The spatiotemporal ginzburg-landau equation: Dissipative solitons & stability

The complex Ginzburg-Landau (GL) equation describes universal wave propagation in dispersive systems that also exhibit competition between amplification and dissipation [1,2]. The balance between dispersive effects (group-velocity dispersion and self-phase modulation), linear gain and nonlinear loss can, in principle, lead to the formation of a stationary wavepacket (or soliton) in the local time frame. Here, we propose a novel two-fold generalization of the traditional GL equation to accommodate additional physical effects: (i) spatiotemporal dispersion [3], and (ii) power-law nonlinearity [4]. Exact analytical bright solitons of the new model have been derived, with asymptotic analysis demonstrating the emergence of well-known solutions [1,2] in a simultaneous multiple limit. Extensive simulations have revealed that, like its conventional counterpart (see Fig. 1), the new class of spatiotemporal dissipative soliton is also susceptible to a blow-up phenomenon (where the zero-amplitude continuous-wave solution is modulationally unstable against background fluctuations of arbitrarily-small magnitude). However, a route to stabilization may be possible by coupling the soliton to a non-dispersing linear wave [5]. References ]1] N. R. Pereira and L. Stenflow, Phys. Fluids 20, 1733 (1977). [2] C. Paré, L. Gagnon, and P. Bélanger, Opt. Commun. 74, 228 (1989). [3] J. M. Christian, G. S. McDonald, T. F. Hodgkinson, and P. Chamorro-Posada, Phys. Rev. Lett. 108, 034101 (2012). [4] J. M. Christian, G. S. McDonald, R. J. Potton, and P. Chamorro-Posada, Phys. Rev. A 76, 033834 (2007). [5] N. Efremidis et al., Phys. Scr. T84, 18 (2000)