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 in the whole sub-critical range 2.1=2; 2. More precisely, ifQc denotes the ground state solution associated to fKdV evolving with velocity c, then, given 0 < cN, we prove the existence of a solution U of fKdV satisfying where 0 j. t cj as t ! C1. 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. [ABSTRACT FROM AUTHOR]