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 $0<c_1< \cdots < c_N$, we prove the existence of a solution $U$ of (fKdV) satisfying $$ \lim_{t\to\infty} \| U(t,\cdot) - \sum_{j=1}^NQ_{c_j}(x-\rho_j(t)) \|_{H^{\frac{\alpha}2}}=0, $$ where $\rho'_j(t) \sim c_j$ as $t \to +\infty$. The proof adapts the construction of Martel in the generalized KdV setting [Amer. J. Math. 127 (2005), pp. 1103-1140]) to the fractional case. The main new difficulties are the polynomial decay of the ground state $Q_c$ and the use of local techniques (monotonicity properties for a portion of the mass and the energy) for a non-local equation. To bypass these difficulties, we use symmetric and non-symmetric weighted commutator estimates. The symmetric ones were proved by Kenig, Martel and Robbiano [Annales de l'IHP Analyse Non Lin\'eaire 28 (2011), pp. 853-887], while the non-symmetric ones seem to be new