Stability limits of spirals and traveling waves in nonequilibrium media

We show that the Eckhaus instability for traveling waves is of convective nature and does not directly lead to absolute instability. As a consequence spiral waves remain stable in a larger range than expected previously, transition to defect-mediated turbulence can be delayed beyond the Benjamin-Feir limit, and the occurrence of phase turbulence can be made plausible. We calculate the onset of absolute instability using the complex Ginzburg-Landau equation and verify the results by simulations.