We consider a system of coupled cubic Schrödinger equations in one space dimension
in the non-integrable case
First, we justify the existence of a symmetric 2-solitary wave with logarithmic distance, i.e. a solution of the system satisfying
$ \lim\limits_{t\to +\infty}\left\| \begin{pmatrix} u(t) \\ v(t)\end{pmatrix} - \begin{pmatrix} e^{ \text{i} t}Q (\cdot - \frac{1}{2} \log (\Omega t) - \frac{1}{4} \log \log t) \\[4pt] e^{ \text{i} t}Q (\cdot + \frac{1}{2} \log (\Omega t) + \frac{1}{4} \log \log t)\end{pmatrix}\right\|_{H^1\times H^1} = 0 $
where
Second, under the conditions
$ \lim\limits_{t\to +\infty}\left\| \begin{pmatrix}u(t) \\ v(t)\end{pmatrix} - \begin{pmatrix}e^{ \text{i} c^2 t}Q_c (\cdot - \frac{1}{(c+1)c} \log (\Omega_c t) )\\[4pt] e^{ \text{i} t} Q (\cdot + \frac{1}{c+1} \log (\Omega_c t))\end{pmatrix} \right\|_{H^1\times H^1} = 0 $
where
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