On small travelling waves to the mass critical fractional NLS

TL;DR

This paper proves the existence of small travelling wave solutions for the mass critical fractional nonlinear Schrödinger equation and describes their asymptotic profiles as the mass approaches zero.

Contribution

It establishes the existence of travelling waves below the ground state mass for the fractional NLS and characterizes their profiles in the small mass limit, revealing new asymptotic behavior.

Findings

Existence of travelling waves for all masses below the ground state.

Complete description of wave profiles in the small mass limit.

Different asymptotic profiles compared to the classical case.

Abstract

We consider the mass critical fractional (NLS). We show the existence of travelling waves for all mass below the ground state mass, and give a complete description of the associated profiles in the small mass limit. We therefore recover a situation similar to the one discovered in [Gerard P.; Lenzmann E.; Pocovnicu O.; Rapha\"el, P., A two soliton with transient turbulent regime for the one dimensional cubic half wave, submitted] for the critical case s = 1, but with a completely different asymptotic profile when the mass vanishes.