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 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, the long-time asymptotic behavior of the solution for the coupled dispersive AB system 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 ).

Long time asymptotic analysis for a nonlocal Hirota equation via the Dbar steepest descent method

“…At the same time, the estimation of Cauchy integral operator in L P space is avoided, and the discontinuous part of Riemann-Hilbert problem(RHP) is rewritten as Dbar problem, which can be solved by integral equation. Because of its advantages, it is widely used in integrable equations, and has made great progress [41][42][43][44][45].…”