solution of
(\ref{gkdve}).
\newline It is worth to observe that,
by substituting (\ref{envelope-solution})
directly in (\ref{gnlse}), separating real and imaginary
parts, and taking into account (\ref{V-definition}) we
get:
\begin{equation}
-\partial\Theta /\partial s = c_0 + u_{0}V~~~,
\label{Theta-solution}
\end{equation}
which, combined with (\ref{solution-continuity}) becomes:
\begin{equation}
\Theta (x,s)~=~\phi_0~-~\left(c_0 + u_{0}^{2}\right)s~+~u_0 x~
+~A_0~\int~\frac{d\xi}{\rho (\xi)}~~~,
\label{Theta-solution-bis}
\end{equation}
where $\phi_0$ is an arbitrary real constant.\newline
Now, let $u(\xi)$ be a positive stationary-profile
solution of (\ref{gkdve}). Thus, $u$ satisfies an
equation similar to (\ref{stationary-gkdve-bis}) and,
provided that (\ref{stationary-gkdve}) and
(\ref{stationary-gkdve-bis}) have still proportional
coefficients, in correspondence of the same boundary
conditions, $u$ is also solution of
(\ref{stationary-gkdve}). Thus, by defining the
quantity
\begin{equation}
\tilde{V}\equiv \frac{\partial\tilde{\Theta} (x,s)}
{\partial x}~~~,
\label{V-tilde}
\end{equation}
with
\begin{equation}
\tilde{\Theta} (x,s)~=~\phi_0~-~\left(c_0 +
u_{0}^{2}\right)s~+~u_0
x~+~A_0~\int~\frac{d\xi}{u(\xi)}~~~,
\label{Theta-tilde}
\end{equation}
it follows that $\rho(\xi)=u(\xi)$ and $\tilde{V}$ are
solutions of the following system of coupled equations:
\begin{equation}
{\frac{\partial u }{\partial s}} + {\frac{\partial
}{\partial x }}
\left(u \tilde{V}\right) =0\,,
\label{madelung-fluid-continuity-bis}
\end{equation}
\begin{equation}
-\left({\frac{\partial \tilde{V}}{\partial s}}\right)u+\tilde{V}{\frac{\partial u}{
\partial s}}+2\left[c_0 (s)-\int\left({\frac{\partial \tilde{V}}{\partial s}}
\right)~dx\right]{\frac{\partial u}{\partial x}}-
\left({\frac{\partial {\cal U}}{
\partial x}}u+2~{\cal U}{\frac{\partial u}{\partial x}}\right)~+~{\frac{
\nu^2}{4}}{\frac{\partial^3 u}{\partial x^3}}~=~0~~~,
\label{c-bis}
\end{equation}

In conclusion, for bright solitonlike solutions,
satisfying the boundary conditions (\ref{u-bright}), $K_0
= A_0
=0$, and the phase of $\Psi$ is linear (see Eq.
(\ref{correspondence-F-ter})~); furthermore, $u(\xi)$ and
the constant $E_0$ play, respectively, the role of
eigenstate and eigenvalue of the GNLSE
(\ref{stationary-gnlse}). In particular, the role played
by the constant $E_0$ suggests the suitability to choose
it as independent parameter instead of $c_0$.
\newline If non-negative solitonlike solutions (for instance,
up-shifted bright solitons, gray solitons) of
(\ref{gkdve}) do not satisfy the above boundary
conditions (\ref{u-bright}), it is useful to introduce
the following positions:
\begin{equation}
u(\xi)=\overline{u}~+~u_1 (\xi)~~~,
\label{u-dark-solution}
\end{equation}
where the constant $\overline{u}$ is
positive~\footnote{Note that in this paper we are dealing
with positive solutions $u(\xi)$, only. This implies that
$|u_1|<\overline{u}$, if $u_1 <0$.} and
\begin{equation}
V(\xi)=V_0~+~V_1 (\xi)~~~.
