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 $\alpha \in (0,2)$, a prescribed coefficient $p^{\ast }(\alpha )$, and a non-linearity $f(u)=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ for $p\in (1,p^{\ast }(\alpha ))$, or $f(u)=u^p$ with an integer $p\in [2;p^{\ast }(\alpha ))$. 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 $\alpha$ 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.