AI 中文总结
研究CMDNLS方程的直接散射理论，通过证明约斯特函数相关性质构造散射系数，确定其在方程流下的演化，揭示传输与反射系数变化规律，还建立加权索伯列夫空间不变性以保障解的正则性。
AI 中文摘要
本文在实轴上建立了卡洛杰罗 - 莫泽导数非线性薛定谔（CMDNLS）方程的直接散射变换。对于加权哈代 - 索伯列夫空间中的势，证明了与拉克斯算子相关的约斯特函数的存在性和唯一性，并构造了包括传输系数和反射系数在内的相应散射系数。在CMDNLS流作用下，确定了这些约斯特函数和散射数据的时间演化，揭示了简单动力学：传输系数不变，反射系数获得纯旋转相位。此外，还建立了流作用下合适加权索伯列夫空间的不变性，保证了解所需正则性的持续性。
英文摘要
This paper establishes the direct scattering transform for the Calogero-Moser derivative nonlinear Schrödinger (CMDNLS) equation on the real line. For potentials in a weighted Hardy-Sobolev space, we prove the existence and uniqueness of the Jost functions associated with the Lax operator, and construct the corresponding scattering coefficients including the transmission coefficient and the reflection coefficient. Under the CMDNLS flow, we determine the time evolution of these Jost functions and scattering data, revealing a simple dynamics: the transmission coefficient is invariant, and the reflection coefficient acquires a purely rotational phase. Furthermore, we establish the invariance of a suitable weighted Sobolev space under the flow, which guarantees the persistence of the required regularity for the solutions.
