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

Abstract

We construct N -soliton solutions for the fractional Korteweg-de Vries (fKdV) equationin the whole sub-critical range α ∈] 1 2 , 2[. More precisely, if Qc denotes the ground state solution associated to fKdV evolving with velocity c, then given 0where ρ ′ j (t) ∼ c j as t → +∞. 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 Qc 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 Linéaire 28 (2011), pp. 853-887], while the non-symmetric ones seem to be new.