Global dynamics of small solutions to the modified fractional Korteweg-de Vries and nonlinear Schr\"{o}dinger equations

TL;DR

This paper proves the global existence and modified scattering of small solutions for a range of fractional dispersive equations, specifically the modified fractional KdV and NLS equations, with a logarithmic phase correction.

Contribution

It establishes the first comprehensive results on global solutions and modified scattering for the full fractional range -1<α<1, excluding zero, for these equations.

Findings

Global existence of small solutions proved for -1<α<1, α≠0.

Modified scattering with logarithmic phase correction demonstrated.

Results cover both modified fKdV and fNLS 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.