Long-time asymptotic behavior of a mixed schrödinger equation with weighted Sobolev initial data

Abstract

We apply $\bar{\partial }$ steepest descent method to obtain sharp asymptotics for a mixed schrödinger equation $$q_t+i{q}_{xx}-ia(|q{|}^{2}q{)}_x-2b^2|q{|}^{2}q=0,$$ $$q(x,t=0)=q_0(x),$$ under essentially minimal regularity assumptions on initial data in a weighted Sobolev space $q_0(x)\in H^{2,2}(\mathbb{R})$. In the asymptotic expression, the leading order term $\mathcal{O}(t^{-1/2})$ comes from dispersive part $q_t+i{q}_{xx}$ and the error order $\mathcal{O}(t^{-3/4})$ from a $\overline{\partial }$ equation

35 pages