  We apply $\bar{\partial}$ steepest descent method to obtain sharp asymptotics
for a mixed schr\"{o}dinger equation $$ q_t+iq_{xx}-ia (\vert q \vert^2q)_x
-2b^2\vert q \vert^2q=0,$$ $$q(x,t=0)=q_0(x),$$ 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 $q_t+iq_{xx}$ and the error
order $\mathcal{O}(t^{-3/4})$ from a $\overline\partial$ equation

Analysis of PDEs;Exactly Solvable and Integrable Systems