Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations

We consider the generalized Korteweg-de Vries equation $\partial_t u = -\partial_x(\partial_x^2 u + f(u))$,where$f(u)$isanoddfunctionofclass$C^3$\mathrm{.}Undersomeassumptionson$f$, this equation admits \emph{solitary waves}, that is solutions of the form $u(t,x)=Q_v(x-vt-x_0)$, for $v$ in some range $(0,v_{\ast })$. We study pure two-solitons in the case of the same limit speed, in other words global solutions $u(t)$ such that \begin{equation} \label{eq:abstract} \tag{$\ast$} \lim_{t\to\infty}|u(t) - (Q_v(\cdot - x_1(t)) \pm Q_v(\cdot - x_2(t)))|{H^1} = 0, \qquad $\text{with}$\quad\lim{t \to \infty}x_2(t) - x_1(t) = \infty. \end{equation} Existence of such solutions is known for $f(u)=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ with $p\in \mathbb{Z}\setminus \{5\}$ and $p > 2$. We describe the~dynamical behavior of any solution satisfying \eqref{eq:abstract} under the assumption that $Q_v$ is linearly unstable (which corresponds to $p>5$ for power nonlinearities). We prove that in this case the sign in \eqref{eq:abstract} is necessarily “$+$”, which corresponds to an attractive interaction. We also prove that the~distance $x_2(t)-x_1(t)$ between the solitons equals $\frac{2}{\sqrt{v}}\log (\kappa t)+o(1)$ for some $\kappa =\kappa (v)>0$.