Soliton solutions of the one-dimensional (1D) complex Ginzburg-Landau equations (CGLE) are analyzed. %e have developed a simple approach that applies equally to both the cubic and the quintic CGLE. This approach allows us to find an extensive list of soliton solutions of the COLE, and to express all these solutions explicitly. In this way, we were able to classify them clearly. %'e have found and analyzed the class of solutions with fixed amplitude, revealed its singularities, and obtained a class of solitons with arbitrary amplitude, as well as some other special solutions. The stability of the solutions obtained is investigated numerically.

Time-localized solitary wave solutions of the one-dimensional complex Ginzburg-Landau equation ͑CGLE͒ are analyzed for the case of normal group-velocity dispersion. Exact soliton solutions are found for both the cubic and the quintic CGLE. The stability of these solutions is investigated numerically. The regions in the parameter space in which stable pulselike solutions of the quintic CGLE exist are numerically determined. These regions contain subspaces where analytical solutions may be found. An investigation of the role of group-velocity dispersion changes in magnitude and sign on the spectral and temporal characteristics of the stable pulse solutions is also carried out. ͓S1063-651X͑97͒10504-9͔ PACS number͑s͒: 42.65. Tg, 42.25.Bs, 42.55.Wd

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A nonlinear theory describing the long-term dynamics of unstable solitons in the generalized nonlinear Schrodinger (NLS) equation is proposed. An analytical model for the instability-induced evolution of the soliton parameters is derived in the framework of the perturbation theory, which is valid near the threshold of the soliton instability. As a particular example, we analyze solitons in the NLS-type equation with two power-law nonlinearities. For weakly subcritical perturbations, the analytical model reduces to a second-order equation with quadratic nonlinearity that can describe, depending on the initial conditions and the model parameters, three possible scenarios of the longterm soliton evolution: (i) periodic oscillations of the soliton amplitude near a stable state, (ii) soliton decay into dispersive waves, and (iii) soliton collapse. We also present the results of numerical simulations that con6rm excellently the predictions of our analytical theory.

Nonlinear theory describing the instability-induced dynamics of dark solitons in the generalized nonlinear Schrödinger equation is presented. Equations for the evolution of an unstable dark soliton, including its transformation into a stable soliton, are derived using a multiscale asymptotic technique valid near the soliton instability threshold. Results of the asymptotic theory are applied to analyze dark solitons in physically important models of optical nonlinearities, including competing, saturable, and transiting nonlinearities. It is shown that in all these models dark solitons may become unstable, and two general ͑bounded and unbounded͒ scenarios of the instability development are investigated analytically. Results of direct numerical simulations of the generalized nonlinear Schrödinger equation are also presented, which confirm predictions of the analytical approach and display main features of the instability-induced dynamics of dark solitons beyond the applicability limits of the multiscale asymptotic theory. ͓S1063-651X͑96͒09707-3͔