| Contents |
| We consider the Cauchy problem for a system of nonlinear Schr\"odinger equations of the form
\begin{align*}
\begin{cases}
i\partial_t u+\frac{1}{2m}\Delta u=\lambda v \overline{u},\\
i\partial_t v+\frac{1}{2M}\Delta v=\mu u^2.
\end{cases}
\end{align*}
In this study, we consider the analytic smoothing property for the above system under the mass resonance
condition
$M = 2m$
for suffciently small Cauchy data with exponential decay
in space dimensions
$n\geq3.$
We prove the global existence in framework of the critical Sobolev space
$\dot{H}^{n/2-2}$
for
$n\geq4$
and
$1/2$
order Sobolev type space defined by the generator of Galilei transform
$x+i\frac{t}{m}\nabla$
for
$n=3.$ |
|