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 $${\partial }_{t}u-{\partial }_x\left(|D{|}^{\alpha }u-u^2\right)=0,$$ in the whole sub-critical range $\alpha \in ]\frac{1}{2},2[$. More precisely, if $Q_c$ denotes the ground state solution associated to fKdV evolving with velocity $c$, then given $0<c_1<\cdots <c_N$, we prove the existence of a solution $U$ of (fKdV) satisfying $$\underset{t\to \infty }{\lim }\Vert U(t,\cdot )-\sum _{j=1}^N{Q}_{c_j}(x-{\rho }_j(t)){\Vert }_{H^{\frac{\alpha }{2}}}=0,$$ where ${\rho }_j^{\prime }(t)\sim c_j$ as $t\to +\infty$. 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 Linéaire 28 (2011), pp. 853-887], while the non-symmetric ones seem to be new.