Taohua Luo | Searcharxiv

arXiv2026

Long-time asymptotics of the integrable defocusing Wadati-Konno-Ichikawa equation with a finite-genus algebro-geometric background

We study the finite-genus algebro-geometric solutions of the Wadati-Konno-Ichikawa (WKI) equation with the saturable nonlinearity and long-time asymptotic behaviors of their short-range perturbations. First, for both the focusing and defocusing reductions, we formulate the finite-genus Baker-Akhiezer functions as explicitly solvable the matrix Riemann-Hilbert (RH) problems on the complex spectral plane and obtain theta-function representations together with the reconstruction formulae for the WKI field and the reciprocal coordinate. We then consider the Cauchy problem of the defocusing WKI equation on a finite-genus algebro-geometric background. We construct the scattering data and RH problem, and perform a Deift-Zhou nonlinear steepest descent analysis. The space-time plane is divided into two transition regions, a Zakharov-Manakov (ZM) region, and a fast-decay region. The leading term is a phase-shifted finite-genus WKI solution. The transition corrections are governed by a Painlev\'e-XXXIV model, while the ZM radiation is described by parabolic-cylinder functions. The reciprocal-coordinate asymptotics are obtained simultaneously.

nlin.SI↗

arXiv2026

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

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 Painlev\'e-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.

nlin.SI↗