Nonlinear wave-number selection in gradient-flow systems

The selection of a final periodic state (wave pattern), out of a family of such states, is shown to be governed generically by defects for the lowest order gradient-flow model, the generalized Kuramoto-Sivashinsky equation. Such defects arise when the nonlinear dispersion relationship of the periodic states couples with the flow-inducing Galilean zero mode, in a manner unique to gradient dynamics, to trigger a modulation instability and a self-similar, finite-time evolution toward jumps in the local wave-number gradient and mean thickness. This coupled modulation instability is much stronger than the classical phase modulation instability. The jumps at these defects then serve as wave sinks whose strength relaxes in time. Due to such consumption of wave peaks (nodes) at the relaxing defects, the bulk wave number away from the defects decreases in time until a unique stable periodic state is reached whose speed is equal to its differential flow rate with respect to change in thickness. We estimate the defect formation dynamics and the final relaxation toward equilibrium analytically, and compare them favorably to numerical results.