  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.
