^{1}

^{2}

The transition regimes of solitons in four-photon resonant processes in the case of two-photon absorption of the fundamental radiation are numerically investigated. The standard system of equations for the amplitudes of probability of finding the system in state with certain energy is used to derive the expression for the induced polarization in the nonlinear medium. As for the equations for the amplitudes of the optical pulses, the general case is considered in which both the amplitudes and phases are space-time dependent. We focus on the finite difference methods and the case of simultaneously propagating solitons at all frequencies of the interacting waves (simultons). The obtained results indicate that upon certain threshold conditions all interacting pulses become the solitons of Lorentzian shape. The numerical analysis has also shown that the soliton amplitudes significantly depend on the ratio between the nonlinear polarizability at the fundamental frequency <i>ω</i><sub>0</sub> and that of combination of <i>ω</i><sub>0</sub> and the trigger-field frequency <i>ω</i><sub>1</sub>(2<i>ω</i><sub>0</sub> + <i>ω</i><sub>1</sub>). In the second part of the paper, we apply the method of phase planes to show that at typical values of parameters, the solitons are stable.

Solitons or self-reinforcing solitary waves can emerge spontaneously in a physical system in which some energy is fed in, for instance as thermal energy or by an excitation with an electromagnetic wave, even if the excitation does not match exactly the soliton solution. Therefore, if a system possesses the necessary properties to allow the existence of solitons, it is highly likely that any large excitation will indeed lead to their formation [

In our previous research [

The second topical problem in modern nonlinear optics is the production of coherent and frequency-tunable radiation in the far ultraviolet (UV) and infrared (IR). In these spectral areas, solid materials have broad absorption bands and this narrows down the application of nonlinear crystals for the generation of electromagnetic radiation. Possible ways of overcoming those difficulties are related with the utilization of nonlinear phenomena in gases and metal vapors. The resonant four-photon interaction (RFPI) in the case of two-photon resonance is one of them. Among the advantages of gases are the presence of narrow resonances and possibility of continuous variation of density, width of spectral line, length of the medium, etc. [

The present paper is devoted to the computer simulation of transition regimes of RFPI solitons in the case of two-photon resonance. The basic equations describing this process are given in Section 2 [

Let us assume that two optical pulses with frequencies

To find the system of equations that governs the processes of propagation of optical pulses with frequencies

where

We next use (1) and the theory of perturbations [

order

To obtain the system of equations for

where

The expression for the polarization induced by the superposition of nonlinear waves is defined by

We introduce (3) into (7) and find that the expression for the induced polarization becomes

where

The system of Equations (4) and (5) can now be rewritten in terms of

where

To make the system (8) - (10) complete we add Maxwell’s equations for the all real “slowly-varying amplitudes”

where

^{3},

We carried out the computer simulation of the system (8) - (10) and (11) - (18) for the following parameters of electromagnetic radiation and medium (gases):

To investigate the stability of solitons we perform the summation of the Equations (12) (14) (16) and (18) for phases

where

The behaviour of the latter system is analyzed in terms of phase planes. As an example,

at 11 different initial conditions for

The space-time evolution of the optical pulses by using the computer simulation of transition regimes of four- photon resonant parametric processes in case of two-photon resonance is investigated. The computer simulation was based on application of the finite difference methods to the system of nonlinear equations modeling the foregoing interactions. It is shown that at certain boundary conditions (those result from the “area theorem” (see, e.g. [

or

M. Arif,J. Hussain,I. Hussain,S. Kumar,G. Bhati,Vladimir Feshchenko,Galina Feshchenko, (2016) Computer Simulation of Transition Regimes of Solitons in Four-Photon Resonant Parametric Processes in Case of Two-Photon Resonance. American Journal of Computational Mathematics,06,267-274. doi: 10.4236/ajcm.2016.63028