## 5 Scientific Novelty and Scientific and Practical Significance of the Research Results
### The Main Areas of Practical Application of the Results  
#### 5.2.1 Development of Nonlinear Optics and Laser Technology  
The methodology proposed in this article for solving and analyzing the properties of the nonlinear Fokker-Planck equation for the process of ion-relaxation polarization in dielectrics can be applied in the development of quasi-classical schemes for analytical studies of the kinetics of nonlinear optical-electrical and optical processes generated by exposing a crystal to high-intensity electromagnetic radiation, which is most effective when irradiating crystals with laser radiation. At high intensity of laser radiation, when the modulus of the electric field strength of the wave is comparable to the microscopic intra-atomic field (approximately 10,000 MV/m), nonlinear optical effects are generated in the substance: nonlinear light absorption (self-illumination and self-darkening of the medium); frequency conversion (harmonic generation, parametric generation, and stimulated light scattering); effects of self-action of light (self-focusing and self-defocusing of beams); and multiphoton processes. Nonlinear optical effects are similar (both in terms of the electrophysical mechanism and in mathematical terms) to nonlinear polarization effects generated in ion-molecular dielectrics in the process of ion-relaxation polarization in the region of ultra-high temperatures (550-1550 K) and ultra-strong fields (10-1000 MV/m) and in the region of ultra-low temperatures (1-10 K) and weak fields (0.1-1 MV/m) [1; 2; 3; 4]. Nonlinear effects of frequency conversion and self-action of light arise due to the fact that when a medium is polarized by an electromagnetic wave (radiation) of high intensity, a nonlinear component of polarization is generated (a multiple of the third, fifth, seventh, etc. degree of the electric field strength of the wave), which rises with increasing intensity of the electromagnetic wave. The presence of such a nonlinear component is explained by the anharmonicity of the oscillations of particles of the medium in the field of a powerful light wave. Because of this anharmonicity, the response of the environment to a harmonic external electric field becomes nonlinear, that is, it ceases to repeat the shape of the external influence. Thus, in the electric field re-radiated by the atoms and molecules of the substance, new harmonic components appear-generation of higher optical harmonics of optical radiation incident on the medium occurs. Nonlinear polarization of the medium is also the reason for the dependence of the refractive index on the intensity of the electromagnetic wave. If the refractive index increases with increasing intensity of the light wave, then the rays bend toward the beam axis, and when a certain critical power is exceeded, the phenomenon of self-focusing of light is observed, i.e., the diverging light beam becomes convergent. The opposite process of self-action of light is also possible: self-defocusing of the beam occurs if the refractive index decreases with increasing wave intensity. The most widespread process in practical applications is the generation of the second harmonic (a multiple of the third power of the electric field of the wave) in \"nonlinear crystals\" with specially selected properties. For example, infrared (IR) radiation from an Nd:YAG laser with a wavelength of 1.064 \(\upmu\)m in a KDP (potassium dihydrogen phosphate) crystal is converted into green light with a wavelength of 0.532 \(\upmu\)m [1,4]. It should be noted that the KDP crystal, in terms of its crystal structure type and properties, belongs to ferroelectric hydrogen-bonded crystals (HBC) and is characterized by a rectangular hysteresis loop (RHL) inherent to materials of this class, which is typical for high-quality proton semiconductors that also have nonlinear electrophysical properties (for the previously indicated ranges of field and temperature parameters). It is obvious that the manifestation of both optical and electrophysical nonlinearities in this crystal is quite natural.