\label{V-dark-solution}
\end{equation}
Thus, the following boundary are imposed:
\begin{equation}
\lim_{\xi\rightarrow {\pm}\infty}u_1 (\xi)~=~0~~~,
\label{u-dark-condition}
\end{equation}
and, correspondingly
\begin{equation}
\lim_{\xi\rightarrow {\pm}\infty}V_1 (\xi)~=~0~~~.
\label{V-dark-condition}
\end{equation}
It follows from (\ref{solution-continuity}) that ($\rho
= u$):
\begin{equation}
A_0~=~-\left(u_0-V_0\right)\overline{u}~~~,
\label{A-zero-dark}
\end{equation}
with $V_0 \neq u_0$, which implies that $A_0 \neq 0$~\footnote {The case
$V_0 =u_0$ corresponds to $A_0 =0$ which selects bright solitons.}.
Consequently, according to (\ref{correspondence-F-bis}) the phase of
solution $\Psi$ is not linear, because the last term is not vanishing.
Additionally, since $A_0\neq 0$, it is clear that, in general (whether
$K_0$ is zero or not zero), solution $u(\xi)$ of Eq. (\ref{u-equation}) and
the constant $E_0
= c_0 +u_{0}^{2}/2$ do not play the
role of eigenstate and eigenvalue of the energy,
respectively. This last cirmustance can be obtained only
by choosing
$$
K_0 \equiv - \frac{A_{0}^{2}}{2}~~~.
$$
It should be stressed that this particular choice of
$K_0$ corresponds to standard bright envelope solitonlike
solutions having linear phase (i.e. $K_0 =-A_{0}^{2}/2=
0$) and satisfying the boundary conditions
(\ref{u-bright}) or corresponds to envelopes having
nonlinear phase (i.e. $K_0
=-A_{0}^{2}/2\neq 0$) that are eigenstates of the energy.
\newline Let us suppose that a finite value
of $\xi$, say $\xi_0$ ($|\xi_0|<\infty$), such that $u(\xi_0)=0$, exists.
Thus, in order to keep limited solution for $V$, Eq.
(\ref{solution-continuity}) implies that $A_0
=0$ ($u_0 =V_0$). Consequently, for any $\overline{u}>0$,
the solitonlike solutions, satifying the relationships
(\ref{u-dark-solution})-(\ref{V-dark-condition}),
satisfy, additionally, both the condition
$u(\xi_0)=0$~\footnote {$u_1$ must be negative and they
are called dark solitons.} and the (\ref{u-equation})
with $K_0 =0$. This means that they are eigenstates of
the energy. Correspondingly, dark envelope solitonlike
solutions of (\ref{gnlse-bis}) have only a linear phase
(in $x$ and in $s$).
\newline The results presented above may suggest a
method for finding stationary-profile solutions of an equation belonging in
one family (whether GNLSE or GKdVE) if stationary-profile solutions of the
associated equation, belonging in the other family (whether GKdVE or
GNLSE), are known. \newline It should be noted that in principle Eq.
(\ref{gnlse-bis}) must be compared with the particular nonlinear
Schr\"{o}dinger equation that we are actually going to solve. In particular,
the potential $U\left[\right|\Psi|^2]$ must be compared with the one of Eq.
(\ref{gnlse-bis}) (i.e. ${\cal U}\left[u=|
\Psi|^2\right]$~) given by (\ref{functional-solution}~). If
$U\left[\right|\Psi|^2]$ has a regular asymptotic behaviuor
($lim_{\xi\rightarrow{\pm}\infty}|U\left[\right|\Psi|^2]|<\infty$), the above
comparison implies that $K_0 =0$. Consequently, if $u(\xi)>0$ is a
solitonlike solution satisfying the relationships
(\ref{u-dark-solution})-(\ref{V-dark-condition}) with $\overline{u}>0$, it
follows that $A_0 \neq 0$ ($u_0\neq V_0$). Consequently, $u(\xi)$ can be an
up-shifted bright solitonlike solution (with $u_1 >0$) or a gray
solitonlike solution (with $u_1<0$).

