  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.
