Physics

Based on the hydrodynamic description, the dynamics of nonlinear cylindrical waves in an isentropic plasma are investigated. The problem is considered in an electrostatic formulation for a two-dimensional plasma medium where ions form a stationary background. Proceeding from the particular, exact solution of hydrodynamic equations, we obtain the system of differential equations which describes the electron’s dynamics, taking into account the finite temperature of electrons. Moreover, we find the conditions when this system is reduced to the generalized Ermakov–Pinney equation which was used for analyzing electron dynamics. In the present calculations, a parabolic-in-radius temperature profile was used, associated with an electron density varying only with time. In the framework of the model that worked out, the influence of initial conditions and thermal effects on the regular and singular dynamics of excited waves are discussed. It is shown that the development of singular behavior due to intrinsic nonlinearity is avoided by taking into account thermal effects and the initial rotation of the electron flow.

The effect of the curvature of the boundary of semi-infinite cold plasma on the parameters and properties of surface oscillations localized near this boundary is considered. An analytical description of various cases of the impact of static deformation of the plasma boundary on the characteristics of the oscillating surface charge is obtained, and the results of the exact numerical solution of the initial equations are found to confirm the reliability of the derived analytical formulas. A significant role of the boundary perturbation shape in the formation of the spatial distribution of surface oscillation parameters is revealed. With the help of analytical formulas and precise numerical calculations, a description of this nonlinear interaction is presented. The availability of such a description is crucial both for determining the possibility of using the examined effect for specific applications and, on the other hand, for exciting it in plasma, which requires knowledge of the field structure features.

By extending the four-dimensional semi-Riemann geometry to higher-dimensional Finsler/Hamilton geometry, the canonical quantization of the fundamental metric tensor of general relativity, i.e., an approach that tackles a geometric quantity, is derived. With this quantization, the smooth continuous Finsler structure is transformed into a quantized Hamilton structure through the kinematics of a free-falling quantum particle with a positive mass, along with the introduction of the relativistic generalized uncertainty principle (RGUP) that generalizes quantum mechanics by integrating gravity. This transformation ensures the preservation of the positive one-homogeneity of both Finsler and Hamilton structures, while the RGUP dictates modifications in the noncommutative relations due to integrating consequences of relativistic gravitational fields in quantum mechanics. The anisotropic conformal transformation of the resulting metric tensor and its inverse in higher-dimensional spaces has been determined, particularly highlighting their translations to the four-dimensional fundamental metric tensor and its inverse. It is essential to recognize the complexity involved in computing the fundamental inverse metric tensor during a conformal transformation, as it is influenced by variables like spatial coordinates and directional orientation, making it a challenging task, especially in tensorial terms. We conclude that the derivations in this study are not limited to the structure in tangent and cotangent bundles, which might include both spacetime and momentum space, but are also applicable to higher-dimensional contexts. The theoretical framework of quantization of general relativity based on quantizing its metric tensor is primarily grounded in the four-dimensional metric tensor and its inverse in pseudo-Riemannian geometry.