2021/12/21 by Arnaud Eychenne, Eychenne, Arnaud · 2 citations
Mathematics · #35Q35 Secondary: 35B40 #37K40 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary: 35Q53 #math.AP #msc:35B40 #msc:35Q35 #msc:35Q53 #msc:37K40
paper · pdf · doi:10.48550/arxiv.2112.11278
arxiv created 2022/10/23 · arxiv updated 2022/10/25
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∈]\frac12,2[. 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 limt→∞ ‖ U(t,⋅) - ∑j=1NQcj(x-ρj(t)) ‖H\fracα2=0, where ρ'j(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.