Continuous-wave instabilities and multi-pulse structures in a complex Ginzburg-Landau equation with a power-law nonlinearity of arbitrary order

We study a generalized complex Ginzburg–Landau equation with linear gain, spectral filtering, dispersion and a power-law nonlinearity of arbitrary order γ, which describes laser propagation in media with Kerr (γ = 2) and higher-order non-Kerr nonlinearities. A travelling-wave reduction yields a three-dimensional dynamical system whose singular points are the continuous waves (CWs). For any γ, the CWs lie on a universal locus fixed by the gain and γ, inside a frequency band whose edges do not depend on γ. We show that the linear stability problem of any CW maps exactly onto that of a plane wave of the cubic equation with an effective gain and an effective wavenumber. Consequently, the Benjamin–Feir–Newell condition 1 + νμ < 0 destabilizes all the CWs independently of γ, the growth rate of the zero-wavenumber CW scales as γ −1, and the band of Eckhaus-stable wavenumbers widens with γ. In the unstable regime, numerical simulations of the reduced system with Luther’s sixth-order Runge–Kutta scheme reveal stationary multi-pulse bursts with phase slips for a zero velocity, and periodic moving pulse trains, born on a stable limit cycle, for a nonzero velocity. In both cases, increasing γ clamps the pulse peaks and regularizes the dynamics.