  In this paper, we mainly investigate the long-time asymptotic behavior of the
solution for the coupled dispersive AB system with weighted Sobolev initial
data, which allows soliton solutions via the Dbar steepest descent method.Based
on the spectral analysis of Lax pair, the Cauchy problem of the coupled
dispersive AB system is transformed into a Riemann-Hilbert problem, and its
existence and uniqueness of the solution is proved by the vanishing lemma. The
stationary phase points play an important role in the long-time asymptotic
behavior. We demonstrate that in any fixed time cone $\mathcal{C}\left(x_{1},
x_{2}, v_{1}, v_{2}\right)=\left\{(x, t) \in \mathbb{R}^{2} \mid x=x_{0}+v t,
x_{0} \in\left[x_{1}, x_{2}\right], v \in\left[v_{1}, v_{2}\right]\right\}$,
the long-time asymptotic behavior of the solution for the coupled dispersive AB
system can be expressed by $N(\mathcal{I})$ solitons on the discrete spectrum,
the leading order term $\mathcal{O}(t^{-1 / 2})$ on the continuous spectrum and
the allowable residual $\mathcal{O}(t^{-3 / 4})$.
