  We consider a system of coupled cubic Schr\"odinger equations in one space
dimension \begin{equation*} \begin{cases} i \partial_t u + \partial_x^2 u
+(|u|^2 + \omega |v|^2) u =0\\ i \partial_t v + \partial_x^2 v+ (|v|^2 + \omega
|u|^2) v=0 \end{cases}\quad (t,x)\in {\bf R}\times{\bf R}, \end{equation*} in
the non-integrable case $0 < \omega < 1$.
  First, we justify the existence of a symmetric 2-solitary wave with
logarithmic distance, more precisely a solution of the system satisfying \[
\lim_{t\to +\infty}\left\| \begin{pmatrix} u(t) \\ v(t)\end{pmatrix} -
\begin{pmatrix} e^{it}Q (\cdot - \frac{1}{2} \log (\Omega t) - \frac{1}{4} \log
\log t) \\ e^{it}Q (\cdot + \frac{1}{2} \log (\Omega t) + \frac{1}{4} \log \log
t)\end{pmatrix}\right\|_{H^1\times H^1} = 0\] where $Q = \sqrt{2}{\rm sech}$ is
the explicit solution of $ Q'' - Q + Q^3 = 0$ and $\Omega>0$ is a constant.
This result extends to the non-integrable case the existence of symmetric
2-solitons with logarithmic distance known in the integrable case $\omega=0$
and $\omega=1$. Such strongly interacting symmetric $2$-solitary waves were
also previously constructed for the non-integrable scalar nonlinear
Schr\"odinger equation in any space dimension and for any energy-subcritical
power nonlinearity.
  Second, under the conditions $0<c<1$ and $0<\omega < \frac 12 c(c+1)$, we
construct solutions of the system satisfying \[ \lim_{t\to +\infty}\left\|
\begin{pmatrix}u(t) \\ v(t)\end{pmatrix} - \begin{pmatrix}e^{i c^2 t}Q_c (\cdot
- \frac{1}{(c+1)c} \log (\Omega_c t) ) \\ e^{i t} Q (\cdot + \frac{1}{c+1} \log
(\Omega_c t))\end{pmatrix} \right\|_{H^1\times H^1}=0\] where $Q_c(x)=cQ(cx)$
and $\Omega_c>0$ is a constant. Such logarithmic regime with non-symmetric
solitons does not exist in the integrable cases $\omega=0$ and $\omega=1$ and
is still unknown in the non-integrable scalar case.
