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In this paper, mathematical models for the analysis of processes of generation and resonance scattering of wave packets on a transversely inhomogeneous, isotropic, cubically polarisable, non-magnetic, linearly polarised (E polarisation) medium with a non-linear, layered dielectric structure, and methods of their numerical simulation are considered. In general, electromagnetic waves in a non-linear medium with a cubic polarisability can be described by an infinite system of non-linear differential equations. In the study of particular non-linear effects it proves to be possible to restrict the examination to a finite number of equations, and also to leave certain terms in the representation of the polarisation coefficients, which characterise the physical problem under investigation. Here we investigate the situation where the incident field consists of a packet of three waves oscillating with single, double and triple frequency. An intense field at the basic frequency leads to the generation of the third harmonic, i.e. of a field at the triple frequency. In this case it is possible to reduce the mathematical model to a system of two equations, where only the non-trivial terms in the expansion of the polarisation coefficients are taken into account (see Angermann & Yatsyk (2010)). The consideration of a weak field at the double frequency or at both the double and triple frequencies allows to analyse its influence on the generation process of the third harmonic. In this situation, the mathematical model consists of three differential equations. The rigorous formulation finally leads to a system of boundary-value problems of Sturm-Liouville type, which can be equivalently transformed into a system of one-dimensional non-linear integral equations (defined along the height of the structure) with respect to the complex Fourier amplitudes of the scattered fields in the non-linear layer at the basic and multiple frequencies. In the paper both the variational approach to the approximate solution of the system of non-linear boundary-value problems of Sturm-Liouville type (based on the application of a finite element method) and an iterative scheme of the solution of the system of non-linear integral equations (based on the application of a quadrature rule to each of the non-linear integral equations) are considered. 8

Generation and Resonance Scattering of Waves on Cubically Polarisable Layered Structures