The Eckhaus and zigzag instability criteria in gradient/skew-gradient dissipative systems

The concept of gradient/skew-gradient structure—an extension of free energy—is introduced. In dissipative systems with this structure, the Eckhaus and zigzag instability criteria can be represented in terms of first integral and free energy per unit length for steady states. Moreover, an exact relation between these criteria is presented. These results are valid for steady states of arbitrary profile and amplitude, which implies that the instability criteria are valid for steady states far from threshold. Furthermore, our results support the marginal stability hypothesis for roll solutions in the wavenumber selection problem in gradient systems. The derivation for the results is simple and rigorous.