The dynamics and stability of continuous-wave and multi-pulse structures are studied theoretically, for a generalized model of passively mode-locked fiber laser with an arbitrary nonlinearity. The model is characterized by a complex Ginzburg-Landau equation with saturable nonlinearity of a general form ($I^m/(1+Gamma I)^n$), where $I$ is the field intensity, $m$ and $n$ are two positive real numbers and $Gamma$ is the optical field saturation power. The analysis of fixed-point solutions of the governing equations, reveals an interesting loci of singular points in the amplitude-frequency plane consisting of zero, one or two fixed points depending upon the values of $m$ and $n$. The stability of continuous waves is analyzed within the framework of the modulational-instability theory, results demonstrate a bifurcation in the continuous-wave amplitude growth rate and propagation constant characteristic of multi-periodic wave structures. In the full nonlinear regime these multi-periodic wave structures turn out to be multi-pulse trains, unveiled via numerical simulations of the model nonlinear equation the rich variety of which is highlighted by considering different combinations of values for the pair ($m$,$n$). Results are consistent with previous analyses of the dynamics of multi-pulse structures in several contexts of passively mode-locked lasers with saturable absorber, as well as with predictions about the existence of multi-pulse structures and bound-state solitons in optical fibers with strong optical nonlinearity such as cubic-quintic and saturable nonlinearities.
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