Breaking electromagneticwaves in a strongly nonlinear medium

Background. Waves in the nonlinear medium without dispersion are described, as a rule, with first-order quasi-linear equations, characteristic of the problems of gas, liquid and plasma hydrodynamics. However, for such branches of physics, as the theory of electromagnetic waves in nonlinear medium, the description of waves is based on Maxwell's equations, which are second-order hyperbolic equations. This paper shows a close link between these equations. In this regard, there arises a question about the existence of a link between the processes, which are described by first- and second-order quasi-linear hyperbolic equations. The aim of this study is to construct exact solutions of non-linear equations of electromagnetic waves dynamics, including the medium with Kerr nonlinearity when there is no dispersion. The analysis of these decisions is carried out. Materials and methods. The main method used in the work is the construction of solutions for Maxwell's equations for waves in nonlinear dielectrics without dispersion as solutions for first-order quasilinear hyperbolic equations. The method was first developed for the equations in arbitrary finite dmension, and then applied to the problem of electromagnetic waves propagation in the medium with Kerr nonlinearity. The study is based on exact solutions for Maxwell's equations and sound waves equations for a wide range of functional dependencies of the medium parameters on the amplitude. Results. New exact solutions for arbitrary dimension of the coordinate space for the nonlinear equations under study are found. The possibility of an arbitrary trajectory of the wave front propagation is established. The existence of the phenomenon of wave breaking and the formation of shock waves in such media is demonstrated. Various types of wave propagation modes for different types of initial distribution symmetries are considered. The processes of energy dissipation in the formation of discontinuous solutions are analysed. Conclusions. The equations for the optical and acoustic pulses allow for classes of exact solutions that are both solutions of quasi-linear equations. The Cauchy problem solution set for one-dimensional quasi-linear equations is the same as that for equations of parabolic approximation of Maxwell's equations used in optics, and sound wave equations in acoustics. In the multidimensional case, there are complex processes that are connected with solutions in the riverton form. The formation of electromagnetic shock waves is accompanied by intense dissipation of wave energy when their amplitude approaches critical values. For a Kerr medium (with a cubic nonlinearity), the amplitude critical values exist for any positive values of the Kerr nonlinearity parameter.