ON DYNAMICS OF SOLUTIONS OF THE GENERALIZED NONLINEAR SCHRODINGER-EQUATION

We consider a generalized nonlinear Schrodinger equation: iq(t) + q(xx) + 2k\q\2q - ikb(\q\2q)x = 0 at various initial conditions. For the purpose of practical applications in optics, we study the following initial value problems: decay of the sech-pulses; dynamics of randomly modulated solitons, and evolution of nonsoliton random pulses. The first of the above problems is solved exactly. Conditions of the sech-pulse decay are stated. For the solution of the second problem, a perturbation theory is developed. Statistical characteristics of both a soliton under the Gaussian initial fluctuations, and noise generated in the system are obtained. Soliton parameters are distributed according to the Gaussian law. The relationship between correlation characteristics of the noise and those of the linear beam in the problem of the Fraunhofer diffraction on a gap is established. The asymptotic formula describing a nonsoliton solution at sufficiently large times is stated. The Fokker-Plank equation for probability density of the amplitude and phase of the reflection coefficient is derived. The drift velocity of the mean peak intensity is calculated. A case of weakly nonlinear pulse propagation is considered in the framework of a perturbation theory. All results are compared with those obtained earlier for the nonlinear Schrodinger equation.