On the direct scattering theory for the Calogero-Moser derivative nonlinear Schrödinger equation

Sun Ruoci; Wang Deng-shan; Zhao Yi

On the direct scattering theory for the Calogero-Moser derivative nonlinear Schrödinger equation

Analysis of PDEs 2026-07-13 v1

Authors: Sun Ruoci , Wang Deng-shan , Zhao Yi

Abstract

This paper establishes the direct scattering transform for the Calogero-Moser derivative nonlinear Schr\"{o}dinger (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.

Keywords

Calogero-Moser derivative nonlinear Schrödinger equation direct scattering transform Jost functions Lax operator scattering coefficients weighted Sobolev space

Cite

@article{arxiv.2607.11123,
  title  = {On the direct scattering theory for the Calogero-Moser derivative nonlinear Schrödinger equation},
  author = {Sun Ruoci and Wang Deng-shan and Zhao Yi},
  journal= {arXiv preprint arXiv:2607.11123},
  year   = {2026}
}