A Study of the Complications regarding the Derivation of Amplitude Equations via Weakly Nonlinear Expansions for Non-self-adjoint Partial Differential Equations

Overview & Synopsis

The interplay between non-normality and nonlinearity has been the focus of numerous works but also contention in Fluid Mechanics. In this thesis, we explore the relationship between non-normality and nonlinearity by considering the failure of firstorder approximations derived via weakly nonlinear expansions (WNLE) to capture saturation characteristics of non-self-adjoint systems. The first-order approximation achieved via WNLE is of the form of the leading eigenvector multiplied by an amplitude, which is governed by an amplitude equation. Authors have used various approaches to improve upon this approximation, such as going higher in order in the approximation, which requires additional assumptions owing to the non-uniqueness of higher order terms, as well as also building higher order amplitude equations to describe the temporal development of the leading eigenvector. The latter approach can be seen as a way to circumvent the nonuniqueness. However, by still only approximating with the leading eigenvector, even though its temporal development is elaborated by a higher order amplitude equation, spatial structures different from the leading eigenvector are neglected. In this thesis, we choose two test-cases to explore the phenomena described above, namely the real nonselfadjoint Ginzburg-Landau equation (RnsaGL) and the Complex non-self-adjoint Ginzburg-Landau (CnsaGL) equation. We choose these test cases because the linear operators in each case generate strongly continuous semigroups, but, also in the real case, there exists a quasi-basis structure. This allows us to expand the solution of the RnsaGL into eigenvalue-eigenvector pairs, which is not always possible for nonself-adjoint linear operators as the eigenvectors’ forming an orthonormal basis is not guaranteed as it is in the self-adjoint case. We begin the thesis by consolidating the research done by other authors in the literature review and in one of the core chapters where we derive higher order amplitude equations for the RnsaGL. It is demonstrated for the RnsaGL that higher order amplitude equations have a smaller radius of convergence, which essentially limits the usefulness of the approximation. We also argue against higher order amplitude equations by projecting the solution of our test cases onto the zeroth eigenvector, where we find that, for increasing non-normality, less of the overall solution is projected onto the zeroth eigenvector. In this way, no matter how elaborate the amplitude equation, it is incapable of representing the entire system. Following the consolidation of previous research, we derive higher order approximations using an assumption that is different to the work of previous authors; namely that there can be no linear contributions to the approximation at higher order terms. We see that in the case of the RnsaGL, this assumption can manifest by ensuring that the higher order terms are orthogonal to the direct eigenvector or the adjoint eigenvector (in the case of the CnsaGL, we can only choose the direct eigenvector). After this, we derive error bounds in order to quantify the difference between the solutions of the test cases and their corresponding approximations. The derivation of these error bounds is facilitated by the fact that the linear operators in our test cases generate strongly continuous semigroups. These error bounds are theoretical tools to show the existence of a possible radius of convergence rather than telling us what the radius of convergence is. We therefore discuss how to turn these error bounds from theoretical tools to something that can be used for application. Lastly, we profit from the existence of the quasi-basis by using a stochastic averaging technique where noise is put on the stable modes. This stochastic averaging technique is different from WNLE as no eigenmodes are neglected as a first step, but the saturation characteristics are still underestimated in the non-self-adjoint case.