\section{Applications}
To give some explicit
examples, this method will be applied here to the family
of equations represented by (\ref{mnlse}) and
(\ref{mkdve}) (MNSE and MKdVE, respectively), restricting
the investigation to the solitonlike solutions,
only.\newline First of all, the simplest case concerning
$\beta =1$ in (\ref{mnlse}), which defines the standard
NLSE \cite{9}, and $\gamma=1$ in (\ref{mkdve}), which
defines the standard KdVE \cite{9}, has been already
considered in Ref.\cite{1}, where envelope solitons
(bright, gray/dark) of the NLSE have been found starting
from the solitonlike solution (bright, gray or dark) of
the associated KdVE and using a sort of correspondence
${\cal H}$, described above. Here, $\beta$ and $\gamma$
are allowed to be arbitrary positive real numbers.
\newline As first example, let us look for bright
envelope solitons of (\ref{mnlse}). To this end, it is
worth to observe that, for $au_0>0$ and $p_0 <0$, Eq.
(\ref{mkdve}) admits the following positive solitonlike
solution (for any positive real $\gamma$):
\begin{equation}
u(\xi)=
\left[\frac{au_0\left(\gamma +2\right)\left(\gamma+1
\right)}{2 |p_0|}\right]^{1/\gamma}~\mbox{sech}^{2/\gamma}~
\left[\frac{\gamma\sqrt{au_0}}{|\nu|}\xi\right]~~~.
\label{u-soliton}
\end{equation}
($\gamma=1$ recovers the bright KdV soliton \cite{9}). It should be pointed
out that, since $\gamma$ is an arbitrary positive real number, solutions
(\ref{u-soliton}) represent an extension of the usual ones that in the
literature have been given only for the case of $\gamma$ positive integer
\cite{5}-\cite{7}. They satisfy the boundary conditions (\ref{u-bright}).
Consequently, solutions (\ref{u-soliton}) are bright solitons of the MKdVE
\footnote{It should be noted that, only in the case of $\gamma$ even
integer, if $u$ is a positive solution of (\ref{u-soliton}), thus $-u$ is a
negative solution. In this case, since $u$ is a bright soliton, solution
$-u$ could be  called {\it anti-bright soliton}.}. According to the
(\ref{gkdve}), for the case under discussion we can write:
\begin{equation}
G\left[u\right]~=~p_0u^{\gamma}~~~.
\label{G-of-u}
\end{equation}
Provided that the following identification
\begin{equation}
u~=~\rho~=~|\Psi|^2
\label{identification}
\end{equation}
is made, the substitution of (\ref{G-of-u}) in
(\ref{gnlse-bis}), gives the following associated MNLSE:
\begin{equation}
i\nu{\frac{\partial\Psi}{\partial
s}}~+~{\frac{\nu^2}{2}}{\frac{
\partial^2\Psi}{\partial x^2}}~-
~\frac{~p_0}{\gamma +2}|\Psi|^{2\gamma}~ \Psi~=~0~~~,
\label{gnlse-ter}
\end{equation}
where (\ref{A-zero-bright}) and (\ref{K-zero-bright})
have been used (note that $K_0 =0$). We must identify the
potential of this equation with the one of the MNLSE
(\ref{mnlse}), which is the equation that we are going to
solve. Thus, after a comparison between (\ref{mnlse}) and
(\ref{gnlse-ter}), we easily get:
\begin{equation}
\nu=\alpha~,~~~~~\gamma =\beta~,~~~~~ p_0 =
(\beta+2)q_0~~~.
