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We consider a system of coupled cubic Schr\"odinger equations in one spacedimension \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*} inthe non-integrable case $0 < \omega < 1$.  First, we justify the existence of a symmetric 2-solitary wave withlogarithmic distance, more precisely a solution of the system satisfying \[\lim_{t\to +\infty}\left\| \begin{pmatrix} u(t) \\ v(t)\end{pmatrix} -\begin{pmatrix} e^{it}Q (\cdot - \frac{1}{2} \log (\Omega t) - \frac{1}{4} \log\log t) \\ e^{it}Q (\cdot + \frac{1}{2} \log (\Omega t) + \frac{1}{4} \log \logt)\end{pmatrix}\right\|_{H^1\times H^1} = 0\] where $Q = \sqrt{2}{\rm sech}$ isthe 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 symmetric2-solitons with logarithmic distance known in the integrable case $\omega=0$and $\omega=1$. Such strongly interacting symmetric $2$-solitary waves werealso previously constructed for the non-integrable scalar nonlinearSchr\"odinger equation in any space dimension and for any energy-subcriticalpower nonlinearity.  Second, under the conditions $0<c<1$ and $0<\omega < \frac 12 c(c+1)$, weconstruct 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-symmetricsolitons does not exist in the integrable cases $\omega=0$ and $\omega=1$ andis still unknown in the non-integrable scalar case.

Construction of 2-solitons with logarithmic distance for the one-dimensional cubic Schrodinger system