Long-time Asymptotic Behavior of the coupled dispersive AB system in Low Regularity Spaces

TL;DR

This paper analyzes the long-time behavior of solutions to the coupled dispersive AB system with low regularity initial data, revealing soliton dominance and specific decay rates using Riemann-Hilbert problem techniques.

Contribution

It introduces a spectral analysis approach and applies the Dbar steepest descent method to characterize the asymptotic behavior of the coupled dispersive AB system.

Findings

Solution decomposes into solitons and dispersive waves.

Leading order decay rate is $\mathcal{O}(t^{-1/2})$ on the continuous spectrum.

Residual decay is $\mathcal{O}(t^{-3/4})$.

Abstract

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 $\mathcal{O}(t^{-3 / 4})$.