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

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):

∂tu + ∂x(−|D|αu + u3) = 0.

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 |D|α , we refine some weighted commutator estimates.