(color online) The figures show the two edge states and explain how these states can be induced by the parabolas and the walls. From top to bottom, the blue solid lines represent two edge states of the nonlinear Schrödinger equation (1) (a): Ψ₈ and (b): Ψ₁₆. The black circle lines show the wave functions of the nonlinear Schrödinger equation (1) with the triangle potential conformed by the wall and the parabolas near the boundaries (yellow thick solid lines). The periodic potentials V(x) are shown by the yellow dashed solid lines.