Formation of the bright and dark solitons in N-core of the optical coupler is the important problem for the creation of
effective optical lines of the communication. This problem is connected to the finding of the solution of the nonlinear
Schrodinger equation taking into account the interaction between the cores of the fiber.
Basing on elementary mathematical model of N-core of the optical fiber coupler with the linear interaction between the
next cores the conditions of formation of the solitons in 2 - core fiber coupler is investigated.
For the certain type of the soliton solution the necessary and sufficient conditions of existence of the solitons are
received. The parametrical space and the physical sense of the received conditions is defined.
It is shown, that for the certain parameters of optical systems, the solitons spread with identical speed and width, but
the amplitudes of the interacting solitons can be both one polarity, and opposite.
The method based on solution of the system of the connected Schrodinger equations determines the necessary and
sufficient conditions of the existence of solitons and it can be used for research of problems of formation and propagation of the bright (or dark) solitons in N- core optical couplers.
The coherent nonlinear interaction of three-four electrostatic waves is considered analytically and numerically. The excitation and nonlinear evolution of the electrostatic two-stream or Farley-Buneman (FB) plasma instabilities in the collisional plasma are investigated. Basing on the magnetohydrodynamics (MHD) system equations for collisional plasmas with magnetized electrons and non-magnetized ions, the evolution equations for the electron density are obtained..
It is shown that three-four wave coherent interaction is one of the effective physical mechanism which leads to nonlinear stabilization of the growing high-frequency waves and to the formation of local density structures of the charged particles. The different scenarios of the stabilization of the two-stream instability (as a periodic, multi-periodic, nonlinear cnoidal, shock waves and quasi-stochastic behavior) are possible in the collisional plasma. If the plasma system is in the weakly supercritical regime, the explosive instability may also occur. The different examples of the nonlinear temporal evolution of the FB instability are represented. The result of the numerical models allows analyzing feasibility of scenarios of the two-stream plasma instability in the plasma.