For the traditional Kolmogorov spectrum of turbulence, the energy cascade covers a wide range of spectra from low wave numbers, involved in the vortex energy being pumped in, to quite high wave numbers, when vortex energy is dissipated. This type of fully developed wave turbulence is popular in the discipline of traditional turbulence research, which was presented [V. E. Zakharov, V. S. L’vov and G. Falkovich, Kolmogorov spectra of turbulence I. Wave turbulence (1992; Zbl 0786.76002)]. In the present book, wave turbulence under parametric excitation and application to magnetics, described as opposite limiting cases against the Kolmogorov cascade turbulence mentioned above, are discussed taking into account the turbulence energy pumping and dissipating occurring at the same scale of wave numbers. The evolution of a parametric instability of waves in media under strong external excitation exhibits some types of parametric wave turbulence, such as the characteristics of a hot plasma in the media under powerful laser beam or in the ferromagnet in a strong microwave field.
Since the parametric wave turbulence can be regarded as an essential part of nonlinear wave physics, first the classical Hamiltonian formalism is described in detail; then the formulation of the so-called \(S\)-theory is presented, which describes the turbulence of parametrically excited waves in the framework of mean-field approximation, similar to approximation in Curie-Weiss theory of a molecular field, etc. Finally, the advanced \(S\)- theory takes into account the self-consistent interaction of wave pairs and the scattering of individual waves, and their interactions with inhomogeneities of the media. Numerous experimental results on parametric wave turbulence are also presented. This book is surely useful for any person, who pursues an introduction to the idea of nonlinear wave turbulence in solid state physics, hydrodynamics, plasma and magnet physics.