2. Conclusions

Here we represent some examples where hydrodynamic methods and solutions may be useful for different problems in nonlinear optics. In these examples, the medium itself has the first degree of importance in realization of all mentioned phenomena. Indeed, the main future of the Vavilov-Cherenkov radiation is that the medium is the source of this radiation instead of any kinds of bremsstrahlung radiations by moving charged particles. The VCR theory presented here for the first time takes into account the real mechanism of VCR by the medium itself, excited by a sufficiently fast electron. It can also be shown only from the microscopic theory, but not from the macroscopic one stated in [16]. The first step in this direction was made in [47] also on the basis of the Abraham theory where it is proposed that the Vavilov-Cherenkov radiation is emitted by the medium in a nonequilibrium polarization state which is arising due to the parametric resonance interaction of the medium with a fast-charged particle.

Hydrodynamic Methods and Exact Solutions in Application to the Electromagnetic Field Theory… DOI: http://dx.doi.org/10.5772/intechopen.80813

The second example, which is represented here, also gives new perspectives on the basis of the new exact solution (in the Euler variables) for n-dimensional Hopf equation because this equation is known as the possible model for weak nonlinear optic problems [46]. The importance of the new solution is connected with its Euler form in dependence from space variables, which are not represented in the solution of the Burgers-Hopf equation well known before (see [45] and others).
