Eigenvalue analysis of the Lax operator for the one-dimensional cubic nonlinear defocusing Schr\"odinger equation

TL;DR

This paper analyzes the eigenvalues of the Lax operator for the 1D cubic defocusing nonlinear Schrödinger equation, using a novel unitary transformation to simplify the spectral problem based on potential parameters.

Contribution

It introduces a new unitary matrix that simplifies the spectral analysis of the Lax operator for this equation, enabling explicit characterization of eigenvalues.

Findings

Eigenvalues are characterized based on potential amplitude and phase velocity.

The analysis reduces to studying a unitarily equivalent operator.

Examples illustrate potentials with specific amplitude and phase velocity.

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.