mathematical physics

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

arXiv:2607.11123

summary

The paper develops the direct scattering transform for the Calogero‑Moser derivative nonlinear Schrödinger equation on the real line, proving existence of Jost functions, constructing scattering data, and describing its simple time evolution.

Abstract

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.

Topics & keywords

#integrable systems#scattering theory#nonlinear Schrödinger equation#Calogero‑Moser model#partial differential equations#spectral analysisJost functionstransmission coefficientreflection coefficientweighted Hardy‑Sobolev spaceLax operatordirect scattering transform
On the direct scattering theory for the Calogero-Moser derivative nonlinear Schrödinger equation · wovepaper