A difference scheme of the predictor–corrector type is constructed for solving nonlinear dispersive equations of wave hydrodynamics with an increased order of approximation of the dispersion relation, based on splitting of the original system of equations into a hyperbolic system and a scalar equation of the elliptic type. A dissipation and dispersion analysis of the new scheme is performed, a condition for its stability is obtained, and a formula for the phase error is written and analyzed. Parameters are found at which the phase characteristics of the difference scheme, the nonlinear-dispersive model approximated by it, and the full model of potential flows have the same order of accuracy.

Properties of dispersion relations for two new fully nonlinear weakly dispersive shallow water models are studied. For these models, it is possible to use certain parameters to obtain the fourth, sixth, or eighth order of accuracy of phase velocity approximation of the three-dimensional potential flow model. For the hierarchy of shallow water models, with the assumption of slight changes in the bottom shape, formulas are obtained that establish a relationship between the variation rate of the wave amplitude and that of the fluid layer thickness; also, the dependences of the amplitude and length of an incoming wave on the water region depth are derived. It is shown that the new model of the fourth-order long-wave approximation with the eighth order of accuracy of the dispersion relation provides the best approximation of considered characteristics in the case of bottoms of both horizontal and variable shape.

Построена разностная схема типа предиктор-корректор для решения нелинейно-дисперсионных уравнений волновой гидродинамики с повышенным порядком аппроксимации дисперсионного соотношения, основанная на расщеплении исходной системы уравнений на гиперболическую систему и скалярное уравнение эллиптического типа. Выполнен диссипативный и дисперсионный анализ новой схемы, получено условие ее устойчивости, выписана и проанализирована формула для фазовой ошибки. Найдены параметры, при которых достигается одинаковый порядок точности фазовых характеристик разностной схемы, аппроксимируемой ею нелинейно-дисперсионной модели и полной модели потенциальных течений. A difference scheme of the predictor-corrector type is constructed for solving nonlinear dispersion equations of wave hydrodynamics with a high order of approximation of the dispersion relation, based on splitting of the original system of equations into a hyperbolic system and a scalar equation of the elliptic type. A dissipation and dispersion analysis of the new scheme is performed, a condition for its stability is obtained, and a formula for the phase error is written and analyzed. Parameters are found at which the phase characteristics of the difference scheme, the nonlinear-dispersive model approximated by it, and the full model of potential flows have the same order of accuracy.

Twenty-seven patients with pain due to the imbalance of the body were examined. Clinical evaluation of patients before and after treatment included a general orthopedic examination, computer optical topography, surface electroneuromyography. The main group (17 patients) received injections of the drug lantoks in spastic muscles, the control group received a standard set of treatment to relieve spasticity (massage, exercise therapy, FTL, pharmacotherapy). The treatment gave a significant difference between the main and control groups in the results of changing the parameters of posture and electrophysiological parameters of muscle as well as the duration of pain relief. The results of the study confirm a significant impact of the local muscle relaxation on the parameters of posture. The high efficacy of "Lantoks" in the reconstruction of functional parameters in the groups of spastic muscles was shown.

Under weaker than in the paper by Green A.E., Naghdi P.M. (J. Fluid Mech. 1976) restrictions on the velocity of a three-dimensional vortex fluid flow above a moving bottom, nonlinear dispersive shallow water equations are derived for an asymptotic description of flows with a free boundary. Orders of approximation for the basic hydrodynamic quantities and equations, appeared in the reduction of the 3D-model to an approximate model, are determined. The laws of change for the total energy and the potential vortex in the obtained nonlinear dispersive model are found.

Nonlinear dispersion shallow water equations are derived, which describe propagation of long surface waves on a spherical surface with allowance for rotation of the Earth and mobility of the ocean bottom. Derivation of these equations is based on expanding the solution of hydrodynamic equations on a sphere in small parameters depending on the relative thickness of the water layer and dispersion of surface waves.

Nonlinear dispersive shallow water equations on a sphere are obtained without using the potential flow assumption. Boussinesq-type equations for weakly nonlinear waves over a moving bottom are derived. It is found that the total energy balance holds for all obtained nonlinear dispersive equations on a sphere.