AI 中文总结
本文针对连接两个不同亏格1周期态的阶梯状数据的散焦非线性薛定谔方程，通过Jost解确定散射数据并重组，建立了其对应全暗孤子气体黎曼-希尔伯特问题的存在性与唯一性，分离了两周期背景的效应。
AI 中文摘要
我们研究阶梯状数据的散焦非线性薛定谔方程，该数据连接两个通常不同的亏格1周期态。从与左右背景相关联的Jost解出发，我们确定对应的散射数据，并通过标量共轭和适当的对称化对其进行重组。所得公式将逆问题识别为全暗孤子气体黎曼-希尔伯特问题，且实轴上附有辐射跳跃。通过这种方式，两个周期背景的效应被分离，阶梯状周期散射与全气体构造之间建立了精确联系。假设不存在离散特征值且散射数据具有适当正则性，我们建立了相关黎曼-希尔伯特问题的存在性与唯一性。
英文摘要
We consider the defocusing nonlinear Schrödinger equation for step-like data connecting two, in general different, genus-one periodic states. Starting from the Jost solutions associated with the left and right backgrounds, we determine the corresponding scattering data and reorganize them through a scalar conjugation and a suitable symmetrization. The resulting formulation identifies the inverse problem with a full dark-soliton gas Riemann--Hilbert problem supplemented by a radiative jump on the real axis. In this way, the effects of the two periodic backgrounds are separated, and a precise link is established between step-like periodic scattering and the full-gas construction. Assuming the absence of discrete eigenvalues and appropriate regularity of the scattering data, we establish the existence and uniqueness of the associated Riemann--Hilbert problem.
Comments45 pages, 2 figures
