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.

The nonlinear crystal with sharp-step dependence of dielectric function on the electric field covered by shielding thin film is considered. A change in the signs of the nonlinearity coefficient is induced by the electric field. Different sets of boundary conditions are used to calculate the exact solutions of the formulated equation. New types of nonlinear surface waves are found and the conditions of their existence are derived analytically. In certain cases, some types of surface waves are characterised by free-varying wave frequency and effective refractive index. Other types of surface waves can propagate with a certain effective refractive index and arbitrary frequencies. In addition, new types of waves propagating only with fixed effective refractive index and fixed frequency are described.

The steady-state diffusion processes are investigated theoretically. Exact solutions of the one-dimensional steady-state quasi-linear diffusion equation with smooth step dependence of the diffusion coefficient on concentration in various smooth step functions (linear step, exponential step, logistic, power saturation step) are obtained. The proposed theory is applied to describe peculiarities of diffusion-activated recrystallisation at steady-state stage at long times of diffusion annealing. The diffusant concentration monotonically decreases with anincreasing depth of diffusant penetration. The smallest penetration depth of the diffusant is the characteristic of the logistic model. The analytical estimations of the diffusion coefficient with known recrystallisation front position are given. The position of recrystallisation front decreases monotonically with an increase in the model parameter, which corresponds to the smoothness of the step and the rate of change of the diffusion coefficient.