Strongly interacting multi-solitons with logarithmic relative distance for gKdV equation

We consider the following class of equations of (gKdV) type $$\partial_t u + \partial_x (\partial_x^2 u + |u|^{p-1}u) = 0, \quad p\mbox{ integer},\quad t,x \in \mathbb{R}$$ with mass sub-critical ($2< p<5$) and mass super-critical nonlinearities ($p> 5$). We prove the existence of 2-solitary wave solutions with logarithmic relative distance, i.e. solutions $u(t)$ satisfying $$\left\|u(t)- \bigg( Q (\cdot - t - \log (ct)) + \sigma Q (\cdot - t + \log (ct))\bigg)\right\|_{H^1}\to 0 \ \ \mbox{as} \ \ t\to +\infty,$$ where $c=c(p)> 0$ is a fixed constant, $\sigma = -1$ in sub-critical cases and $\sigma = 1$ in super-critical cases. For the integrable case ($p=3$), such solution was known by integrability theory. This regime corresponds to strong attractive interactions. For sub-critical $p$, it was known that opposite sign traveling waves are attractive. For super-critical $p$, we derive from our computations that same sign traveling waves are attractive.