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

Abstract: This paper concerns the modified fractional Korteweg-de Vries (modified fKdV) and nonlinear Schr\"{o}dinger (modified fNLS) equations, with the dispersions |D|^{\alpha}\partial_x and |D|^{\alpha+1}, 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 -1<\alpha<1, \alpha\neq 0 for both the modified fKdV and fNLS equations.