Long-time asymptotic behavior for the defocusing Hirota equation on a finite-genus algebro-geometric background

Abstract

In this paper, we investigate the long-time asymptotics for the solution of the Cauchy problem of the defocusing Hirota equation on a finite-genus algebro-geometric background in the whole $(x,t)$-half-plane, whose method is mainly based on a Riemann-Hilbert (RH) formulation and Deift-Zhou nonlinear steepest descent method. The critical values of the phase function in the associated RH problem divide the space-time plane into four regions, in which the leading-order term is given by a phase-shifted finite-genus algebro-geometric solution. The subleading behavior depends on the region: the correction is of order $t^{-1/3}$ and is governed by a Painlevé-XXXIV model RH problem in the transition regions; the leading radiation is of order $t^{-1/2}$ in the Zakharov--Manakov region; and the error is $O(t^{-1})$ in the fast-decay region. These results can also be extended to other higher-order members of the AKNS hierarchy.

40 pages, 9 figures