Wavenumber Selection in Pattern Forming Systems

Pattern forming systems are characterized by the emergence of a band of stable spatially periodic states as a control parameter is varied. Wavenumber selection refers to the evolution of such systems to one of these states at long times, irrespective of initial conditions. Numerical studies of pattern forming phenomena indicate that the presence of noise is a mechanism for wavenumber selection at long times. We investigate this for the noisy Stabilized Kuramoto Sivashinsky (SKS) equation. Computational difficulties restricted earlier numerical simulations of this equation to small system sizes and a narrow range of control parameters^[1]. Our aim now is two-fold: to determine whether wavenumber selection occurs for larger system sizes and to do so for a broader range of control parameter values. With the use of spectral methods of integration, we have been able to simulate larger system sizes and obtain a crude probability distribution of final states. We present our results for various system sizes and demonstrate a possible connection to large deviation theory^[2]. The drawbacks of our approach and possible improvements are also discussed.
[1] D. Obeid, J. M. Kosterlitz, B. Sandstede. Phys. Rev. E 81, 066205 (2010)
[2] H. Touchette, Physics Reports 478 (2009)