Three-wave interaction of helicoidal plasma waves

In this paper we study a propagation of nonlinear helicoidal wave in plasma. We consider the hydrodynamic approximation of a single ion (electron) plasma. In this approximation, one of the main plasma parameters, Knudsen number, is much smaller than one (Kn Lt 1) so the length of free path of plasma particles is negligible compared to the space scale of the wave (or the frequency compared to collision frequency). In this case, collisions play main role and plasma may be considered as a conductive fluid. In the case of two-dimensional plasma, we derive model equation of interaction between circular polarized helicoidal waves and give expressions for coupling constants. Next we go to the equations that, describe the directed quasi-one-dimensional waves. Going down to the pulses interaction the case of the so-called three-wave structure of the field with corresponding carrier frequencies and wave numbers. The equations for envelopes are obtained when the nonlinear resonance holds. To generate solutions we also use a form of the binary Darboux transformation (bDT) applying it to the three-wave interactions. We start, from the generalized pair of linear equations (Lax pair) being a compatibility condition for considered equation. The Lax representation allows to construct N-soliton solutions (having in mind three-component solitons) via iterations of the bDT. The solutions has been analyzed numerically, the results are shown on pictures. The novel feature of the study, in the context of the soliton theory, is the account, of the wave asynchronism. We reproduce the known results of the three-wave interaction (without asynchronism) theory with the special attention paid to the physical origin of the system's parameters. Then we compare the results to the problem with asynchronism and absorption, that yields the waves are nonsymmetric form ("inclined" solitons)