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We analyze the 1D cubic nonlinear stationary Schr\"odinger equation on a ringwith 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 ellipticfunctions, and eigenvalues for the ground state and first few excited energylevels. All six independent Jacobi elliptic functions are found to be solutionsof some boundary condition. We also provide a way to map all eigenfunctionssatisfying $\delta$/$\delta'$ conditions to any other general boundarycondition or point-like potential.

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