Aim. Development of a microscopic theory of nonlinear thermodiffusophoresis that combines quantum corrections, critical fluctuations, and anomalous transport regimes for systems with strong temperature gradients.

Methodology. Methods of nonequilibrium statistical mechanics (nonequilibrium statistical operator), renormalization group analysis for critical phenomena, and fractional calculus for describing anomalous transport are used.

Results. Within the framework of the developed theory, generalized transport equations are derived, including a non-local memory kernel K (r, t; T), which explicitly depends on the temperature. Anomalous behavior of the thermodiffusion coefficient near the critical point is established, described by the scaling 𝐷_T~|𝑇 – 𝑇_c𝑇|^-𝛾 with an effective exponent 𝛾 = 1,24 + 0,17 where the addition of 0,17 is due to hydrodynamic interactions. Regimes of anomalous transport with fractional exponents are discovered and classified, where the root-mean-square displacement of particles follows the law ⟨∆𝑟²⟩~𝑡^α with the exponent α that smoothly varies from 0,7 (subdiffusion) to 1.5 (superdiffusion) depending on the magnitude of the temperature gradient. For nanoscale systems at low temperatures, we have obtained explicit expressions for quantum corrections to the system's Hamiltonian that account for tunneling effects and the nonlocality of the temperature field.

Research implications include creating a fundamental basis for the design of microfluidic devices, nanoparticle control in biomedicine, and the development of new materials with thermally controlled properties.

Aim. The dynamics of classical fields, which leads to photon production in nonlinear crystals, obey nonlinear equations. These equations have soliton solutions and describe the formation of shock-wave profiles.

Methodology. This paper presents a mathematical description of parametric scattering processes based on coupled-wave equations and the nonlinear Schrödinger equation.

Results. The relationship between the shock wave type solution in the Burgers equation and the formation of the bi-photon generation front is shown.

Research implications. The mathematical apparatus and physical analogies from the theory of shock waves and solitons are extremely useful for describing and understanding how and under what conditions new photons are produced.

Aim: analytical development of the method of total differentials, created earlier for the numerical solution of hyperbolic systems of partial differential equations of the first order and the construction of a complete system of integrals of the equations of motion.

Methodology. The method consists in the fact that the systems of first-order partial differential equations are reduced to relationships between total differentials of gas-dynamic variables along different directions using linear transformations. The integration procedure is applied to the resulting transformed systems and the analysis and synthesis of the obtained results is performed.

Results. A complete system of integrals is obtained for the Euler system of gas dynamics equations. Based on their analysis, a new result was obtained on the influence of plane waves and on the hypersurfaces formed by the intersection points of these waves, covering the characteristics. A new idea is obtained about the causes of numerical instability of solutions of general gas dynamics equations by the method of characteristics. A new understanding of the causes of numerical instability of solutions to general gas dynamics equations is obtained using the method of characteristics. Based on the application of the method to the Laplace equation, the Cauchy integral theorem is obtained.

Research implications. Complete integrals of Euler equations give an idea of how its solution is generally structured. The expressions obtained in the work can also be used to construct difference schemes of high resolution and high order of approximation. Such schemes are most suitable for describing transition regimes of flow of viscous heat-conducting gas.

Aim is to investigate, in a nonlinear formulation, the instability of the boundary of two perfect gases with different heat capacities when an intense shock wave falls on it.

Methodology. Numerical modeling within the Euler equations was applied to use these calculation results with the data obtained within the weakly nonlinear approximation and the experiment.

Results. It is shown that for irregular refraction in the problem, self-organization of the subsonic flow of a perfect gas occurs, which made it possible to obtain a solution without additional measures to determine one of the adiabatic indices. Against the background of small-scale turbulence generated by the Richtmyer – Meshkov instability, explosive instability was detected in the problem.

Research implications. The process of self-organization of gas flow, which allowed the determination of the adiabatic index in one gas, was discovered for the first time in a refraction problem. The results presented in the article explain, at a nonlinear level, the mechanism for the formation of finger-like structures during the interaction of a shock wave with a light-heavy gas boundary, and complement the data obtained experimentally and theoretically within the framework of a weakly nonlinear approach.

Aim. To study the dispersion of the refractive index anisotropy Δn and the order parameter S of the liquid crystal mixture ZhK-1289 in the temperature range of –60 to +60°C and to verify the applicability of the Landau-de Gennes model.

Methodology. Interference spectroscopy was employed. Transmission spectra of a planar LC cell were measured at temperatures ranging from –60°C to +60°C. The birefringence Δn was determined from the positions of interference maxima. The dependence S(T) was calculated based on Δn(T).

Results. The dependences Δn(λ) were established throughout the entire existence range of the nematic phase. The refractive index anisotropy decreases with increasing temperature and wavelength. The order parameter decreases from 0.75 at –40°C to 0.23 at +61°C. The critical exponent β = 0.23 ± 0.01 is close to 0.25, confirming the model.

Research implications. Data on Δn(λ, T) and S(T) in the low-temperature nematic phase of ZhK-1289 were obtained, and the Landau-de Gennes model was validated. The results are important for designing thermally stable LC devices.