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

Abstract

We consider a system of coupled cubic Schrödinger equations in one space dimension \begin{equation*} \begin{cases} i \partial_t u + \partial_x^2 u +(|u|^2 + ω|v|^2) u =0\\ i \partial_t v + \partial_x^2 v+ (|v|^2 + ω|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 \[ \lim_{t\to +\infty}\left\| \begin{pmatrix} u(t) \\ v(t)\end{pmatrix} - \begin{pmatrix} e^{it}Q (\cdot - \frac{1}{2} \log (Ωt) - \frac{1}{4} \log \log t) \\ e^{it}Q (\cdot + \frac{1}{2} \log (Ωt) + \frac{1}{4} \log \log t)\end{pmatrix}\right\|_{H^1\times H^1} = 0\] where $Q=\sqrt{2}\mathrm{sech}$ is the explicit solution of $Q^{\prime \prime }-Q+Q^3=0$ 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 Schrödinger 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 \[ \lim_{t\to +\infty}\left\| \begin{pmatrix}u(t) \\ v(t)\end{pmatrix} - \begin{pmatrix}e^{i c^2 t}Q_c (\cdot - \frac{1}{(c+1)c} \log (Ω_c t) ) \\ e^{i t} Q (\cdot + \frac{1}{c+1} \log (Ω_c t))\end{pmatrix} \right\|_{H^1\times H^1}=0\] 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.