%0 Journal Article %1 daniels2008nonlinear %A Daniels, P. G. %A Ho, D. %A Skeldon, A. C. %D 2008 %J iMA Journal of Applied Mathematics %K 76e09-stability-and-instability-of-nonparallel-flows %N 6 %P 10.1093/imamat/hxn032 %T Nonlinear Wavelength Selection in a Channel %V 73 %X A method is described for calculating nonlinear steady-state patterns in channels taking into account the effect of an end wall across the channel. The key feature is the determination of the phase shift of the nonlinear periodic form distant from the end wall as a function of wavelength. This is found by analysing the solution close to the end wall, where Floquet theory is used to describe the departure of the solution from its periodic form and to locate the Eckhaus stability boundary. A restricted band of wavelengths is identified, within which solutions for the phase shift are found by numerical computation in the fully nonlinear regime and by asymptotic analysis in the weakly nonlinear regime. Results are presented here for the two-dimensional Swift-Hohenberg equation but in principle the method can be applied to more general pattern-forming systems. Near onset, it is shown that for channel widths less than a certain critical value the restricted band includes both subcritical and supercritical wavelengths, whereas for wider channels only subcritical wavelengths are allowed. Key words: convection; nonlinear systems; pattern selection.

@article{daniels2008nonlinear, abstract = {A method is described for calculating nonlinear steady-state patterns in channels taking into account the effect of an end wall across the channel. The key feature is the determination of the phase shift of the nonlinear periodic form distant from the end wall as a function of wavelength. This is found by analysing the solution close to the end wall, where Floquet theory is used to describe the departure of the solution from its periodic form and to locate the Eckhaus stability boundary. A restricted band of wavelengths is identified, within which solutions for the phase shift are found by numerical computation in the fully nonlinear regime and by asymptotic analysis in the weakly nonlinear regime. Results are presented here for the two-dimensional Swift-Hohenberg equation but in principle the method can be applied to more general pattern-forming systems. Near onset, it is shown that for channel widths less than a certain critical value the restricted band includes both subcritical and supercritical wavelengths, whereas for wider channels only subcritical wavelengths are allowed. Key words: convection; nonlinear systems; pattern selection.}, added-at = {2021-09-25T06:36:49.000+0200}, author = {Daniels, P. G. and Ho, D. and Skeldon, A. C.}, biburl = {https://www.bibsonomy.org/bibtex/2eb5a5d092e9754e8cfe2eede86a7cb65/gdmcbain}, interhash = {4f614bea4d6b49649b9fcbd2fd9a5ee9}, intrahash = {eb5a5d092e9754e8cfe2eede86a7cb65}, journal = {iMA Journal of Applied Mathematics }, keywords = {76e09-stability-and-instability-of-nonparallel-flows}, number = 6, pages = {10.1093/imamat/hxn032}, timestamp = {2021-09-25T06:36:49.000+0200}, title = {Nonlinear Wavelength Selection in a Channel}, volume = 73, year = 2008 }