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\title{INTERACTION OF TWO-DIMENSIONAL SPATIAL OPTICAL SOLITONS}
\author{V.A. Smirnov, L.A. Nesterov, N.N. Rosanov, N.V. Vyssotina}
{{\it Institute of Laser Physics,\\
Research Center "S.I. Vavilov State Optical Institute"\\
Birzhevaia linia 12, S.~Petersburg, 199034, Russia}\\
Phone 812 2189982, Fax 812 2181093, E-mail rosanov@ilph.spb.su
}

We present systematic analytical investigation and computer simulation
of stability and interaction of spatial transversal
2-d solitons in self-focusing medium with saturable nonlinearity.
 The formation of such spatial patterns is of great
interest as an example of compensation of diffraction by self-focusing.
The solitons of this kind are fundamental cylindrically symmetrical
mode of nonlinear wave equation which is stable in respect to small
perturbations in the medium with saturation . In the present work
it was shown for the first time that mode with topological index
M=1 nearest to the fundamental one is unstable. It decays
to some fundamental solitons. This circumstance allows us to believe
that only the fundamental mode can be considered as the spatial soliton,
that in its turn, awakes interest in studying the properties of that mode.
The analysis performed has revealed nonmonotonic dependence of effective
radius of the soliton on its power and the existence of minimum radius of
the soliton. In the case of the central (complanar) soliton collisions
there is a critical angle which divides the regions of strong and weak
interaction. The region of weak interaction corresponds to sufficiently
large angles of collisions described by two nonlinear parabolic equations
for coupled waves. In the work these equations were obtained and
analytical method of their solution on the basis of perturbation theory
was developed. This approach allows to determine correctly
asymptotical values of soliton parameters after their mutual scattering.
The investigations which have been carried out showed that, in the case of
central collision of solitons, they are not scattered and behave as
traditional solitons. The situation is changed principally in the
region of small angles of collisions.
Preliminary research of interaction and merging of such solitons was
performed in [1].
In this case  nonlinear interaction plays a
leading part in a process of collision. Exchange of power takes place at
asymmetrical collisions of the small power solitons. The exchange efficiency
depends on
collision angle and phase difference of solitons. The third soliton appears
after collision of solitons with higher power. Its power depends on collision
 angle, but
actually phase difference between beams on the boundary of interaction region
is of decisive importance. The differences of the colliding solitons phases
define
direction and efficiency of power exchange and appearance of a new soliton.
Subsequent increasing of the soliton power could result in appearance of
three new solitons.

Investigation of spatial soliton collisions demonstrates a wide variety of the
collision results and possibility of control of the final solitons
(after collision) parameters by
means of change of power, collision angle and phase difference of the
colliding solitons.

These investigations were partially supported by the International Scientific
Foundation, grant R53300.

\begin{thebibliography}{99}
\bibitem{rosanov1} 
N.N.~Rosanov, V.A.~Smirnov and V.A.~Vyssotina. {\it Numerical
Simulations of Interaction of Bright Spatial Solitons in Medium with
Saturable Nonlinearity}, Chaos, Solitons and Fractals, 1994, 4(8/9), 1767-1782.
\end{thebibliography}

\end{document}
