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

Arnaud Eychenne

doi:10.4171/rmi/1396

JournalsrmiVol. 39, No. 5pp. 1813–1862

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

Arnaud Eychenne
University of Bergen, Norway

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Abstract

We construct $N$ -soliton solutions for the fractional Korteweg–de Vries (fKdV) equation

\partial_{t} u - \partial_{x} (∣ D ∣^{α} u - u^{2}) = 0,

in the whole sub-critical range $α \in (1/2, 2)$ . More precisely, if $Q_{c}$ denotes the ground state solution associated to (fKdV) evolving with velocity $c$ , then, given $0 < c_{1} < \dots < c_{N}$ , we prove the existence of a solution $U$ of (fKdV) satisfying

t \to \infty lim U (t, \cdot) - j = 1 \sum N Q_{c_{j}} (x - ρ_{j} (t))_{H^{α /2}} = 0,

where $ρ_{j}^{'} (t) \sim c_{j}$ 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.

Cite this article

Arnaud Eychenne, Asymptotic $N$ -soliton-like solutions of the fractional Korteweg–de Vries equation. Rev. Mat. Iberoam. 39 (2023), no. 5, pp. 1813–1862

DOI 10.4171/RMI/1396