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 $\mathcal{C}\left(x_1,x_2,v_1,v_2\right)=\left\{(x,t)\in {\mathbb{R}}^2\mid x=x_0+vt,x_0\in \left[x_1,x_2\right],v\in \left[v_1,v_2\right]\right\}$, 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 $\mathcal{O}(t^{-3/4})$.

References in corpus (7)

• Long-Time Asymptotics for the Camassa-Holm Equation
• Rogue-wave solutions of a three-component coupled nonlinear Schrodinger equation
• Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent
• Double and triple poles solutions for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with zero/nonzero boundary conditions
• Long-time asymptotics and stability for the sine-Gordon equation
• Soliton resolution for the Hirota equation with weighted Sobolev initial data
• High-order soliton matrix for the third-order flow equation of the Gerdjikov-Ivanov hierarchy through the Riemann-Hilbert method