Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction

In this paper we study several \begin{document}$ L^2 $\end{document} -constrained variational problems associated with a three component system of nonlinear Schrodinger equations with three wave interaction. We consider the existence and the orbital stability of minimizers for these variational problems. We also investigate an asymptotic expansion of the minimal energy and the asymptotic behavior of a minimizer for the variational problem when the attractive effect of three wave interaction is sufficiently large.