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

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 $$\underset{t\to +\mathrm{\infty }}{\lim }{\parallel \left(\begin{array}{c}u(t)\\ v(t)\end{array}\right)-\left(\begin{array}{c}{e}^{it}Q(\cdot -\frac{1}{2}\log (\mathrm{\Omega }t)-\frac{1}{4}\log \log t)\\ e^{it}Q(\cdot +\frac{1}{2}\log (\mathrm{\Omega }t)+\frac{1}{4}\log \log t)\end{array}\right)\parallel }_{H^1\times H^1}=0$$ where $Q=\sqrt{2}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}$ is the explicit solution of $Q^{\mathrm{\prime }\mathrm{\prime }}-Q+Q^3=0$ and $\mathrm{\Omega }>0$ 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 $0<\omega <\frac{1}{2}c(c+1)$, 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 ${\mathrm{\Omega }}_c>0$ 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.