Coherent structures emerging from turbulence in the nonlocal complex Ginzburg-Landau equation

The nonlocal complex Ginzburg-Landau equation (NCGLE) has been recently derived as a general model for electrochemical systems close to a supercritical Hopf bifurcation [V. Garcia-Morales and K. Krischer, Phys. Rev. Lett. 100, 054101 (2008)]. We carry out the stability analysis of plane waves for arbitrary Fourier numbers providing the generalized Eckhaus criterion for stability to long-wavelength fluctuations in the NCGLE. We also show that coherent structures (standing waves, heteroclinic orbits) arise in the NCGLE at intermediate coupling ranges from states which are turbulent under local coupling. These results are substantiated through simulations of the full NCGLE and bifurcation analysis of the truncated NCGLE which preserves the symmetry of the observed patterns. We briefly discuss the effect of the nonlocal coupling on other localized structures (Bekki-Nozaki holes) found in the NCGLE.