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 derivative(t)u - partial derivative(x )(|D|(alpha)u -u(2)) = 0,in the whole sub-critical range alpha is an element of (1/2, 2). More precisely, if Q(c) denotes the ground state solution associated to fKdV evolving with velocity c, then, given 0 < c(1) < center dot center dot center dot < c(N), we prove the existence of a solution U of fKdV satisfying lim(t ->infinity) || U(t, center dot)- Sigma(N)(j=1) Q(cj) (x -rho(j) (t))||(alpha/2)(H) = 0,where p '(j) (t) similar to c(j) as t -> +infinity. 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 & eacute;aire 28 (2011), pp. 853-887], while the non-symmetric ones seem to be new.