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arXiv · 2608.06889

The direct scattering problem for the defocusing nonlinear Schr\"odinger equation with step-like periodic background

Dinghao Zhu
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Abstract

We consider the defocusing nonlinear Schr\"odinger 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.

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Dinghao Zhu. 2026-08-07. The direct scattering problem for the defocusing nonlinear Schr\"odinger equation with step-like periodic background. https://arxiv.org/abs/2608.06889

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