  In this work, we employ the $\bar{\partial}$-steepset descent method to study
the Cauchy problem of the coupled dispersive AB system with initial conditions
in weighted Sobolev space $H^{1,1}(\mathbb{R})$,
  \begin{align*} \left\{\begin{aligned} &A_{xt}-\alpha A-\beta AB=0,\\
&B_{x}+\frac{\gamma}{2}(|A|^2)_t=0,\\ &A(x,0)=A_0(x),~~~~B(x,0)=B_0(x)\in
H^{1,1}(\mathbb{R}). \end{aligned}\right. \end{align*} Begin with the Lax pair
of the coupled dispersive AB system, we successfully derive the solutions of
the coupled dispersive AB system by constructing the basic Riemann-Hilbert
problem. By using the $\bar{\partial}$-steepset descent method, the long-time
asymptotic behaviors of the solutions for the coupled dispersive AB system are
characterized without discrete spectrum. Our results demonstrate that compared
with the previous results, we increase the accuracy of the long-time asymptotic
solution from $O(t^{-1}\log t)$ to $O(t^{-1})$.
