NONLINEAR WAVE-NUMBER SELECTION IN GRADIENT-FLOW SYSTEMS

The selection of a final periodic state (wave pattern), out of a famil y of such states, is shown to be governed generically by defects for t he lowest order gradient-flow model, the generalized Kuramoto-Sivashin sky equation. Such defects arise when the nonlinear dispersion relatio nship of the periodic states couples with the flow-inducing Galilean z ero mode, in a manner unique to gradient dynamics, to trigger a modula tion instability and a self-similar, finite-time evolution toward jump s in the local wave-number gradient and mean thickness. This coupled m odulation instability is much stronger than the classical phase modula tion 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 de fects decreases in time until a unique stable periodic state is reache d whose speed is equal to its differential flow rate with respect to c hange in thickness. We estimate the defect formation dynamics and the final relaxation toward equilibrium analytically, and compare them fav orably to numerical results.