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

Arnaud Eychenne

• We construct N-soliton solutions for the fractional Korteweg–de Vries (fKdV) equation @tu @x.jDj ˛u u 2 / D 0; in the whole sub-critical range ˛ 2 .1=2; 2/. More precisely, if Qc denotes the ground state solution associated to fKdV evolving with velocity c, then, given 0 < c1 < < cN , we prove the existence of a solution U of fKdV satisfying lim t!1 U.t; / X N jD1 Qcj .x j .t // H˛=2 D 0; 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.