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Motivated by recent experiments and numerical simulations of the positive column of a neon glow discharge we investigate the Eckhaus instability of traveling waves. Compared to the classical results the plasma system shows some peculiarities, e.g., an asymmetric stability region and strong selection of periodic patterns. These complex phenomena may be explained by a transition from supercritical to subcritical Hopf bifurcation near the critical point. In the weak nonlinear region the wave dynamics is approximated by a quintic Ginzburg-Landau equation supplemented by nonlinear gradient terms. Starting from a hydrodynamic model the coefficients of this equation, which depend on the plasma parameters, are calculated. The stability properties of plane wave solutions are discussed for an infinitely long discharge as well as for finite ones. The theoretical results show most of the properties that are observed in real experiments.