  In this paper, we consider the initial value problem of a specific system of
cubic nonlinear Schr\"{o}dinger equations. Our aim of this research is to
specify the asymptotic profile of the solution in $L^{\infty}$ as $t \to
\infty$. It is then revealed that the solution decays slower than a linear
solution does. Further, the difference of the decay rate is a polynomial order.
This deceleration of the decay is due to an amplification effect by the
nonlinearity. This nonlinear amplification phenomena was previously known for
several specific systems, however the deceleration of the decay in these
results was by a logarithmic order. As far as we know, the system studied in
this paper is the first model in that the deceleration in a polynomial order is
justified.
