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

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): partial differential tu + partial differential x(-|D|alpha u + 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|alpha, we refine some weighted commutator estimates. (c) 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).