V. I. Karpman's research while affiliated with Hebrew University of Jerusalem and other places

A theory of envelope whistler solitons beyondthe approximation based on the nonlinear Schrödinger (NLS) equation is developed. It is shown that such solitons must emanate radiation due to the continuos transformation of trapped whistler modes into other modes that cannot be trapped in the duct, produced by the soliton (such modes are not described by the NLS equation). An equation governing the decrease of soliton amplitude due to the loss of trapped radiation is derived. The soliton radiation increases with the decrease of the soliton size and, therefore only weak solitons have sufficiently large lifetime. The theory is extended to the whistler spiral wave beams which, according to the NLS equation, must be liable to the self-focusing. It is shown that when the wave beam becomes sufficiently narrow, the self-focusing is replaced by the defocusing because of big radiation losses. These predictions are confirmed by numerical experiments. Possible generalizations to other gyrotropic media are briefly discussed.

General Schrödinger equation describing the propagation of whistler waves along the magnetic field is obtained. On the basis of this equation and the magnetohydrodynamic plasma equations supplemented by the ponderomotive force of the whistler wave field, the self-focusing of whistler waves is considered. A family of stationary solutions describing the structures of self-focused wave beams both in density crests and troughs is found. It is shown that the beam propagating in density crest decays due to the wave tunnelling. Therefore, only the density troughs may result from the well-developed self-focusing of whistler wave parallel to the magnetic field. This conclusion as well as some others seem to be in agreement with laboratory experiments.

The nonstationary evolution of nonlinear Langmuir waves is considered. In particular the process of soliton formation is studied in the case where the perturbation propagates with a speed close to that of sound. The influence of electron collisions on the motion of the soliton is also described.

This paper is a review of results obtained by the authors in the field of non-linear monochromatic wave-particles interaction in the inhomogeneous plasma. The explicit expressions for the nonlinear growth rates and frequency shifts are derived and investigated for the cases of weak and strong inhomogeneity (the physical difference between the two cases is that in the first case there are trapped as well as untrapped particles; in the second one there are only untrapped particles). The modulational instability connected with the frequency shift, arising due to the nonlinearity and inhomogeneity, is considered.

A perturbation theory for nonlinear waves based on the inverse scattering method is presented. The theory is applied to the description of soliton evolution in the presence of permanent perturbation. It is shown that small perturbation leads to three main effects: (i) a slow change of soliton parameters; (ii) a deformation of its shape (iii) formation of a soliton "tail" which is a small amplitude wave packet with growing length. All these effects are investigated in detail for the Korteweg-de Vries, modified Korteweg-de Vries and nonlinear Schrödinger equations to which perturbation terms of general form are added. It is show, in particular, that for the last equation, in contrast to the previous two, the tails do not appear for perturbations of a very broad type.

The collapse (blow up) instability depends on the strength of nonlinearity and the number of space dimensions. Considering, as typical examples, the systems described by generalized nonlinear Schrödinger and Korteweg - de Vries equations we investigate the role of high order dispersive effects in collapse-type instabilities. Conditions of their stabilization with account of high order dispersion are derived.

The resonant radiation of a soliton described by the nonlinear Schrödinger equation with third and fourth order derivative terms is investigated by different methods. The results are in good agreement between themselves.

The extended third-order nonlinear Schrödinger equation and its solutions is studied on the basis of Galileo's transformation and generalized Galileo's invariance.

The extended third-order nonlinear Schrdinger equation and its solutions are studied on the basis of Galilean transformation and generalized Galilean invariance.

Instabilities and dispersion branches for electrostatic electron-cyclotron waves in a plasma consisting of several components having different electron temperatures are investigated analytically and numerically. It is shown that, under certain conditions, a cold component may significantly change the structure of the dispersion curves and the conditions for instability even when its relative concentration is comparatively small. At the same time the maximum growth rate may be much larger than that in the absence of the cold fraction. The presence of a hot component with temperature anisotropy Tperpendicular to /T/sub ///<2 leads, as a rule, to the widening of the range of unstable frequencies and to a decrease in the growth rate.