Nonlinear localized states near the interface with nonlinear response between the medium with a parabolic index spatial profile and Kerr-type medium

The interface with nonlinear response separating the parabolic graded-index and the Kerr nonlinear media are considered. Exact solutions to the nonlinear Schrödinger equation with nonlinear short-range potential and a parabolic spatial profile are found applying to the theoretical description of the stationary states localized near interface with nonlinear properties. Localized states with continuous/discrete energy spectrum are described by the Whittaker function/Hermite polynomials in the medium with a parabolic profile of characteristic and the hyperbolic cosine (sine) in the medium with a self-focusing/defocusing Kerr nonlinearity. The field localization length is wider in the case of a self-focusing nonlinearity than in the case of a defocusing one. The maximum of the wave function is located in a nonlinear medium in the case of a self-focusing nonlinearity and at the interface in the case of a defocusing one. It is shown the possibility of a motion closer to the interface (or away from it) the maximum intensity of the localized state by changing the values of the interface response parameters at the fixed localization energy. A growth of the width of the parabolic graded-index layer adduced an increase in the maximum height of localized states of discrete spectrum and theirs localization length in the graded-index layer, but it had almost no effect on the profile of localized states of the continuous spectrum.