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

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.