The long-time asymptotic behaviors of the solutions for the coupled dispersive AB system with weighted Sobolev initial data

TL;DR

This paper employs the $ar{ ext{D}}$-steepest descent method to analyze the long-time asymptotic behavior of solutions to the coupled dispersive AB system with weighted Sobolev initial data, improving the accuracy of asymptotic estimates.

Contribution

It introduces a novel application of the $ar{ ext{D}}$-steepest descent method to the coupled dispersive AB system, achieving more precise long-time asymptotic results without discrete spectrum.

Findings

Long-time asymptotics characterized as $O(t^{-1})$

Improved accuracy over previous $O(t^{-1} ext{log} t)$ results

Solutions derived via Riemann-Hilbert problem construction

Abstract

In this work, we employ the $\bar{\partial}$-steepset descent method to study the Cauchy problem of the coupled dispersive AB system with initial conditions in weighted Sobolev space $H^{1,1}(\mathbb{R})$, \begin{align*} \left\{\begin{aligned} &A_{xt}-\alpha A-\beta AB=0,\\ &B_{x}+\frac{\gamma}{2}(|A|^2)_t=0,\\ &A(x,0)=A_0(x),~~~~B(x,0)=B_0(x)\in H^{1,1}(\mathbb{R}). \end{aligned}\right. \end{align*} Begin with the Lax pair of the coupled dispersive AB system, we successfully derive the solutions of the coupled dispersive AB system by constructing the basic Riemann-Hilbert problem. By using the $\bar{\partial}$-steepset descent method, the long-time asymptotic behaviors of the solutions for the coupled dispersive AB system are characterized without discrete spectrum. Our results demonstrate that compared with the previous results, we increase the accuracy of the long-time asymptotic solution from $O(t^{-1}\log t)$ to $O(t^{-1})$.