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A finite-difference scheme is presented for the solution of the problem on the interaction of a femtosecond laser pulse with glass in the approximation of nonlinear Maxwell equations supplemented by equations of the hydrodynamic type for conduction electrons. The model takes into account all the basic physical processes involved in this interaction. An axially symmetric problem is considered. The construction of the scheme allows for the specifics of the problem, which ensures the efficiency of the developed method. The use of the scheme is illustrated by the results of modeling the interaction between glass and femtosecond laser pulses of an ordinary Gaussian shape with linear polarization and doughnut pulses with radial and azimuthal polarization. Significant differences in the dynamics of the interaction between glass and these three types of pulses are revealed.

Представлена конечно-разностная схема для решения задач о взаимодействии фемтосекундного лазерного импульса со стеклами в приближении нелинейных уравнений Максвелла, дополненных уравнениями гидродинамического типа для электронов проводимости. В модели учтены все основные физические процессы, имеющие место в этом взаимодействии. Рассматривается аксиально-симметричная задача. При построении схемы учитываются особенности задачи, что обеспечивает высокую эффективность разработанного метода. В качестве примера использования схемы приведены результаты моделирования взаимодействия со стеклом фемтосекундных лазерных импульсов обычной гауссовой формы с линейной поляризацией и импульсов тороидальной формы с радиальной и азимутальной поляризациями. Выявлены существенные различия в динамике взаимодействии со стеклом этих трех видов импульсов.

Generalization of deep convection equations for gas for an arbitrary medium is suggested. Some details of solvability and applicability conditions for these equations and selection of equilibrium pressure are considered.

In this work we consider an experimentally observed effect of significant increasing of the residual heat in metal targets at their irradiation with femtosecond laser pulses in an ambient gas in respect to the vacuum conditions. Numerical modelling of heating of a platinum target by femtosecond laser pulses in argon under normal conditions has been performed taking into account gas breakdown in the focussing region of the laser beam in front of the target. The applied model is based on a combination of a thermal model describing heating and phase transitions in irradiated samples and a hydrodynamic model to describe motion of the ambient gas perturbed by laser irradiation as a result of multiphoton ionization. The hot ambient gas is shown to heat efficiently the irradiated sample. The hydrodynamic processes in the ambient gas play an important role in heating.

New instability conditions for some classes of nonlinear dynamical systems of an arbitrary order are considered. These conditions reduce the investigation of instability of the initial nonlinear system to the investigation of instability of the linearized system. In this connection, the considered conditions are similar to the hypothesis of the instability theorem of the first Lyapunov method (the stability investigation by the first approximation), but they are applicable to more complicated nonlinear systems, because it takes into account the non-autonomy of the linearized system and the fact that the nonlinear terms in the right-hand sides of the equations of the initial system belong not only to the class of analytical functions. For different classes of the nonlinear systems, sufficient instability conditions are presented.

A method for studying the stability of the equilibrium points of linearized, nonlinear dynamic systems of arbitrary order is considered. The method is based on the fact that, due to the nature of the mutual arrangement of the trajectories of the corresponding linearized system, and the boundaries of some simply-connected, bounded neighborhood of its equilibrium point, one can judge the asymptotic stability and instability of both this point and the equilibrium point of the nonlinear system. Necessary and sufficient conditions of asymptotic stability and sufficient conditions of instability of equilibrium points of linear systems are given. Together with the theorems of the first Lyapunov method, these conditions determine the sufficient conditions of asymptotic stability and instability of equilibrium points of nonlinear systems. In some cases, the proposed conditions may turn out to be preferable to the known ones.

The relaxation of a magnetic field perturbation periodic in two directions is studied in the electron magnetohydrodynamic approximation. It is shown that the relaxation is accompanied by the coalescence of magnetic cells. When transport coefficients are sufficiently low, the nonlinear stage of the coalescence process is independent of their values. This effect may help to explain why the duration of the crash phase of sawtooth oscillations in tokamaks is anomalously short.

A study is made of the propagation of an Alfven pulse and the superposition of an Alfven and a magnetosonic pulse in the vicinity of the magnetic null line (the X point). It is shown that, on long time scales, the poloidal components of the velocity and magnetic field relax to steady-state distributions similar to those in the case of a magnetosonic pulse. In the essentially nonlinear problem under investigation, the steady-state distributions of the toroidal components of the velocity and magnetic field are found to be close to those in the corresponding linear problem. It is established that two-dimensional effects play an important role in the evolution of the forming current sheets.

The so-called reduced magnetohydrodynamics, which deals with the motion of incompressible fluids and is usually applied to describe plasma flows in a strong toroidal magnetic field, has a number of drawbacks and, in some cases, fails to produce correct results. The equations proposed here are simpler than the original MHD equations and are free of these drawbacks. These equations, like reduced MHD equations, make it possible to remove from consideration fast magnetosonic waves and to introduce the vector potential for the poloidal magnetic field. However, our equations differ from the reduced MHD equations in that they completely incorporate slow magnetosonic waves, the specific features of the toroidal geometry, and the effects of the toroidal velocity.

The problem of the tearing instability is solved numerically in cylindrical geometry by using the unreduced two-fluid MHD model. It is shown that the duration of the nonlinear stage of the tearing instability in a hot plasma is rather sensitive to such factors as the initial radial density and temperature profiles, the initial ion-to-electron pressure ratio, and the longitudinal thermal conductivity. Depending on these factors, the two-fluid effects (primarily, the Hall effect) can either greatly hasten the magnetic reconnection process (in comparison to that in the one-fluid MHD model) or greatly slow it. An illustrative explanation of the results obtained is given.