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This talk is devoted to a one of the most interesting and rapidly developing areas of modern nonlinear physics and mathematics - the analytical and advanced numerical study of the structure and stability of two- and three-dimensional solitons in dispersive complex media described by the generalized system which includes the Kadomtsev-Petviashvili and derivative nonlinear Schrodinger classes of equations and takes into account the generalizations relevant to various complex physical media, associated with the effects of high-order dispersion corrections, influence of dissipation and instabilities. This is consistent representation of the both early known and new original results obtained by author and also some generalizations in theory and numerical simulation of nonlinear waves and solitons in complex dispersive media. The analysis of stability of solutions is based on study of transformational properties of system Hamiltonian.

© 2017, Pleiades Publishing, Ltd. The structure and dynamics of ion-acoustic waves in an unmagnetized plasma, including the case of weakly relativistic collisional plasma (when it is necessary to take into account the high energy particle flows which are observed in the magnetospheric plasma), are studied analytically and numerically on the basis of a model of the Kadomtsev-Petviashvili (KP) equation. It is shown that, if the velocity of plasma particles approaches the speed of light, the relativistic effects start to strongly influence on the wave characteristics, such as its phase velocity, amplitude, and characteristic wavelength, with the propagation of the twodimensional solitary ion-acoustic wave. The results can be used in the study of nonlinear wave processes in the magnetosphere and in laser and astrophysical plasma.

© 2017 COSPAR The results of analysis and numerical simulation of evolution and interaction of the N-vortex structures of various configuration and different vorticities in the continuum including atmosphere, hydrosphere and plasma are presented. It is found that in dependence on initial conditions the regimes of weak interaction with quasi-stationary evolution and active interaction with the “phase intermixing”, when the evolution can lead to formation of complex forms of vorticity regions, are realized in the N-vortex systems. For the 2-vortex interaction the generalized critical parameter determining qualitative character of interaction of vortices is introduced. It is shown that for given initial conditions its value divides modes of active interaction and quasi-stationary evolution. The results of simulation of evolution and interaction of the two-dimensional and three-dimensional vortex structures, including such phenomena as dynamics of the atmospheric synoptic vortices of cyclonic types and tornado, hydrodynamic 4-vortex interaction and also interaction in the systems of a type of “hydrodynamic vortex – dust particles” are presented. The applications of undertaken approach to the problems of such plasma systems as streams of charged particles in a uniform magnetic field B and plasma clouds in the ionosphere are considered. It is shown that the results obtained have obvious applications in studies of the dynamics of the vortex structures dynamics in atmosphere, hydrosphere and plasma.