City Research Online

Abstract

This thesis is concerned with solutions of the one-dimensional and two-dimensional Swift-Hohenberg equation as a model of nonlinear convection. In particular it is concerned with the influence of lateral boundaries on nonlinear solutions.

We start by giving a linear stability analysis for the one-dimensional case and use this as a basis for finding one-dimensional nonlinear periodic solutions. We also study the bifurcation structure and stability of nonlinear mode interactions.

We use Floquet theory to analyse, in a spatial sense, the departure of the nonlinear solutions from their periodic form and locate the Eckhaus boundary for the one-dimensional case.

We then use the Floquet analysis to find nonlinear solutions of the Swift-Hohenberg equation in the presence of a lateral boundary and determine the restriction imposed by the boundary on wavenumber selection.

In the two-dimensional case, we obtain linear stability results for the solution in a channel of finite width and use this as a basis for finding nonlinear solutions which are periodic along the channel.

We then use Floquet theory to analyse, in a spatial sense, the departure of nonlinear solutions from their periodic form and to locate the two-dimensional equivalent of the Eckhaus boundary.

Finally, the Floquet theory is used as the basis of an approximate method of finding the restriction on wavenumber selection imposed by the presence of a lateral boundary across the channel.

Publication Type:	Thesis (Doctoral)
Subjects:	Q Science > QA Mathematics
Departments:	School of Science & Technology > Mathematics School of Science & Technology > School of Science & Technology Doctoral Theses Doctoral Theses

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