Long-time asymptotic behavior of the coupled dispersive AB system in low regularity spaces

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