Long-time asymptotic behavior of the modified Schr\"{o}dinger equation via Dbar-steepest descent method

Abstract

In this paper, we consider the Cauchy problem for the modified NLS equation. Using nonlinear steepest descent method and combining the Dbar-analysis, we show that inside any fixed cone, the long time asymptotic behavior of the solution for the modified NLS equation can be characterized with an soliton on discrete spectrum and leading order aasymptotic term on continuous spectrum up to an residual error order O(t^{-3/4}).