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

In this paper, the ∂̄-steepest descent method and Bäcklund transformation are employed to study the asymptotic stability of solitons for the Cauchy problem of the focusing Hirota equation. It is shown that the solution of the Cauchy problem can be expressed in terms of the solution of a Riemann–Hilbert (RH) problem. The solution of the RH problem is further decomposed into a pure radiation solution and a solitons solution obtained by using the ∂̄-steepest descent method and Bäcklund transformation, respectively. As a consequence, the asymptotic stability of solitons for the Hirota equation is established.