Existence of multi-solitary waves with logarithmic relative distances for the NLS equation

We construct in this paper global (for $t \geq 0$) and bounded solutions $u(t)$ for the nonlinear Schr\"odinger equation $$i \partial_t u + \Delta u + |u|^{p-1} u = 0, \quad t \in \mathbb{R}, x \in \mathbb{R}^d$$ in mass sub-critical cases ($1 < p < 1 + \frac{4}{d}$) and mass super-critical ($1 + \frac{4}{d} < p < \frac{d+2}{d-2}$) such that $u(t)$ decomposes asymptotically into two solitary waves with logarithmic distance $$\|u(t) - e^{i \gamma (t)} \sum_{k=1}^2 Q(\cdot - x_k(t))\|_{H^1} \to 0$$ and $$|x_1(t) - x_2(t)| \sim 2 \log t, \quad \mbox{as}t \to + \infty.$$ The logarithmic distance is related to strong interactions between solitary waves. In the integrable case ($d=1$ and $p=3$) the existence of such solutions has been shown in [14].