Stationary real solutions of the nonlinear Schr\"odinger equation on a ring with a defect

TL;DR

This paper classifies all real stationary solutions of the nonlinear Schrödinger equation on a ring with a defect, using Jacobi elliptic functions, and explores boundary conditions and mappings between solutions.

Contribution

It provides a comprehensive analysis of solutions under all delta and delta prime boundary conditions, including explicit eigenfunctions and eigenvalues, and introduces a method to relate different boundary conditions.

Findings

All six Jacobi elliptic functions are solutions for some boundary condition.

Explicit eigenfunctions and eigenvalues are computed for various boundary conditions.

A mapping method relates solutions across different boundary conditions.

Abstract

We analyze the 1D cubic nonlinear stationary Schr\"odinger equation on a ring with a defect for both focusing and defocusing nonlinearity. All possible $\delta$ and $\delta'$ boundary conditions are considered at the defect, computing for each of them the real eigenfunctions, written as Jacobi elliptic functions, and eigenvalues for the ground state and first few excited energy levels. All six independent Jacobi elliptic functions are found to be solutions of some boundary condition. We also provide a way to map all eigenfunctions satisfying $\delta$/$\delta'$ conditions to any other general boundary condition or point-like potential.