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

We construct 2-solitary wave solutions with logarithmic distance to the nonlinear Schrödinger equation,

$\mathrm{i}{?}_{t}u+\mathrm{\Delta }u+|u{|}^{p?1}u=0,t\in \mathbb{R},x\in {\mathbb{R}}^d,$

in mass-subcritical cases $1<p<1+\frac{4}{d}$ and mass-supercritical cases $1+\frac{4}{d}<p<\frac{d+2}{d?2}$, i.e. solutions $u(t)$ satisfying

${6u(t)?{\mathrm{e}}^{\mathrm{i}\gamma (t)}\sum _{k=1}^{2}Q(??x_k(t))6}_{H^1}\to 0$

and

$|x_1(t)?x_2(t)|～2\log ?t,\text{as}t\to +\infty ,$

where Q is the ground state. 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 is known by inverse scattering (E. Olmedilla, Multiple pole solutions of the nonlinear Schrödinger equation, Physica D 25 (1987) 330–346; T. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34 (1972) 62–69). The mass-critical case $p=1+\frac{4}{d}$ exhibits a specific behavior related to blow-up, previously studied in Y. Martel, P. Raphaël (Strongly interacting blow up bubbles for the mass critical NLS, Ann. Sci. Éc. Norm. Supér. 51 (2018) 701–737).