Asymptotic $N$-soliton-like solutions of the fractional Korteweg-de Vries equation

We construct $N$-soliton solutions for the fractional Korteweg-de Vries (fKdV) equation $$ \partial_t u - \partial_x\left(|D|^{\alpha}u - u^2 \right)=0, $$ in the whole sub-critical range $\alpha \in]\frac12,2[$. More precisely, if $Q_c$ denotes the ground state solution associated to fKdV evolving with velocity $c$, then given $0