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

TL;DR

This paper investigates the decay properties of solitary waves in fractional Korteweg-de Vries equations, providing asymptotic expansions and analyzing how dispersion and non-linearity influence wave behavior.

Contribution

It offers new asymptotic descriptions of solitary wave decay for fractional KdV equations, utilizing kernel analysis and complex analysis techniques.

Findings

Asymptotic decay rates depend on dispersion coefficient and non-linearity.

Second order asymptotics are derived for positive solutions.

Kernel formulation effectively describes wave decay behavior.

Abstract

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