arXiv subjects

Arnaud Eychenne

Publications and source records attributed to Arnaud Eychenne.

4 recordsLinked to original sources

arXiv2022

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 $$ \partial_t u - \partial_x\left(|D|^αu - u^2 \right)=0, $$ in the whole sub-critical range $α\in]\frac12,2[$. 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 $$ \lim_{t\to\infty} \| U(t,\cdot) - \sum_{j=1}^NQ_{c_j}(x-ρ_j(t)) \|_{H^{\fracα2}}=0, $$ 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.

math.AP↗

arXiv2022

Decay of solitary waves of fractional Korteweg-de Vries type equations

We study the solitary waves of fractional Korteweg-de Vries type equations, that are related to the $1$-dimensional semi-linear fractional equations: \begin{align*} \vert D \vert^αu + u -f(u)=0, \end{align*} with $α\in (0,2)$, a prescribed coefficient $p^*(α)$, and a non-linearity $f(u)=\vert u \vert^{p-1}u$ for $p\in(1,p^*(α))$, or $f(u)=u^p$ with an integer $p\in[2;p^*(α))$. Asymptotic developments of order $1$ at infinity of solutions are given, as well as second order developments for positive solutions, in terms of the coefficient of dispersion $α$ and of the non-linearity $p$. The main tools are the kernel formulation introduced by Bona and Li, and an accurate description of the kernel by complex analysis theory.

math.AP↗

arXiv2022

Strongly interacting solitary waves for the fractional modified Korteweg-de Vries equation

We study one particular asymptotic behaviour of a solution of the fractional modified Korteweg-de Vries equation (also known as the dispersion generalised modified Benjamin-Ono equation): \begin{align}\tag{fmKdV} \partial_t u + \partial_x (-\vert D \vert^αu + u^3)=0. \end{align} The dipole solution is a solution behaving in large time as a sum of two strongly interacting solitary waves with different signs. We prove the existence of a dipole for fmKdV. A novelty of this article is the construction of accurate profiles. Moreover, to deal with the non-local operator $\vert D \vert^α$, we refine some weighted commutator estimates.

math.AP↗

arXiv2018

On the stability of 2D dipolar Bose-Einstein condensates

We study the existence of energy minimizers for a Bose-Einstein condensate with dipole-dipole interactions, tightly confined to a plane. The problem is critical in that the kinetic energy and the (partially attractive) interaction energy behave the same under mass-preserving scalings of the wave-function. We obtain a sharp criterion for the existence of ground states, involving the optimal constant of a certain generalized Gagliardo-Nirenberg inequality.

math-ph↗