Global dynamics of small solutions to the modified fractional Korteweg-de Vries and fractional cubic nonlinear Schrödinger equations

Abstract

This paper concerns the modified fractional Korteweg-de Vries (modified fKdV) and fractional cubic nonlinear Schrödinger (fNLS) equations, with the dispersions and respectively. We prove the global existence of small solutions for both the Cauchy problems to the modified fKdV and fNLS equations, with a modified scattering which has a logarithmic phase correction. Our results cover the full range for both the modified fKdV and fNLS equations.