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

TL;DR

This paper analyzes the long-time behavior of solutions to a nonlocal Hirota equation using the Dbar steepest descent method, transforming the problem into a Riemann-Hilbert problem and revealing unique effects of nonlocality.

Contribution

It introduces a novel application of the Dbar steepest descent method to a nonlocal integrable equation, extending spectral analysis techniques.

Findings

Long-time asymptotics derived for the nonlocal Hirota equation.

The leading order term is influenced by the imaginary part of a spectral parameter.

Absence of discrete spectrum simplifies the asymptotic analysis.

Abstract

In this paper, we mainly focus on the Cauchy problem of an integrable nonlocal Hirota equation with initial value in weighted Sobolev space. Through the spectral analysis of Lax pairs, we successfully transform the Cauchy problem of the nonlocal Hirota equation into a solvable Riemann-Hilbert problem. Furthermore, in the absence of discrete spectrum, the long-time asymptotic behavior of the solution for the nonlocal Hirota equation is obtained through the Dbar steepest descent method. Different from the local Hirota equation, the leading order term on the continuous spectrum and residual error term of $q(x,t)$ are affected by the function $Im\nu(z_j)$.