  In this work, we employ the $\bar{\partial}$ steepest descent method in order
to study the Cauchy problem of the cgNLS equations with initial conditions in
weighted Sobolev space $H^{1,1}(\mathbb{R})=\{f\in L^{2}(\mathbb{R}): f',xf\in
L^{2}(\mathbb{R})\}$. The large time asymptotic behavior of the solution
$u(x,t)$ and $v(x,t)$ are derived in a fixed space-time cone
$S(x_{1},x_{2},v_{1},v_{2})=\{(x,t)\in\mathbb{R}^{2}: x=x_{0}+vt,
~x_{0}\in[x_{1},x_{2}], ~v\in[v_{1},v_{2}]\}$. Based on the resulting
asymptotic behavior, we prove the solution resolution conjecture of the cgNLS
equations which contains the soliton term confirmed by
$|\mathcal{Z}(\mathcal{I})|$-soliton on discrete spectrum and the
$t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to
$O(t^{-\frac{3}{4}})$.
