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) equation ∂t​u−∂x​(∣D∣αu−u2)=0,

in 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 0<c1​<⋯<cN​, we prove the existence of a solution U of (fKdV) satisfying t→∞lim​ ​U(t,⋅)−j=1∑N​Qcj​​(x−ρj​(t)) ​Hα/2​=0,

where ρj′​(t)∼cj​ 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.