Eigenvalue cut-off in the cubic-quintic nonlinear Schrodinger equation

TL;DR

This paper proves that eigenvalues of localized stationary solutions in the 2D+1 cubic-quintic nonlinear Schrödinger equation have an upper cut-off, using mathematical inequalities and identities, and compares eigenstates near zero eigenvalues to those of the cubic case.

Contribution

It provides a rigorous proof of the eigenvalue cut-off phenomenon and analyzes the behavior of eigenstates near zero eigenvalues in the cubic-quintic nonlinear Schrödinger equation.

Findings

Eigenvalues have an upper cut-off value.

Eigenstates near zero eigenvalues resemble those of the cubic Schrödinger equation.

Theoretical proof using inequalities and identities.

Abstract

Using theoretical arguments, we prove the numerically well-known fact that the eigenvalues of all localized stationary solutions of the cubic-quintic 2D+1 nonlinear Schrodinger equation exhibit an upper cut-off value. The existence of the cut-off is inferred using Gagliardo-Nirenberg and Holder inequalities together with Pohozaev identities. We also show that, in the limit of eigenvalues close to zero, the eigenstates of the cubic-quintic nonlinear Schrodinger equation behave similarly to those of the cubic nonlinear Schrodinger equation.