We describe analytically the nonlinear surface waves at the interface between two nonlinear media with different characteristics. We use one-dimensional nonlinear Schrödinger equation with cubic nonlinearity differing on the opposite sides of the interface. We take into account the interaction of excitations with media interface. We consider the interaction of the wave with the interface using the local potential approximated by Dirac delta function. We derive and analyse three types of dispersion equations determining the surface wave frequencies. We propose two approaches to determine the flux depending on the choice of one of the possible control parameters. We calculate the energy flux of the surface waves and analyse the influence of intensity interaction of excitations with interface and difference of media characteristics on the opposite sides of the interface.