Conserved energy approximation to wave scattering by a nonlinear interface

An alternative approach to the investigation of the space-time evolution of an electromagnetic field in a nonlinear medium is presented. An adiabatic theory to obtain the dynamic picture of reflection and penetration of electromagnetic radiation into a nonlinear medium is devised. It is based upon the energy balance equation and uses the nonlinear modes of the stationary problem like trial functions. Filamentation and soliton generation are analyzed and the model easily predicts the giant nonlinear Goos-H\"anchen beam shifts that emerge from previously numerically intensive investigations. The method is nonperturbative and is straightforward to apply to linear-nonlinear interfaces.