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

We consider a system of coupled cubic Schrödinger equations in one space dimension

$ \begin{equation*} \begin{cases} \text{i}\partial_t u + \partial_x^2 u +(|u|^2 + \omega |v|^2) u = 0\\ \text{i}\partial_t v + \partial_x^2 v+ (|v|^2 + \omega |u|^2) v = 0 \end{cases}\quad (t,x)\in \mathbb{R}\times \mathbb{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, 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 $ Q = \sqrt{2} $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 $ ([15,33]). Such strongly interacting symmetric $ 2 $-solitary waves were also previously constructed for the non-integrable scalar nonlinear Schrödinger equation in any space dimension and for any energy-subcritical power nonlinearity ([20,22]).

Second, under the conditions $ 0<c<1 $ and $ 0<\omega < \frac 12 c(c+1) $, we construct solutions of the system satisfying

$ \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 $ 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.