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\title{Instabilities of one-dimensional stationary solutions of the cubic nonlinear Schr\"{o}dinger equation}
\date{Draft - \today}
\author{Roger J. Thelwell\dag, John D. Carter\ddag, Bernard Deconinck\dag \\ \\
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  \dag Department of Applied Mathematics& \ddag Mathematics Department \\
  \, University of Washington & \,  Seattle University\\
  \, Seattle, WA 98195 & \, Seattle, WA 98122
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\begin{abstract}
We consider the two-dimensional cubic nonlinear Schr\"{o}dinger (NLS) equation,
which admits a large family of one-dimensional bounded traveling-wave 
solutions.
\end{abstract}

\section{Introduction}
\label{s:Intro}

The cubic nonlinear Schr\"{o}dinger (NLS) equation in two spatial dimensions is
given by
\begin{equation}
\label{E:NLS}
i\psi_t + \alpha \psi_{xx} + \beta \psi_{yy} + |\psi|^2 \psi = 0.
\end{equation}
The NLS equation is said be {\em focusing} or
{\em attractive} in the $x$-dimension if $\alpha >0$.  If $\alpha < 0$,
NLS is said to be {\em defocusing} or {\em repulsive} in the 
$x$-dimension.  Similarly, the sign of $\beta$ leads to focusing 
or defocusing in the $y$-dimension~\cite{sulemsulem}.
The NLS equation is called {\em hyperbolic} if 
$\alpha \beta <0$ and {\em elliptic} if $\alpha \beta > 0$. 

Equation \eqref{E:NLS} admits a large family of one-dimensional bounded traveling-wave solutions.
All such solutions, modulo Lie group symmetries~\cite{sulemsulem}, may be written in the form~\cite{carr1,carr2}
\begin{equation}
\label{E:soln_form}
\psi(x,t)=\phi(x) e^{i\theta(x) + i \lambda t},
\end{equation}
where $\phi(x)$ and $\theta(x)$ are real-valued functions, and $\lambda$ is a
real constant.  Solutions of the form \eqref{E:soln_form} are possible if
\begin{subequations}
\label{E:defs}
\begin{eqnarray}
\phi^2(x)&=& \displaystyle{\alpha \left(-2 k^2 \mbox{sn}^2(x,k)+B \right),}\label{E:phi_def} \\
\theta(x)&=& \displaystyle{c\int_0^x\phi^{-2}(\xi)d\xi,}  \label{E:theta_def} \\
\lambda&=& \displaystyle{\frac{1}{2}\alpha (3B-2(1+k^2)),} \label{E:omega_def} \\
c^2&=& \displaystyle{-\frac{\alpha^2}{2} B(B-2k^2)(B-2),} \label{E:c_def}
\end{eqnarray}
\end{subequations}
where $c$ is a real constant.

\begin{figure}
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   fig 1 & fig 2 \\
   (a) &  (b)
  \end{tabular}
  \caption{Admissible parameter space  for solutions of the form given in \eqref{E:soln_form} for (a) focusing ($\alpha = 1$) and 
(b) defocusing ($\alpha = -1$) regimes.  The gray interior region corresponds to NTP solutions.\label{F:parameter_space}}
 \end{center}
\end{figure}
Figure \ref{F:parameter_space} includes the region of $(k,B)$-parameter 
space corresponding to NTP solutions of the NLS equation.  

\section{Summary}

In this paper, we considered the spectral instability of one-dimensional 
traveling-wave nontrivial-phase (NTP) solutions of the cubic nonlinear Schr\"{o}dinger equation.

\subsection*{Acknowledgements}
We gratefully recongnize support from the National Science Foundation: NSF-DMS 0139093.

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