% 9th Japan-Slovenia Seminar on Nonlinear Science (JPSI9) 
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Nonlinear dynamics of wave packets in Bose-Einstein condensates
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E. Schr\"{o}dinger  and \underline{K. Nakamura}$^{1}$
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Dublin Institute for Advanced Studies, Dublin 4, Ireland
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$^{1}$Department of Applied Physics, Osaka City University, Osaka 558-8585, Japan

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\noindent
Role of nonlinearity in the dual (wave-and-particle) aspect
of wave-packet dynamics in Bose-Einstein condensates (BECs)
is considered.  
The analysis is based on the nonlinear
Schr{\"o}dinger equation  (Pethick and Smith (2002)):
\begin{equation}
i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}
\nabla^2\psi + V\psi + U|\psi|^2\psi \quad .
\end{equation}
Following the spirit of our latest work (Nakamura (2007)),
we shall investigate nonlinear dynamics of BECs under a variety of situations.  
Firstly a wave nature of  
BECs falling through the double
slits is analyzed. Most of 
the activities so far around BEC analog of Young's interference
experiment had recourse to a double-well trap leading to the
2nd-order interference between initially separated two wave
packets (WPs) with uncontrolled independent phases. By contrast
we mimic the BEC analog of the 1st-order interference
(: a single electron interference).  The subject is identified
with a search for the fate of a soliton showing a head-on
collision with a hard-walled obstacle of finite size.
We explore the splitting of the wave packet and its
reorganization to form an interference pattern.  Particular
attention is paid to the role of gravity and repulsive
nonlinearity in the fringe pattern.

Secondly we examine a particle picture for interacting 
wave packets (WPs) and explore the emergence of chaos
in soliton molecules.

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\noindent{\bf References}
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C.J. Pethick and H. Smith, 2002
{\it Bose-Einstein Condensation in Dilute Gases}, 
Cambridge University Press.
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K. Nakamura, 
2007 {\it   Prog. Theor. Phys. Suppl.} {\bf 166},  179.

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