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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 masssub-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 asymptoticallyinto 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 tostrong interactions between solitary waves. In the integrable case ($d=1$ and$p=3$) the existence of such solutions has been shown in [14].

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