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

Abstract

In this paper, we mainly investigate the long-time asymptotic behavior of the solution for coupled dispersive AB systems with weighted Sobolev initial data, which allows soliton solutions via the Dbar steepest descent method. Based on the spectral analysis of Lax pairs, the Cauchy problem of coupled dispersive AB systems is transformed into a Riemann–Hilbert problem, and the existence and uniqueness of its solution is proved by the vanishing lemma. The stationary phase points play an important role in determining the long-time asymptotic behavior of these solutions. We demonstrate that in any fixed time cone Cx1,x2,v1,v2=(x,t)∈R2∣x=x0+vt,x0∈x1,x2,v∈v1,v2, the long-time asymptotic behavior of the solution for coupled dispersive AB systems can be expressed by N(I) solitons on the discrete spectrum, the leading order term O(t−1/2) on the continuous spectrum, and the allowable residual O(t−3/4).