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

TL;DR

This paper investigates the asymptotic behavior of solutions to the fractional modified Korteweg-de Vries equation, focusing on the existence and construction of dipole solutions composed of strongly interacting solitary waves.

Contribution

It introduces the first construction of dipole solutions for the fractional mKdV and develops refined weighted commutator estimates for the non-local operator.

Findings

Existence of dipole solutions for the fractional mKdV.

Construction of accurate solitary wave profiles.

Refined estimates for non-local operator commutators.

Abstract

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^\alpha u + 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 $\vert D \vert^\alpha$, we refine some weighted commutator estimates.