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

Abstract

We study the solitary waves of fractional Korteweg-de Vries type equations, that are related to the 1- dimensional semi-linear fractional equations: |D|αu + u − f (u) = 0, with α ∈ (0, 2), a prescribed coefficient p∗(α), and a non-linearity f (u) = |u|p−1 u for p ∈ (1,p∗(α)), or f (u) = up with an integer p ∈ [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.