Dissipative solitons of a spatiotemporal Ginzburg-Landau equation

Complex Ginzburg-Landau (GL) equations play a fundamental role in our understanding of universal wave propagation in generic systems with dispersion, nonlinearity, gain, and loss. Over three decades ago, stationary bright-type solitons were derived by Pereira and Stenflo (PS solitons) for a cubically-nonlinear GL equation [Phys. Fluids, 20, 1733 (1977)]. Darktype solutions followed a short time later, introduced by Nozaki and Bekki (NB solitons) [J. Phys. Soc. Jpn. 53, 5365 (1984)]. In photonics, such self-localizing optical wavepackets can typically arise when group-velocity dispersion is balanced by self-phase modulation, and nonlinear losses (e.g., from two-photon absorption) are compensated by linear gain (e.g., from doping in the host medium). In the corresponding GL model, linear gain tends to introduce instability (growth of infinitesimally-small perturbations) such that all its solutions are rendered unstable in the long term.