\label{parameters}
\end{equation}
Finally, by virtu of (\ref{correspondence-F-ter}) we can
write the following bright envelope solitonlike solutions
of (\ref{mnlse}) for any real positive $\beta$:
\begin{eqnarray}
& &\Psi (x,s)~=~\left[\frac{|E_0|\left(\beta +1
\right)}{|q_0|}\right]^{1/2\beta}\mbox{sech}^{1/\beta}~
\left[\frac{\beta\sqrt{2|E_0|}}{|\alpha|}\xi\right]{\times}\nonumber \\ &
&{\times}\exp\left\{\frac{i}{\alpha}
\left[\phi_0-\left(E_0 + u_{0}^{2}/2\right)s+u_0 x
\right]\right\}~~~,
\label{mnlse-solution}
\end{eqnarray}
where the first of (\ref{identities}), (\ref{u-soliton})
and (\ref{parameters}) have been used, and where $E_0<0$
and $q_0<0$. TABLE I and TABLE II display the plots of
$\rho $ and $\sqrt{\rho}$, as function of $\xi
/\Delta$, for some values of the parameter $\beta$,
$E_0$, $\alpha$, and $q_0$, according to
(\ref{mnlse-solution}).
\newline As second example, let us find envelope
solitonlike solutions for the following MNLSE with a
quartic potential given by (\ref{sum}). Since ($\rho
=|\Psi|^2$)
$$U=U\left[\rho\right]=q_1\rho+q_2\rho^2~, $$
according to the theory
presented above, it is easy to see that we have to use
the correspondence ${\cal H}$ starting from the following
MKdVE:
\begin{equation}
a~{\frac{\partial u}{\partial s
}}~-~\left[p_0\left(u-\overline{u}\right)^2+\delta_0\right]
{\frac{\partial u}{\partial
x}}~+~\frac{\nu^2}{4}~{\frac{\partial^3 u}{\partial
x^3}}~=~0~~~,
\label{mkdve-quadratic}
\end{equation}
where $p_0$, $\overline{u}$, and $\delta_0$ are constant
to be determined. Thus, according to (\ref{gnlse-bis})
the following MNLSE can be written:
\begin{equation}
i\nu{\frac{\partial\Psi}{\partial
s}}~+~{\frac{\nu^2}{2}}{\frac{
\partial^2\Psi}{\partial x^2}}~-
~\left[K_0|\Psi|^{-4}+\frac{p_0}{4}|\Psi|^4-
\frac{2p_0\overline{u}}{3}|\Psi|^2+
\frac{1}{2}\left(\delta_0+p_0\overline{u}^2\right)\right]~
\Psi~=~0~~~,
\label{mnlse-quartic}
\end{equation}
which must be compared with the one that we have to
solve. Consequently, we get: $\nu =\alpha$, $K_0 =0$,
$p_0
=4q_2$, $\overline{u}=-3q_1/(8q_2)$, and $\delta_0
=-9q_{1}^{2}/16q_2$. \newline
It is easily recognized that (\ref{mkdve-quadratic}) can
be cast as (\ref{mkdve}) for $\gamma =2$. Thus, taking
into account the (\ref{u-soliton}), positive solitonlike
solutions of (\ref{mkdve-quadratic}) are immediately
found:
\begin{equation}
u(\xi)=
\overline{u}~\left[1~+~\epsilon~\mbox{sech}
\left(\xi /\Delta\right)\right]~~~,
\label{u-soliton-quadratic}
\end{equation}
where
$$\epsilon ={\pm}\sqrt{1- 32|q_2|\left(u_0 -
V_0\right)^2/\left(3q_{1}^{2}\right)}~~~,$$
$$\Delta
=|\alpha |/\left(2\sqrt{2|E'_{0}|}\right)~~~,$$
and
$$E'_0 =
-3q_{1}^{2}/\left(64|q_2|\right)~+~\left(u_0 -V_0\right)^2/2~~~,$$
provided that $E'_0 <0$ and $q_2 <0$ and
$$
-\sqrt{\frac{3q_{1}^{2}}{32|q_2|}}+V_0 <u_0 <
\sqrt{\frac{3q_{1}^{2}}{32|q_2|}}+V_0~~~.
$$
Eq. (\ref{u-soliton-quadratic}) shows that we can
distinguish the following four cases.

