The asymptotic stability of solitons in the focusing Hirota equation on the line

TL;DR

This paper proves the asymptotic stability of solitons in the focusing Hirota equation on the line using advanced analytical methods, demonstrating the long-term persistence of soliton solutions.

Contribution

It introduces a novel combination of $ar{ ext{D}}$-steepest descent and B"acklund transformation techniques to analyze soliton stability in the Hirota equation.

Findings

Solitons are asymptotically stable under the Hirota equation.

Decomposition of the solution into radiation and soliton parts.

Application of $ar{ ext{D}}$-techniques and B"acklund transformation.

Abstract

In this paper, the $\overline\partial$-steepest descent method and B\"acklund transformation are used to study the asymptotic stability of solitons to the Cauchy problem of focusing Hirota equation. The solution of the RH problem is further decomposed into pure radiation solution and solitons solution obtained by using $\overline\partial$-techniques and B\"acklund transformation respectively. As a directly consequence, the asymptotic stability of solitons for the Hirota equation is obtained.