Eigenvalue analysis of the Lax operator for the one-dimensional cubic nonlinear defocusing Schr\"odinger equation
Analysis of PDEs
2023-02-02 v2
Abstract
We characterize the location and number of eigenvalues for the Lax operator associated to the one-dimensional cubic nonlinear defocusing Schr\"odinger equation. With the help of a newly discovered unitary matrix, the analysis reduces to the study of the spectral problem for a unitarily equivalent operator, which involves only the amplitude and the phase velocity of the potential. Examples of potentials with special amplitude and phase velocity are investigated.
Keywords
Cite
@article{arxiv.2207.05186,
title = {Eigenvalue analysis of the Lax operator for the one-dimensional cubic nonlinear defocusing Schr\"odinger equation},
author = {Xian Liao and Michael Plum},
journal= {arXiv preprint arXiv:2207.05186},
year = {2023}
}
Comments
The number of eigenvalues for more general cases is characterized. Some interesting examples of potentials with piecewise-constant amplitude and phase velocity functions are included