\bigskip
\noindent
(a). $0<\epsilon <1$ ($u_0 -V_0\neq 0$):
$$u(\xi=0)=\overline{u}(1+\epsilon)~,~~~\mbox{and}~~~
\lim_{\xi\rightarrow {\pm}\infty}u(\xi)=\overline{u}$$
which corresponds to a bright soliton of maximum amplitude
$(1+\epsilon)\overline{u}$ and up-shifted by the quantity $\overline{u}$.
We could call it {\it up-shifted bright soliton}.

\bigskip
\noindent
(b).$-<\epsilon <0$ ($u_0 -V_0\neq 0$):
$$u(\xi=0)=\overline{u}(1-\epsilon)~,~~~\mbox{and}~~~
\lim_{\xi\rightarrow {\pm}\infty}u(\xi)=\overline{u}$$
which is a dark soliton with minimum amplitude $(1-\epsilon)\overline{u}$
and reaching asimptotically the upper limit $\overline{u}$. It corresponds
to a standard {\it gray soliton}.

\bigskip
\noindent
(c). $\epsilon =1$ ($u_0 -V_0 = 0$):
$$u(\xi=0)=2\overline{u}~,~~~\mbox{and}~~~
\lim_{\xi\rightarrow {\pm}\infty}u(\xi)=\overline{u}$$
which corresponds to a bright soliton of maximum amplitude $2\overline{u}$
and up-shifted by the maximum quantity $\overline{u}$. We could call it
{\it upper-shifted bright soliton}.

\bigskip
\noindent
(d).$\epsilon =-1$ ($u_0 -V_0 = 0$):
$$u(\xi=0)= 0~,~~~\mbox{and}~~~
\lim_{\xi\rightarrow {\pm}\infty}u(\xi)=\overline{u}$$
which is a dark soliton (zero minimum amplitude),
reaching asimptotically the upper limit $\overline{u}$.
It correspond to a standard {\it dark soliton}.\newline
TABLE III displays the plots of $\rho $, as a function of
$\xi /\Delta$, for given values of the parameters,
according to (\ref{u-soliton-quadratic}).
\newline Correspondingly, by means of the correspondence
${\cal H}$, we can conclude that the following MNLSE:
\begin{equation}
i\alpha{\frac{\partial\Psi}{\partial
s}}~+~{\frac{\alpha^2}{2}}{\frac{
\partial^2\Psi}{\partial x^2}}~-
~\left[q_1~|\Psi|^2~+~q_2~|\Psi|^4\right]\Psi~=~0~~~,
\label{mnlse-quartic-bis}
\end{equation}
has the following envelope solitonlike solutions:
\begin{eqnarray}
\Psi (x,s)& = &\sqrt{\overline{u}~\left[1~+~\epsilon~\mbox{sech}
\left(\xi /\Delta\right)\right]}\exp\left\{\frac{i}{\alpha}
\left[\phi_0 - A s + u_0 x \right]\right\}{\times}\nonumber\\
& {\times} & \exp\left\{\frac{i B}{\alpha}
\left[\frac{\xi}{\Delta}+\frac{2\epsilon}{\sqrt{1-\epsilon^2}}
\mbox{arctan}\left(\frac{\left(\epsilon -1\right)\mbox{tanh}
\left(\xi /2\Delta\right)}{\sqrt{1-\epsilon^2}}\right)
\right]\right\}~~~,
\label{envelope-soliton-quartic}
\end{eqnarray}
where $\phi_0$ still plays the role of arbitrary
constant,
\begin{equation}
A~=~\frac{15 q_{1}^{2}}{64|q_2|}+\frac{\left(u_0 -
V_0\right)^2}{2} +\frac{u_{0}^{2}}{2}~~~,
\label{A-def}
\end{equation}
and
\begin{equation}
B~=-\frac{|\alpha|\left(u_0 -
V_0\right)}{2\sqrt{2|E'_0|}}~~~~.
\label{B-def}
\end{equation}
\medskip


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\bibitem{3} R. Fedele, H. Schamel and P. K. Shukla,
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